extensions and applications of 1+3 decomposition ... - CiteSeerX

E XTENSIONS AND A PPLICATIONS OF 1 + 3 D ECOMPOSITION M ETHODS IN G ENERAL R ELATIVISTIC C OSMOLOGICAL M ODELLING

A Thesis presented to the University of London for the Degree of D OCTOR OF P HILOSOPHY in Applied Mathematics by

H ENK

VAN

E LST

Wörth (Rhein), Federal Republic of Germany

Date of Oral Examination: November 29, 1996 Queen Mary & Westfield College, London, United Kingdom December 3, 1996

Remember today is the tomorrow you promised yourself yesterday

(Reza Tavakol, early 1970s)

Abstract 1 + 3 “threading” decomposition methods of the pseudo-Riemannian spacetime manifold ( M, g ) and all its geometrical objects and dynamical relations with respect to an invariantly defined preferred timelike reference congruence u/c have been useful tools in general relativistic cosmological modelling for more than three decades. In this thesis extensions of the 1 + 3 decomposition formalism are developed, partially in fully covariant form, and partially on the basis of choice of an arbitrary Minkowskian orthonormal reference frame, the timelike direction of which is aligned with u/c. After introductory remarks, in Chapter 2 first an exposition is given of the general 1 + 3 covariant dynamical equations for the fluid matter and Weyl curvature variables, which arise from the Ricci and second Bianchi identities for the Riemann curvature tensor of ( M, g, u/c ). New evolution equations are then derived for all spatial derivative terms of geometrical quantities orthogonal to u/c. The latter are used to demonstrate in 1 + 3 covariant terms that the spatial constraints restricting relativistic barotropic perfect fluid spacetime geometries are preserved along the integral curves of u/c. The integrability of a number of different special subcases of interest can easily be derived from this general result. In Chapter 3, 1 + 3 covariant representations of two classes of well-known cosmological models with a barotropic perfect fluid matter source are obtained. These are the families of the locally rotationally symmetric (LRS) and the orthogonally spatially homogeneous (OSH) spacetime geometries, respectively. Subcases arising from either dynamical restrictions or the existence of higher symmetries are systematically discussed. For example, models of purely “magnetic” Weyl curvature and, in the LRS case, a transparent treatment of tilted spatial homogeneity can be obtained. The 1 + 3 covariant discussion of the OSH models requires completion. Chapter 4 reviews the complementary 1 + 3 orthonormal frame (ONF) approach and extends it to include the second Bianchi identities, which provide dynamical relations for the physically interesting Weyl curvature variables. Then, possible choices of local coordinates within the 1 + 3 ONF framework are introduced, taking both the 1 + 3 threading and the ADM 3 + 1 slicing perspectives. The 1 + 3 ONF method is employed in Chapter 5 to investigate the integrability of the dynamical equations describing “silent” irrotational dust spacetime geometries, for which the “magnetic” part of the Weyl curvature is required to vanish. Evidence is obtained that these equations may not be consistent in the generic case, but that only either algebraically special or spatially homogeneous classes of solutions may be covered. Furthermore, this chapter uses the extended 1 + 3 ONF dynamical equations to describe LRS models with an imperfect fluid matter source and contrasts the perfect fluid subcase with the results obtained in Chapter 3. In Chapter 6, a brief detour is taken into considering those classical theories of gravitation in which the Lagrangean density of the gravitational field is assumed to be proportional to a general differentiable function f (R) in the Ricci curvature scalar. The generalisations of the relativistic 1 + 3 covariant dynamical equations to the f (R) case are derived and a few examples of applications are commented on. Finally, Chapter 7 investigates in detail features of the dynamical evolution of the cosmological density parameter Ω in anisotropic inflationary models of Bianchi Type–I and Type–V and points out important qualitative changes as compared to the idealised standard FLRW situation. A related analysis employing the same spacetime geometries addresses the occurrence of restrictions on the permissible functional form of the inflationary expansion length scale parameter S as a consequence of the so-called reality condition for Einstein–Scalar-Field configurations. Again, the effect of the (exact) anisotropic perturbations on the FLRW case is thoroughly studied and found to have significant effects. Both cases can be treated as examples of structural instability. This thesis ends with concluding remarks and an appendix section containing the conventions employed and mathematical relations relevant to derivations given in various chapters.

PACS number(s): 04.20.-q, 98.80.Hw, 98.80.Dr, 04.20.Jb

v

Contents Abstract

v

1

1

Introduction

2 1 + 3 Decomposition of ( M, g, u/c ) in Covariant Form 2.1

2.2

2.3

3

11

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.1.1

Covariant spatial 3-scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.1.2

Covariant spatial 3-gradients . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.1.3

Covariant spatial rotation terms of spatial 3-tensors . . . . . . . . . . . . . . .

14

The Ricci identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2.1

Time derivative equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2.2

Constraint equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

The second Bianchi identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.3.1

Time derivative equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.3.2

Constraint equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.4

Comments on the 1 + 3 covariant dynamical equations . . . . . . . . . . . . . . . . .

17

2.5

Vanishing vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.6

Derivation of new evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.6.1

Evolution of spatial 3-gradients . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.6.2

Evolution of further geometrical quantities . . . . . . . . . . . . . . . . . . .

22

2.7

Preservation of constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.8

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

Special Perfect Fluid Spacetime Geometries. I.

31

3.1

Locally rotationally symmetric spacetime geometries . . . . . . . . . . . . . . . . . .

31

3.1.1

The Ricci identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.1.2

The second Bianchi identities . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.1.3

Vanishing vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.1.4

The equations for e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.1.5

The consistency equations and their different cases . . . . . . . . . . . . . . .

35

3.1.6

(ω/c) 6= 0: Rotating solutions (LRS class I) . . . . . . . . . . . . . . . . . . .

38

3.1.7

k 6= 0: Homogeneous orthogonal models with twist (LRS class III) . . . . . .

40

3.1.8

0 = k = (ω/c): The inhomogeneous orthogonal family (LRS class II) . . . . .

44

3.1.9

Higher symmetry subcases: Hypersurface homogeneous models in LRS class II

50

3.1.10 Self-similar OSH models in LRS classes II and III . . . . . . . . . . . . . . .

57

vii

viii

CONTENTS

3.2

3.1.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Orthogonally spatially homogeneous spacetime geometries . . . . . . . . . . . . . . .

60

3.2.1

Solutions with isotropic 3-Ricci curvature . . . . . . . . . . . . . . . . . . . .

67

3.2.2

Solutions with E = 0 (“Pure magnetic”) . . . . . . . . . . . . . . . . . . . . .

67

3.2.3

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

4 1 + 3 ONF Decomposition of ( M, g, u/c ) 4.1

The commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

4.2

The curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

4.2.1

Einstein field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.2.2

Jacobi identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.2.3

“Electric” and “magnetic” parts of the Weyl curvature tensor . . . . . . . . . .

74

4.2.4

Conformal 3-Cotton–York tensor . . . . . . . . . . . . . . . . . . . . . . . .

75

The second Bianchi identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

4.3.1

Bianchi identities for the Weyl curvature tensor . . . . . . . . . . . . . . . . .

75

4.3.2

Bianchi identities for the source terms . . . . . . . . . . . . . . . . . . . . . .

76

4.4

Introducing local coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

4.5

Comments on the 1 + 3 ONF dynamical equations . . . . . . . . . . . . . . . . . . .

79

4.6

Irreducible tracefree decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.3

5

Special Perfect Fluid Spacetime Geometries. II.

85

5.1

So-called “silent” models of the Universe . . . . . . . . . . . . . . . . . . . . . . . .

85

5.1.1

1 + 3 covariant formulation . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

5.1.2

1 + 3 ONF formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

5.1.3

Constraint analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

5.1.4

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Locally rotationally symmetric spacetime geometries . . . . . . . . . . . . . . . . . .

94

5.2.1

LRS class II perfect fluid models . . . . . . . . . . . . . . . . . . . . . . . . .

98

5.2.2

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2

6

69

Generalisation to f (R)-Theories 6.1

6.2

101

The Ricci identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.1.1

Time derivative equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.1.2

Constraint equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

The second Bianchi identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.2.1

Time derivative equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2.2

Constraint equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.3

Vanishing vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

ix

CONTENTS 7

Inflation in Bianchi Type–I and Type–V Models 7.1

7.2

8

109

Anisotropic inflationary 2-fluid model . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.1.1

Dynamical equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.1.2

2 phase/ 2 fluid setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.1.3

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Constraints on anisotropic scalar field models . . . . . . . . . . . . . . . . . . . . . . 121 7.2.1

FLRW ( C = 0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.2.2

OSH Type–I and Type–V ( C 6= 0 ) . . . . . . . . . . . . . . . . . . . . . . . 126

7.2.3

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Conclusions

131

A Sign and Index Conventions

135

B Useful Relations

141

B.1 3-scalar derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 B.2 3-vector derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 B.2.1

3-vector derivatives in 1 + 3 ONF form . . . . . . . . . . . . . . . . . . . . . 143

B.3 3-tensor derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 B.3.1

3-tensor derivatives in 1 + 3 ONF form . . . . . . . . . . . . . . . . . . . . . 146

B.4 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 B.5 Some contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 C 1 + 3 ONF Relations in Explicit Form

149

C.1 1 + 3 ONF dynamical variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 C.1.1

Commutation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

C.1.2

Ricci rotation coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

C.1.3

Riemann curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

C.2 Equation of geodesic deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 C.2.1

Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

C.3 Weyl curvature components relating to null frames . . . . . . . . . . . . . . . . . . . 153 D 1 + 3 ONF Equations in Dimensionless Formulation

157

D.1 The commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 D.2 The curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 D.2.1 Einstein field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 D.2.2 Jacobi identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 D.2.3 “Electric” and “magnetic” parts of the Weyl curvature tensor . . . . . . . . . . 160 D.2.4 Conformal 3-Cotton–York tensor . . . . . . . . . . . . . . . . . . . . . . . . 160 D.3 The second Bianchi identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 D.3.1 Bianchi identities for the Weyl curvature tensor . . . . . . . . . . . . . . . . . 160 D.3.2 Bianchi identities for the source terms . . . . . . . . . . . . . . . . . . . . . . 161 Bibliography

162

Acknowledgements

173

x

CONTENTS

Curriculum Vitae

175

Copies of Published Papers

176

Chapter 1

Introduction As opposed to the three other known fundamental forces of Nature, which, to a compelling degree, are solidly founded on modern quantum field theories of physics, to date the most successful framework for the description of gravitational interactions is still the Theory of General Relativity (GR) proposed by Einstein in 1915 [40]. GR is based on the concepts of Riemannian differential geometry. As such it is a classical (i.e., non-quantum) field theory that treats gravitational phenomena as manifestations of infinitesimal, i.e., “local” curvature properties of the so-called 4-D spacetime manifold ( M, g ), where the metric tensor field g, which encodes information on infinitesimal 4-D physical distances, is determined by the local distribution of the total energy density of all forms of matter present.1 This physical relation is expressed by the Einstein field equations (EFE). Since its publication more than 80 years ago, GR is known on the basis of direct and indirect tests to give very accurate results for analysis and prediction of gravitational interactions on solar system and galactic distance scales. Nevertheless, from the beginning and starting with Einstein, GR was also assumed to provide a valid framework for modelling the Universe in its entirety. Support for this assumption comes from observational indications that gravitation is the only fundamental interaction which initiates physical processes over the vast cosmological distance scales (at least during the largest part of the past history of the Universe). This branch of modern physics is nowadays known as relativistic cosmology. Influenced by the predominant philosophical bias of his time, Einstein himself initially favoured a static and highly symmetric model of the Universe, put forward in 1917 [41]. However, with the work of Friedmann (1922) [57] and Lemaˆıtre (1927) [93] during the following decade, it soon emerged that dynamical models satisfying the EFE were also conceivable. This idea was further strengthened by observations made by Hubble in 1929 [77], which revealed an overall recession of nearby and distant galaxies relative to wo/man-kind’s Earth, i.e., Milky Way based position. Observational data of reasonable quality and quantity, providing relativistic cosmology with reference points to test itself against, only became available from the early 1960s onwards, and improved significantly during the last twenty years. This data demonstrates that after local peculiar velocities of matter accumulations such as galaxies, clusters of galaxies and higher-order structures thereof have been averaged out, the (expanding) Universe looks roughly isotropic to an observer located in our own galaxy, i.e., similar to a high degree in all directions on the celestial sphere. This near-isotropy is best reflected and most strongly supported by the startling uniformity of the so-called cosmic microwave background radiation (CMBR), which is a thermal signal with an almost exact Planckian distribution 1 To be precise, the metric tensor field is determined up to the freedom of applying arbitrary coordinate transformations, and a unique solution arises after boundary conditions have been specified.

1

2

CHAPTER 1. INTRODUCTION

of its spectral energy density, measured in all directions on the sky anywhere on the globe. Its frequencies lie in the microwave band of the spectrum of electromagnetic radiation, i.e., the infrared end, with a temperature recently determined to be at T0 = (2.726 ± 0.10) K ( see, e.g., Partridge (1994) [122] ). Thus, it reveals the features of radiation in thermal equilibrium as generated by a black body emitter source. The CMBR was accidentally discovered by Penzias and Wilson in 1965 [128]. The failure to explain the origin of this all-pervading radiation in terms of thermal processes within our own galaxy, coupled with Hubble’s observational evidence indicating a phase of overall expansion, led to the interpretation that what one actually measured was in fact a remnant thermal signal from an earlier hot dense phase in the distant past of the evolution of the Universe (when it was much smaller in size). Extrapolating the expansion backwards in time this implied that the Universe originated from a state of possibly infinite density — a singular state referred to as the “Big Bang” — in which time, space and all forms of matter had been created. In short, the Universe had a beginning at a finite time back. Ironically the existence of the CMBR as emerging from a Big Bang scenario had been plausibly predicted by Gamow and coworkers in 1948 [59], but their work had remained largely unknown to the scientific community. Combining our Milky Way based observation of a nearly spatially isotropic Universe with the so-called “weak Copernican Principle”, i.e., the assumption that our own galaxy is in no way special as compared to other matter accumulations,2 in particular, as regards its spatial position after the thermal radiation decoupled from any fermionic matter particles, it is generally assumed that the Universe (and physical processes therein) is nearly spatially homogeneous. The observational features mentioned above provide the main ingredients for the standard mathematical setting in general relativistic cosmological modelling. This setting is based on dynamical equations describing an expanding spatially isotropic and spatially homogeneous matter distribution, which were originally derived from the EFE by Friedmann (1922) [57] and Lemaˆıtre (1927) [93]. A more geometrical framework was later worked out in detail by Robertson (1935) [131] and Walker (1936) [162], and hence the standard models of relativistic cosmology are commonly referred to as the FLRW models. In line with the singularity theorems by Hawking and Penrose (1970) [69] these, like most other relativistic cosmological models, possess a Big Bang initial singularity. An important early achievement of these models (in good agreement with observational data), first calculated by Hoyle and Tayler in 1964 [74], was that they successfully accounted for the production rates of the light chemical elements such as 4He and 2H in processes of nucleosynthesis, at a time shortly following after the Big Bang. Two of the most intriguing problems in relativistic cosmology over the last 30 years have been (and still remain to be) to explain dynamically (in terms of the EFE), firstly, the apparent overall smoothness of the Universe, which by the majority of cosmologists is thought to prevail on the largest conceivable physical length scales,3 and, secondly, the formation of complex structures amongst matter accumulations, arising from irregularities in the initial matter distribution, which lead to gravitational instabilities and subsequent collapses. These apparently contradictory features immediately led to the question of the nature of the initial conditions for the classical evolution of the Universe (modelled in terms of GR) at the so-called Planck time, i.e., at about 10−43 s after the Big Bang. A first prominent attempt at providing an explanation to this problem was given by Misner in his so-called “chaotic cosmology” 2 One may want to argue though that wo/man-kind by the very fact of its existence is in a very special position indeed, for no form of life comparable to our human kind could be detected within the observable part of the Universe up to the present time. 3 (which, apart from the CMBR, have not been probed observationally yet)

3 programme in 1968 [113]. The underlying idea of this programme was that the Universe started off from a highly irregular initial state immediately following the Big Bang. In the subsequent expansion phase this irregular initial state was thought to gradually smooth out in the large due to viscous thermodynamical processes amongst matter particles (like, e.g., neutrinos) such that eventually an overall FLRW-like state was developed, which, nevertheless, still allowed for those irregularities that were needed to grow into galaxies and larger structures. In this work the complex “chaotic” initial state was approximated by a simpler spatially homogeneous but anisotropic initial state. However, Stewart (1969) [146] showed by investigating the effectiveness of dissipative processes at removing arbitrary irregularities that there does exist an open set of initial conditions which evolve into irregular states instead. Furthermore, Collins and Hawking (1973) [31] showed that the likelihood of a FLRW-like state to asymptotically emerge from arbitrary spatially homogeneous but anisotropic initial data was negligibly low. These results deflected much of the attention first given to Misner’s suggestion. In the wake of successful quantum field theory models describing and predicting results in elementary particle physics, Guth [66] suggested in 1981 that in the very early phases of its evolution, at about 10−35 s after the Big Bang, the Universe might have undergone an extremely short phase of accelerated, near-exponential growth of all physical length scales, which pushed any sort of irregularities very far outside those regions that are accessible to observations nowadays. That is, the Universe would be extremely inhomogeneous in its overall structure, but way beyond any current scale of detectability. According to this idea, those parts of the Universe that can actually be probed would look spatially very flat. This so-called inflationary model of the early life of the Universe ( see, e.g., Olive (1990) [119] ) has been refined many times since its inception. Its current most popular variant — the so-called “chaotic inflation” scenario of Linde (1983) [98] — assumes the existence of a new, however non-quantised (and so far undetected), massive elementary particle, a self-interacting scalar field matter source, minimally coupled to the gravitational field. In some small FLRW-like local regions of the early Universe the (to be determined) self-interaction potential of the scalar field is assumed for a brief period of time to simulate a dominant positive, hence, anti-gravitating cosmological constant that is thought to be the driving force behind the phase of inflationary expansion. Structure formation in these models is qualitatively explained as originating from quantum fluctuations in the non-quantised scalar field, which inflation magnifies to macroscopic size, thus providing seeds for matter accumulations. If one believes that gravitation ultimately must be linked to thermodynamical considerations, and that the law that the total entropy of a (closed) physical system must be a non-decreasing function — the second law of thermodynamics4 — also applies to the Universe as a whole, a different perspective on the above-mentioned problem of overall FLRW-like smoothness of the Universe can be gained. Penrose (1979/89) [125, 126, 127] suggested that gravitational entropy would manifest itself in the tendency of matter to clump and eventually form black holes, and that the entropy gained by those means would exponentially outweigh any ordinary known form of entropy, generally associated with thermalised matter degrees of freedom. In mathematical terms this feature could be represented by some, as yet unknown, measure of the Weyl curvature of the spacetime manifold ( see also Goode and Wainwright (1985) [64] for related work ). With respect to the Universe, Penrose therefore proposed that at the beginning of its phase of classical evolution, i.e., at the Planck time, an extremely restrictive constraint must have been placed on its Weyl curvature to ensure a state of extreme uniformity (and thus very low entropy), in order for the subsequent dynamics to comply with the implications of the second 4 Some

people consider the second law of thermodynamics to be the most important law of Nature.

4


law of thermodynamics. The proposal was called the “Weyl curvature hypothesis” (WCH). According to Penrose, this constraint can only be hoped to be understood in terms of a true quantum theory of gravitation, which, however, has not been fully discovered yet [125, 126, 127]. It is well-known that the ideal FLRW models have zero Weyl curvature, which, consequently, do not allow for any of those deviations in the matter distribution that are needed to explain structure formation. Hence, for Penrose’s attractive proposal to be of practical interest, a more general form of measure of gravitational entropy must be found. Nevertheless, if some form of the WCH is taken seriously, then obviously neither the “chaotic cosmology” programme, nor the (“chaotic”) inflationary models, are in agreement with the second law of thermodynamics, as both scenarios assume a generic (hence, highly irregular) initial state, corresponding, presumably, to very large values of the Weyl curvature ( see Ref. [127] ). In a semi-classical study, Hu (1983) [76] tried to explain the WCH as the outcome of dynamical processes in the (speculative) pre-Planck era by investigating the effects of cosmological quantum processes such as particle production and phase transitions and their back-reaction on the non-quantised spacetime geometry. The fundamental problem of near uniformity of the observable part of the Universe mentioned in this brief historical review and other related problems of relativistic cosmology have been prominently highlighted during the last decade in popular and semi-popular books by, e.g., Hawking (1988) [68], Penrose (1989) [126], Wheeler (1990) [163], Breuer (1990) [18] and Lightman and Brawer (1990) [97]. The interested non-expert reader may also want to refer to standard textbooks on GR and cosmology of both elementary and advanced level such as, e.g., Schutz (1985) [136], Stephani (1991) [144], Misner, Thorne and Wheeler (1973) [114], Wald (1984) [161], and Hawking and Ellis (1973) [70]. In order to put the work contained in this thesis into context, this introduction now turns to address more specific questions and related developments in relativistic cosmology. In the metric based picture of GR, the EFE are a highly complicated system of ten coupled quasilinear,5 second-order partial differential equations for ten components of g ( see Appendix A ). Owing to their enormous complexity, only rather special exact solutions to this system are known to date, which, in general, were obtained either by assuming particular idealised symmetries of the spacetime manifold, or by imposing algebraic or dynamical restrictions on various of its geometrical quantities. A well-known review of exact solutions to the EFE is the book by Kramer, Stephani, MacCallum and Herlt (1980) [88], an updated version of which is in preparation. Complementary to this work, with special focus on relativistic cosmological models and their physical interpretation, is a recent review by Krasiński (1996) [89] of the known exact solutions modelling spatially inhomogeneous spacetime geometries.6 As it is quite likely that the general solution to the EFE will still be out of reach for yet some time to come7 , for many years three other main lines of research have been pursued to tackle the goal of (fully) revealing the specific features inherent in the non-linearity of the EFE, in particular in the field of relativistic cosmology where the EFE have dynamical character. These are approximate, numerical and qualitative methods, respectively. Various examples of work of broader interest can be listed in this context. The first category contains the linear perturbation methods of FLRW spacetime 5 (nevertheless,

still non-linear) the evolution of the Universe eventually may have to be described by more complex models than the standard FLRW spacetime geometries was hinted at by observations by de Lapparent, Geller and Huchra in 1986 [92], which revealed surprising structures amongst matter accumulations on physical length scales far larger than they were previously assumed to exist. 7 (!) 6 That

5 geometries as initiated by Lifshitz (1946) [96] and extended by Hawking (1966) [67]. Both works raised the important issue of the definition of gauge-invariant (physical) perturbations, as these works did not manage to completely separate out (unphysical) pure coordinate transformation effects. This flaw was latter remedied via a metric based approach by Bardeen (1980) [4], and in a more intuitive geometrical way by Ellis and Bruni (1989) [50] (see also below). FLRW-linearised perturbation analyses are commonly thought to provide a first step to understanding generic non-linear features of the clumping of matter on large scales (as needed to model the formation of galaxies and larger structures of matter) and the propagation of gravitational waves in cosmological settings. Numerical methods in GR were given much wider emphasis only with the advent of powerful computing facilities over the last 10 to 15 years. This field of research is starting to gain a lot of momentum at the present time and opens up quite promising prospects for the future, in particular, as regards exploring properties of fully spatially inhomogeneous cosmological models ( see, e.g., Anninos et al (1991) [1] and Berger and Moncrief (1993) [11] ). Qualitative methods, which so far may have constituted the largest one between the above-mentioned categories, are further divided into various subbranches. Belinskiˇı et al (1970) [9] introduced an approximation scheme into cosmology in which they neglected different source terms in the EFE rated by their dynamical importance during different evolutionary epochs. In their analysis they specifically looked at the possible natures of a dynamical approach to a generic initial singularity and concluded that it would involve chaotic dynamical behaviour. During recent years increasing efforts have also been placed on the search for deterministic chaos in GR ( see, e.g., Hobill et al (1994) [73] ). Hamiltonian methods were the particular tools that Misner (1968) [113] developed and applied in his “chaotic cosmology” programme. The Hamiltonian point of view was later taken up and expanded by Jantzen, Uggla and Rosquist in a systematic investigation of the family of spatially homogeneous spacetime geometries ( see, e.g., Refs. [155] and [156] ). Here, interesting intrinsic properties of the dynamical equations such as, e.g., hidden internal symmetries of higher order were explored in broad detail. The Hamiltonian approach is complementary to the dynamical systems methods that had been employed, e.g, in the analysis of spatially homogeneous cosmological models by Collins and Stewart (1971) [30] and the paper by Collins and Hawking (1973) [31]. In line with this work, the dynamical systems method was used by Wainwright, Hsu, Hewitt and others for a systematic analysis of the past and future asymptotic, as well as intermediate, behaviour of orthogonally spatially homogeneous (OSH) ( see Ellis and MacCallum (1969) [49] ) and some more general spacetime geometries ( see the forthcoming book edited by Wainwright and Ellis [160] ). Unfortunately, the applicability of dynamical systems methods has so far been mostly limited to cases where the evolution equations of a particular model reduce to a system of ordinary differential equations, i.e., where the generically infinite dimensional state space underlying the EFE reduces to a (preferably compact) finite dimensional one. Another dynamical problem that arises relates to the question of the stability of (necessarily) idealised cosmological models under changes in the values and number of their free parameters, or the introduction of physically motivated perturbations to, e.g., their geometry or matter content. Here, investigations of the “fragility” or “structural stability” of a working model may be of interest ( see, e.g., Tavakol and Ellis (1988) [152] and Coley and Tavakol (1992) [29] ). Often it is found that such models are fragile under various changes. A different angle on obtaining information on the qualitative properties of (exact) spacetime geometries of more complex dynamical structure than the standard FLRW models is provided by addressing the important question of the integrability of the underlying dynamical equations. Recently, attention

6


has been focused on the integrability conditions of some classes of spatially inhomogeneous dust models, for which the corresponding solutions to the EFE are generally unknown ( see, e.g., Ellis (1996) [48] and Maartens (1996) [101] ). Once the integrability is established, the content of the related solution space can be probed by trying to transform the dynamical equations into a (flux conserving) first-order symmetric hyperbolic form. Systems of partial differential equations of this form are known to satisfy theorems on existence and uniqueness of solutions ( for a recent discussion see Geroch (1996) [60] ). For practical purposes of general relativistic cosmological modelling, in particular, in relation to the dynamical investigations mentioned in the last paragraph, a so-called covariant 1 + 3 “threading” decomposition of the spacetime manifold ( M, g ) has proved to be very helpful and instructive. In particular, the 1 + 3 decomposition is applied to all geometrical objects defined on ( M, g ), such as the Riemann curvature tensor, and the dynamical equations relating them, as, e.g., the EFE. The 1 + 3 covariant approach was mainly developed during the 1950s and 1960s by Raychaudhuri, Schücking, Ehlers, Sachs and Trümper ( see Ehlers (1961) [37] for many related references ), and thoroughly reviewed by Ellis (1971) [45]. Here an extra structure provided by an invariantly defined (future pointing) preferred timelike congruence with normalised tangent u/c is assumed to exist on ( M, g ), thus leading to an extended setting ( M, g, u/c ). The tangent u/c is often chosen to be aligned (comoving) with the timelike axis of the CMBR rest frame, i.e., that frame in which the CMBR appears to be isotropic to a very high degree. For reasons of methodological ease the matter content of a relativistic cosmological model is commonly represented phenomenologically in terms of a (in general viscous) self-gravitating fluid, as opposed to a more adequate kinetic theory description in terms of the moments of a distribution function over the 8-D phase space of position and momentum of the constituent particles of matter ( see, e.g., Ehlers (1971) [38] ). Rather than focusing on the metric oriented treatment of GR as originally devised by Einstein, in the 1 + 3 covariant approach dynamical relations between geometrical objects of more direct physical interest such as fluid matter and curvature variables are investigated. In particular, dynamical relations for the Weyl curvature variables naturally arise in the 1+3 covariant treatment, thus offering itself to further exploration of the WCH. Following work by Hawking (1966) [67], the 1 + 3 covariant approach was revived in the late 1980s and early 1990s by Ellis, Bruni and others ( see Refs. [50], [51], and [19] ) by establishing a fully covariant and gauge-invariant description of linearised spatially inhomogeneous matter and curvature perturbations of FLRW and other OSH cosmological models. The 1 + 3 covariant formulation of relativistic dynamics in cosmology is also particularly appealing as viewed from a pedagogical perspective. It is important to note that our confidence in extrapolating the applicability of GR as a theory of gravitational interactions to cosmological distance scales mainly derives from the successes the theory has had on much smaller scales, where it can be tested in an experimental sense. As a field theory that describes physical interactions “locally”, one would like to have a precise understanding of the degree to which GR may be considered to be a microscopic or a macroscopic theory of gravitation. Related to this problem is the question of whether the EFE require modification or not when averages are taken over both geometry and matter content on physical length scales that differ by many orders of magnitude. By necessity, some sort of averaging is automatically implied in relativistic cosmological modelling. A number of serious attempts at addressing the averaging problem within GR itself have

7 been made over the last nearly 15 years. The problem as such has been particularly emphasised by Ellis (1984) [46], while one of the first concrete proposals was put forward by Carfora and Marzuoli (1984) [25]. The problems related to averaging were also extensively investigated in a recent series of papers by Zalaletdinov (1992/93) [166, 167], in which he proposed a framework of macroscopic gravity based on a covariant spacetime averaging procedure, which advocates a form of a statistical approach to cosmological modelling. However, a number of important fundamental issues of averaging such as the uniqueness of a particular method remain to be open, and their resolution may have to be found outside the framework of GR, e.g, in non-Riemannian geometries ( see, e.g., Tavakol and Zalaletdinov (1996) [153] ). Indication that the back-reaction on the Hubble expansion of averaging over spatial inhomogeneities could lead to important consequences for structure formation processes and the estimates for the age of the Universe was given in work by, e.g., Bildbauer and Futamase (1991) [13]. The work presented in this thesis has been divided into two main parts. In the larger first part, extensions to the 1 + 3 decomposition formalism are developed, partially in fully covariant form, and partially on the basis of choice of an arbitrary Minkowskian orthonormal reference frame, the timelike direction of which is aligned with the preferred timelike reference congruence u/c. The treatment given here is mainly applied to fluid spacetime geometries with a perfect fluid matter source obeying a barotropic equation of state, but in parts also viscous fluid terms are taken into account. In the smaller second part, aspects of inflationary evolution in simple non-FLRW spacetime geometries are investigated. The outline of the thesis is as follows. Chapter 2 starts with an exposition of the familiar general 1 + 3 covariant dynamical equations for the fluid matter and Weyl curvature variables as they arise from the Ricci and second Bianchi identities for the Riemann curvature of ( M, g, u/c ). Here, the EFE are used algebraically to substitute for the Ricci curvature in terms of the fluid matter variables. This exposition is followed by a derivation of evolution equations for all spatial derivative terms of geometrical quantities, defined in the local rest 3-spaces orthogonal to u/c, that occur both in the spatial constraint equations as well as in some of the temporal propagation equations for barotropic perfect fluid spacetime geometries. These new evolution equations enable one to investigate in covariant terms the consistency of a set of dynamical equations describing specialised barotropic perfect fluid configurations according to the Dirac procedure, where by propagating all spatial constraints along the integral curves of u/c further integrability conditions may be generated. Consistency requires all spatial constraints restricting a specific system to be preserved along u/c. As a result of this derivation it is demonstrated that, as was generally expected but had not been explicitly shown so far, the 1 + 3 covariant dynamical equations describing a general barotropic perfect fluid spacetime geometry are consistent, as are a variety of special subcases that arise from either dynamically restricting the congruence u/c, or by restricting the intrinsic curvature of spacelike 3-surfaces (if well-defined) orthogonal to it. The new evolution equations for the spatial derivatives of the Weyl curvature variables may be of interest, e.g., in considerations of cosmological spacetime geometries containing gravitational radiation, as well as having significance in investigations relating to the WCH. Examples of a new 1+3 covariant representation of prominent barotropic perfect fluid cosmological models found in the literature are discussed in Chapter 3. The cases presented are, firstly, spacetime geometries that allow for a second invariantly defined reference congruence e, which is spacelike and

8


orthogonal to u/c. Furthermore, in these models the components of the Riemann curvature tensor of ( M, g, u/c, e ) and all covariant derivatives thereof are required to be invariant under rotations in the local spacelike 2-surfaces orthogonal to e (and u/c). These are the so-called locally rotationally symmetric (LRS) models, originally analysed by Ellis (1967) [44] and Stewart and Ellis (1968) [147] in terms of a 1+3 orthonormal frame discussion (see below), who confined themselves to using connection components and fluid matter quantities as dynamical variables only. The present discussion obtains the same overall classification as those analyses, but differs from them in the use of a 1 + 3 ( 1 + 1 + 2 ) covariant analysis throughout, giving a fully covariant characterisation of each class. Higher-symmetry subcases, which had not been given in previous treatments of the LRS family as a whole, are then systematically investigated. The second example given in this chapter outlines an attempt at a new 1 + 3 covariant formulation of the well-known family of OSH cosmological models, first discussed and analysed in detail by Ellis and MacCallum (1969) [49] (applying the same technique as Stewart and Ellis). Neither of the two cases of a 1 + 3 covariant treatment of particular cosmological models discussed in this chapter lead to new solutions of the EFE, but interesting dynamical subcases such as spacetime geometries with so-called purely “magnetic” Weyl curvature are obtained, which have not been explicitly derived before. Also, in the LRS case, a transparent treatment of “tilted” spatial homogeneity ( see King and Ellis (1973) [86] ) can be given in terms of an investigation of the functional dependencies between dynamical variables of relevance. Chapter 4 reviews the general 1+3 dynamical equations in Minkowskian orthonormal frame (ONF) formulation and extends them to include the second Bianchi identities in fully expanded form. This extension motivates inclusion of the Weyl curvature, which is of direct physical significance, in the set of dynamical variables. Through the latter step the 1 + 3 ONF dynamical equations are moved onto an equal footing with their 1 + 3 covariant counterparts. The extended 1 + 3 ONF formulation follows work by Pirani (1956) [129], Ellis (1967) [44], and MacCallum (1973) [103, 104] ( see also Wainwright and Ellis (1996) [160] ). Next in this chapter, the choice of local coordinate systems on ( M, g, u/c ) is outlined in both the 1 + 3 “threading” picture ( see, e.g., Jantzen et al (1992) [81] ) and, in the case of an irrotational preferred timelike reference congruence u/c, in the traditional 3 + 1 “slicing” picture of Arnowitt, Deser and Misner (1962) [2]. This allows some comments on the structure of the 1 + 3 ONF dynamical equations to be made. Ending this chapter, a useful decomposition of rank 2 symmetric tracefree tensor fields orthogonal to u/c, as suggested by Wainwright et al ( see Ref. [160] ), is given. Chapter 4 provides a basis for adapting the equivalence problem formalism for invariantly distinguishing between different spacetime geometries, as developed by Cartan (1946) [27] and Karlhede (1980) [84], to cosmological requirements in terms of 1 + 3 variables. Also, the general extended 1 + 3 ONF dynamical equations linking spatial connection components, fluid matter as well as Weyl curvature variables may provide an alternative setting for the rapidly growing and widespread attempts at numerical simulations and analysis of generic features of relativistic cosmological models ( see, e.g., Friedrich (1996) [58] ). In Chapter 5 the extended 1 + 3 ONF formalism is used to investigate the preservation along u/c of an extra spatial constraint that arises under the so-called “silent” Universe assumption for irrotational dust fluids, put forward by Matarrese et al (1994/95) [111, 20, 21]. This assumption restricts the Weyl curvature of the related spacetime manifold to be of purely “electric” character. On the basis of the results of the investigation given in this chapter, it is conjectured that the “silent” Universe assumption is too restrictive as to allow for more general classes of spatially inhomogeneous solutions — classes with

9 a Weyl curvature tensor of algebraic Petrov type I — than the already known Szekeres (1975) [151] dust family and the LRS class II dust models of Ellis (1967) [44]. The latter two both exhibit a degeneracy of the eigenvalues of the fluid rate of shear tensor and of the “electric” part of the Weyl curvature tensor and are therefore of algebraic Petrov type D. Besides these two spatially inhomogeneous classes, OSH models of Bianchi Type–I occur as special “silent” examples that do not show the aforementioned degeneracy. The extended 1 + 3 ONF method is further utilised by considering spacetime geometries with LRS symmetry again — here for an imperfect fluid matter source — and briefly comparing results relating to the perfect fluid subcase to the 1 + 3 ( 1 + 1 + 2 ) covariant discussion given in Chapter 3. In Chapter 6, the 1+3 covariant dynamical equations are generalised to cases where the Lagrangean density of the gravitational field is proportional to an arbitrary differentiable function f (R) in the Ricci curvature scalar. For reasons of generality the form of the stress-energy-momentum tensor of the associated matter source is not further specified. In this way, an extension of previous work by Maartens and Taylor (1994) [102] is obtained, who confined themselves to consider gravitational Lagrangean densities proportional to R + 1 R2 only. Possible applications of the f (R)-type theory are shortly touched upon. This part is in line with considerations initiated by Buchdahl (1970) [22] on generalised Lagrangean theories of gravitation which lead to fourth-order field equations. Inspired by modern quantum field theories, generalised Lagrangean theories of gravitation are thought to be capable of avoiding an initial singularity and generating higher rates of expansion in models of the early evolutionary phases of the Universe as compared to conventional models based on GR. In the latter sense, they offer an alternative to the inflationary (scalar field) models, which historically surfaced later in time relative to the former. In fact, Schmidt (1987) [133] has shown that the field equations of (vacuum) f (R)-type theories are conformally equivalent to those for Einstein–Scalar–Field spacetime geometries ( see also Maeda (1989) [110] ). However, results obtained on the basis of classical higher-order Lagrangean theories of gravitation have not gained widespread acclaim so far. Finally, Chapter 7 contains a detailed investigation of the dynamical evolution of the positive dimensionless cosmological density parameter Ω in spatially homogeneous inflationary fluid models with non-zero rate of shear. In standard FLRW-based inflationary models its present value is said to be very close, if not equal, to unity, Ω ≈ 1. It was, however, pointed out by Ellis (1988) [47] ( see also Hübner and Ehlers (1991) [78] ) that the set of FLRW inflationary models which lead to Ω 6= 1 instead is non-empty, if spatial curvature is not neglected. Here, this work is generalised by employing a simple two-phase, two-fluid model of either Bianchi Type–I or Type–V spacetime geometry in order to analyse the influence of spatial anisotropy on the evolution in the standard FLRW setting. Qualitative changes occur and are discussed. If a homogeneous, self-interacting scalar field matter source minimally coupled to the gravitational field is used as a matter source, restrictions on the permissible functional forms of the expansion length scale parameter can be derived. Here, following practice encountered in a large part of the literature on inflationary models, rather than assuming a particular form of the scalar field self-interaction potential and solving the resultant EFE and scalar field equation of motion, a particular functional form for the growth of physical length scales is assumed, and the EFE are inverted and solved for their source terms instead. This consideration is given again for both Bianchi Type–I and Type–V spacetime geometry and changes as compared to the FLRW case, discussed by Ellis and Madsen (1991) [52], are pointed out.

10


This thesis ends with concluding remarks in Chapter 8. Conventions on notation, signs and indices employed in this work are gathered and established in Appendix A. They are aimed at finding a working compromise between the many different, and partially even counterproductive conventions that occur in the literature. Useful mathematical relations and identities underlying the derivation of material presented in Chapters 2 and 4 are collected in Appendix B. Appendix C contains the explicit expansion of some 1+3 ONF variables and relations, and Appendix D presents the extended 1+3 ONF dynamical equations for arbitrary fluid matter sources in terms of expansion-normalised dimensionless variables and derivative operators, extending the work by Wainwright and collaborators for spatially homogeneous models and those spacetime geometries that allow for an Abelian G2 isometry group with 2-D spacelike orbits ( see Wainwright and Ellis (1996) [160] ). Throughout this work the computer algebra packages REDUCE and CLASSI have been invaluable tools.

Part of the work presented in this thesis was published in the following journals: 1. van Elst H and R Tavakol: Evolution of the Density Parameter in Inflationary Cosmology in the Presence of Shear and Bulk Viscosity, Phys. Rev. D 49 (1994), 6460. [ Chapter 7 ] 2. van Elst H, P K S Dunsby and R Tavakol: Constraints on Inflationary Solutions in the Presence of Shear and Bulk Viscosity, Gen. Rel. Grav. 27 (1995), 171. [ Chapter 7 ] 3. Rippl S, H van Elst, R Tavakol and D Taylor: Kinematics and Dynamics of f (R) Theories of Gravity, Gen. Rel. Grav. 28 (1996), 193. [ Chapter 6 ] 4. van Elst H and G F R Ellis: The Covariant Approach to LRS Perfect Fluid Spacetime Geometries, Class. Quantum Grav. 13 (1996), 1099. [ Chapter 3 ] 5. van Elst H and C Uggla: General Relativistic 1 + 3 Orthonormal Frame Approach Revisited, QMW Preprint QMW–AU–96005 (1996), gr-qc/9603026. [ Chapters 4 and 5 and Appendix D ] To provide evidence of work which is not further taken account of in this thesis, in agreement with the examination regulations of the University of London copies of papers 1. – 3. as well as of • van Elst H, J E Lidsey, R Tavakol: Quantum Cosmology and Higher-Order Langrangian Theories, Class. Quantum Grav. 11 (1994), 2483. have been bound in at the end.

Chapter 2

1 + 3 Decomposition of ( M, g, u/c ) in Covariant Form In geometrical treatments of general relativistic astrophysics and cosmology, the mathematical implication of the presence of a distribution of matter sources always allows the definition of a (future pointing) preferred timelike vector field on the spacetime manifold ( M, g ), which is locally unique. In the standard phenomenological fluid description of the matter content, this timelike vector field can locally be chosen to represent the average 4-velocity of all constituent particles of the fluid matter source, or, alternatively, as the timelike normal congruence to the local 3-surfaces of constant (positive) total energy density within the fluid ( i.e., in the notation defined below, the local 3-surfaces given by the condition Xµ = 0 ). If the latter option is chosen, however, one needs to be careful, as these 3-surfaces may change their causal character from spacelike to timelike (and vice versa for their normals). Given this preferred timelike vector field on ( M, g ), the associated normalised timelike vector field u/c ( such that uµ /c uµ /c = − 1 ⇒ uν /c (∇µ uν /c) = 0 ) determines tensors U and h, respectively, projecting parallel and orthogonal to u/c according to U µ ν := − uµ /c uν /c

hµ ν := δ µ ν + uµ /c uν /c .

(2.1)

Note that the components of the metric tensor field g and its inverse g−1 are related by g µρ gρν = δ µ ν . It follows that U µρ U ρν hµ ρ h ρ ν

= U µν = hµ ν

U µ ν uν /c = uµ /c hµ ν uν /c = 0

U µµ hµµ

= =

1 3.

(2.2)

The extra structure on the spacetime manifold, as provided by the preferred timelike reference congruence u/c ( extending the former to a setting ( M, g, u/c ) ), lends itself to a covariant 1+3 “threading” decomposition of ( M, g, u/c ), all its geometrical objects, and the dynamical equations relating them. This approach to GR was first introduced and developed in the 1950s and early 1960s by Raychaudhuri, Schücking, Ehlers, Sachs and Trümper ( see Ehlers (1961) [37] for references ). Ellis (1971) [45] later gave a nice review of the underlying methods and focused on applications of the 1 + 3 covariant approach in relativistic cosmology. As much of the work presented in this thesis is based on the 1 + 3 covariant formulation, in the first part of this chapter notations and all relevant relations, together with part of their derivation, are exhibited in detail.

11

CHAPTER 2. 1 + 3 DECOMPOSITION OF ( M, g, u/c ) IN COVARIANT FORM

12

It is helpful to introduce the standard notation (Aµν )˙/c := uρ /c (∇ρ Aµν ) for a directional covariant derivative of any tensor A along the integral curves of the preferred timelike reference congruence u/c. Then, taking directional covariant derivatives of the projection tensors, as defined in Eq. (2.1), both along and orthogonal to u/c, and projecting back into the local rest 3-spaces orthogonal to u/c, one obtains the simple relations

hσµ

hµρ hνσ (U ρσ )˙/c =

0

(2.3)

hµρ hνσ (hρσ )˙/c =

0

(2.4)

= 0

(2.5)

= 0.

(2.6)

hτν

hκρ

(∇σ Uτ κ )

hσµ hτν hκρ (∇σ hτ κ )

That is, in particular, the orthogonal projection tensor h provides a covariantly constant 3-metric tensor field in the local rest 3-spaces orthogonal to u/c. Next, the covariant derivative of u/c itself may be 1 + 3 decomposed into its irreducible parts. On defining u˙ µ /c2

:= hµν (uν /c)˙/c

σµν /c := [ hρ(µ hσν) − (Θ/c)

(2.7) 1 3

hµν hρσ ] (∇ρ uσ /c) = σ(µν) /c

:= hµν ∇µ uν /c

(2.8) (2.9)

ω µ /c := − 21 µνρσ (∇ν uρ /c) uσ /c ,

(2.10)

which leads to ∇µ uν /c = − uµ /c u˙ ν /c2 + σµν /c +

1 3

(Θ/c) hµν − µνρσ ω ρ /c uσ /c ,

(2.11)

one obtains dynamical variables, respectively, representing the (non-gravitational) acceleration vector field, u˙ µ , the symmetric tracefree rate of shear tensor field, σµν ( σ µµ = 0 ), the (volume) rate of expansion scalar field, Θ, and the vorticity vector field, ω µ , associated with the intrinsic distortions and twists of the integral curves of the preferred timelike reference congruence u/c. Frequently, the vorticity vector is defined in terms of an antisymmetric tensor field ωµν by1 ω µ /c := − 12 µνρσ ωνρ /c uσ /c

⇔

ωµν /c = − µνρσ ω ρ /c uσ /c ,

(2.12)

referred to as the vorticity tensor. The symmetric stress-energy-momentum tensor of the matter sources, in the phenomenological fluid description is given by Tµν = µ uµ /c uν /c + 2 q(µ /c uν) /c + p hµν + πµν ,

(2.13)

1 Note that the vorticity vector and tensor as defined here both have the opposite sign compared to definitions often found in the literature. This arises from the fact that here, e.g., in Eq. (2.10), ∇ is used for denoting the covariant derivative operator (rather than the semicolon symbol ‘;’), and signs have been adapted accordingly.

13 when 1+3 decomposed into its irreducible parts with respect to u/c. Generically, Eq. (2.13) represents a viscous (“imperfect”), self-gravitating fluid in a state away from thermal equilibrium, comoving with the preferred timelike reference congruence u/c (and thus defining it 1 + 3 invariantly). The definitions µ(u/c)

:= Tµν uµ /c uν /c

(2.14)

qµ /c(u/c)

:= − Tρν hρµ uν /c

(2.15)

p(u/c) πµν (u/c)

:=

1 3

Tµν h

µν

:= [ hρ(µ hσν) −

(2.16) 1 3

hµν hρσ ] Tρσ = π(µν)

(2.17)

provide dynamical variables related to the fluid matter sources. These are, respectively, the total energy density scalar field, µ, the energy current density vector field, q µ , the isotropic pressure scalar field, p, and the anisotropic pressure tensor field, πµν .2 A so-called “perfect fluid” has vanishing energy current density and anisotropic pressure in the local rest 3-spaces orthogonal to u/c, 0 = q µ /c = πµν . On the other hand, if one instead chooses the normals of the local 3-surfaces of constant total energy density as a preferred timelike reference congruence, then Eq. (2.13) can serve to represent the irreducible 1 + 3 decomposition of the stress-energy-momentum tensor of a non-comoving perfect fluid with respect to these normals ( see, e.g., King and Ellis (1973) [86] ). As in this example, in general the average 4velocity of a (relativistic) fluid is “tilted” with respect to its surfaces µ = const. Of course, a complete description requires specification of the thermodynamical properties of a particular fluid matter source. Comments on this aspect are made below. By use of the projection tensors defined in Eq. (2.1), the Weyl conformal curvature tensor ( cf. Eq. (A.22) in Appendix A ) may be 1 + 3 decomposed into symmetric tracefree, so-called “electric” and “magnetic” parts. Thus: Eµν (u/c) Hµν (u/c)

:= Cρκσλ hρµ uκ /c hσν uλ /c = E(µν) := (−

1 2 ρκξη

C ξησλ ) hρµ

u

κ

/c hσν

λ

u /c = H(µν) ,

( 0 = E µµ = H µµ ), which leads to3 i h C µνρσ = 4 δ [µκ δ ν]λ δ ξ[ρ δ ησ] − µνκλ ξηρσ E κξ uλ /c uη /c h i + 2 µνκλ δ ξ[ρ δ ησ] + δ [µκ δ ν]λ ξηρσ H κξ uλ /c uη /c .

(2.18) (2.19)

(2.20)

The Weyl curvature tensor vanishes if its “electric” and “magnetic” parts vanish, and vice versa, 0 = Eµν = Hµν ⇔ Cµνρσ = 0. The Riemann curvature tensor of ( M, g, u/c ) can now be cast into the form Rµν ρσ

[µ ν] ν] 4πG [µ = C µν ρσ + 2 4πG c4 (µ − p) δ [ρ h σ] + 4 c4 δ [ρ π σ] [µ ν] ν] + 4 4πG c4 δ [ρ ( q /c uσ] /c + u /c qσ] /c ) [µ ν] + 2 4πG c4 (µ + 3 p) δ [ρ u /c uσ] /c

−

2 4πG 3 c4

(µ − 3 p) δ µ [ρ δ ν σ] +

2 3

Λ δ µ [ρ δ ν σ] ,

(2.21)

2 It is still a conceptual problem whether these quantities reflect either microscopic, or rather macroscopic aspects of a distribution of (in general discrete) matter sources ( see, e.g., Tavakol and Zalaletdinov (1996) [153] ). 3 Note the sign error in the first term of the equivalent formula given on p130 of Ellis (1971) [45].

14


where Eq. (A.15) of Appendix A was used, the Ricci curvature tensor was algebraically expressed in terms of the EFE (A.31), and the stress-energy-momentum tensor of an imperfect fluid comoving with u/c was substituted from Eq. (2.13) above. In the following a set of 1 + 3 invariantly defined 3-scalars and various 3-derivative terms is introduced, which prove to be of notational convenience in subsequent chapters and sections.

2.1 Definitions 2.1.1

Covariant spatial 3-scalars

For general purposes the symbol f is introduced as denoting any 1+3 invariantly defined 3-scalar field. In particular, one defines: (ω/c)2

:=

(σ/c)2

:=

E2

:=

ωµ /c ω µ /c 1 2

(u/c ˙ 2 )2

:= u˙ µ /c2 u˙ µ /c2

σ µν /c σ νµ /c 1 2

E µν E νµ

(2.22) H2

:=

1 2

H µν H νµ .

All of these 3-scalars are positive semi-definite, i.e., (ω/c)2 ≥ 0, and similarly for all other 3-scalars. Note also that (ω/c) = 0 ⇔ ω µ /c = 0, and equivalently for the remaining 3-scalars.

2.1.2

Covariant spatial 3-gradients

It is useful to define Fµ := hνµ ∇ν f as denoting the spatial 3-gradient of any 1 + 3 invariantly defined 3-scalar field f . The following symbols will occur frequently throughout, the first two of which were defined in Ellis and Bruni (1989) [50]: Xµ

:=

hνµ ∇ν µ

Zµ

:=

hνµ ∇ν (Θ/c)

Wµ

:=

hνµ ∇ν (ω/c)2

Aµ

:=

hνµ ∇ν (u/c ˙ 2 )2

Sµ

:=

hνµ ∇ν (σ/c)2

Eµ

:=

hνµ ∇ν E 2

SEµ

2.1.3

:= hνµ ∇ν [ σ ρσ /c E σρ ]

(2.23) Hµ SHµ

:=

hνµ ∇ν H 2

:= hνµ ∇ν [ σ ρσ /c H σρ ] .

Covariant spatial rotation terms of spatial 3-tensors

Spatial rotation terms of the symmetric tracefree 3-tensor fields Eµν , Hµν and σµν /c occur in both the evolution as well as the constraint equations of the 1 + 3 covariant formalism (to be discussed below). It is suggested to use the notations: Iµν

:=

− hρ(µ hσν) ρτ κλ (∇τ E κσ ) uλ /c = I(µν)

Jµν

:=

− hρ(µ hσν) ρτ κλ (∇τ H κσ ) uλ /c = J(µν)

Kµν

:= − hρ(µ hσν) ρτ κλ (∇τ σ κσ /c) uλ /c = K(µν) .

(2.24)

Note that these spatial rotation terms themselves are symmetric tracefree 3-tensor fields ( 0 = I µµ = J µµ , K µµ = 0 ).

15

2.2. THE RICCI IDENTITIES

2.2

The Ricci identities

The dynamical equations in the 1+3 covariant approach to GR, including both evolution and constraint equations and linking the geometrical quantities defined in the previous sections, arise from two sets of identities satisfied by the Riemann curvature tensor of ( M, g, u/c ). In this approach the EFE are used algebraically to substitute for the Ricci curvature tensor in terms of fluid matter variables. The first set are the algebraic Ricci identities, Eq. (A.16) in Appendix A. Applying these identities to the preferred timelike reference congruence u/c itself, substituting for the Riemann curvature tensor from Eq. (2.21), invoking the 1 + 3 decomposition, and projecting back into the local rest 3-spaces orthogonal to u/c, one arrives at the following set of equations ( including, for completeness, the cosmological constant, Λ ):4

2.2.1

Time derivative equations 2

2

2

(Θ/c)˙/c = − 31 (Θ/c) + hµν ∇µ u˙ ν /c2 + (u/c ˙ 2 )2 − 2 (σ/c) + 2 (ω/c) − 4πG c4 (µ + 3p) + Λ

hµν (ω ν /c)˙/c = − 23 (Θ/c) ω µ /c + σ µν /c ω ν /c −

(2.25)

1 µνρσ 2

∇ν u˙ ρ /c2 uσ /c

(2.26)

hµρ hνσ (σ ρσ /c)˙/c = − 32 (Θ/c) σ µν /c + h(µρ hν)σ ∇ρ u˙ σ /c2 + u˙ µ /c2 u˙ ν /c2 − σ µρ /c σ νρ /c − ω µ /c ω ν /c − (E µν − −

2.2.2

1 3

4πG c4

2

π µν ) 2

[ hρσ ∇ρ u˙ σ /c2 + (u/c ˙ 2 )2 − 2 (σ/c) − (ω/c) ] hµν . (2.27)

Constraint equations 0

= hµρ hνσ (∇ν σ ρσ /c) −

2 3

Z µ − µνρσ [ ∇ν ωρ /c + 2 u˙ ν /c2 ωρ /c ] uσ /c

µ + 2 4πG c4 q /c

0

= hµν ∇µ ω ν /c − u˙ µ /c2 ω µ /c

0

= Hµν + 2 u˙ (µ /c2 ων) /c + hρ(µ hσν) (∇ρ ωσ /c) − Kµν −

1 3

[ 2 u˙ ρ /c2 ω ρ /c + hρσ ∇ρ ω σ /c ] hµν .

(2.28)

(2.29)

(2.30)

Equations (2.28) - (2.30) are often called, respectively, the divergence equation for the fluid rate of shear, the divergence equation for the fluid vorticity, and the H-constraint. If, e.g., the fluid matter source has vanishing acceleration and vorticity, 0 = (u/c ˙ 2 ) = (ω/c), i.e., the fluid is geodesic and irrotational, then the H-constraint (2.30) reduces to the simple relation 0 = Hµν − Kµν . 4 This

was first done systematically by Ehlers in 1961 [37] ( see also Ellis (1971) [45] ).


16

2.3 The second Bianchi identities The second set of identities for the Riemann curvature tensor of ( M, g, u/c ), from which one derives further dynamical equations for the geometrical quantities in the 1 + 3 covariant approach, are the differential second Bianchi identities. In the form given by Kundt and Trümper (1962) [90], Eq. (A.24) in Appendix A, by applying the same procedure as briefly outlined above, the second Bianchi identities lead to the following set of equations:5

2.3.1

Time derivative equations

hµρ hνσ (E ρσ +

4πG c4

µν µν π ρσ )˙/c = − 4πG + c4 (µ + p) σ /c − (Θ/c) (E

−

4πG c4

1 4πG 3 c4

π µν )

h(µρ hν)σ (∇ρ q σ /c) − 2 4πG ˙ (µ /c2 q ν) /c c4 u

ν)ρ + 3 σ (µρ /c (E ν)ρ − 13 4πG ) + J µν c4 π ρ σ 4πG + 13 4πG ˙ ρ /c2 q ρ /c c4 h σ ∇ρ q /c + 2 c4 u µν σ − 3 σ ρσ /c (E σρ − 13 4πG c4 π ρ ) h − h(µκ ν)ρστ 2 u˙ ρ /c2 H κσ κ + ωρ /c (E κσ + 4πG c4 π σ ) uτ /c (2.31)

(µ ν) µν hµρ hνσ (H ρσ )˙/c = − (Θ/c) H µν + 3 σ (µρ /c H ν)σ + 3 4πG c4 ω /c q /c − I µν ρ − σ ρσ /c H σρ + 4πG c4 ωρ /c q /c h κ − h(µκ ν)ρστ 4πG ˙ ρ /c2 E κσ c4 (∇ρ π σ ) − 2 u

+

4πG c4

σ κρ /c qσ /c

+ ωρ /c H κσ ] uτ /c

(2.32)

hµν (q ν /c)˙/c = − 34 (Θ/c) q µ /c − hµν ∇ν p − (µ + p) u˙ µ /c2 − hµρ hνσ (∇ν π ρσ ) − u˙ ν /c2 π µν − σ µν /c q ν /c + µνρσ ων /c qρ /c uσ /c

(2.33)

µ/c ˙ = − (µ + p) (Θ/c) − hµν ∇µ q ν /c − 2 u˙ µ /c2 q µ /c − σ µν /c π νµ . 5 This

derivation was first obtained by Trümper (unpublished).

(2.34)

2.4. COMMENTS ON THE 1 + 3 COVARIANT DYNAMICAL EQUATIONS

2.3.2

17


= hµρ hνσ ∇ν (E ρσ + −

4πG c4

4πG c4

π ρσ ) −

2 4πG 3 c4

Xµ +

2 4πG 3 c4

(Θ/c) q µ /c

σ µν /c q ν /c − 3 ων /c H µν

+ µνρσ [ σντ /c H τρ − 3 4πG c4 ων /c qρ /c ] uσ /c

0

µ µν = hµρ hνσ (∇ν H ρσ ) + 2 4πG − c4 (µ + p) ω /c + 3 ων /c (E

− µνρσ [

4πG c4

∇ν qρ /c + σντ /c (E τρ +

4πG c4

(2.35)

1 4πG 3 c4

π µν )

π τρ ) ] uσ /c .

(2.36)

Within the set (2.31) - (2.36), Eqs. (2.33) and (2.34) correspond to the results obtained from a covariant 1 + 3 decomposition of the twice-contracted second Bianchi identities given in the form of Eq. (A.33) in Appendix A, where Eq. (2.13) was used to express the stress-energy-momentum tensor. Equations (2.35) and (2.36) in this set are often called the spatial divergence equations for the “electric” and the “magnetic” parts of the Weyl curvature tensor, respectively.

2.4

Comments on the 1 + 3 covariant dynamical equations

On close inspection it becomes obvious that the set of 1 + 3 dynamical equations as constituted by Eqs. (2.25) - (2.36) is incomplete. For example, this set only provides an evolution equation for the 4πG combination (Eµν + 4πG c4 πµν ), but none for (Eµν − c4 πµν ), which is a source term on the righthand

side of the rate of shear evolution equation (2.27). Also, there are no evolution equations for either the isotropic pressure, p, or the fluid acceleration, u˙ µ /c2 . Hence, in order to close this set of equations, further assumptions about the fluid matter source fields are required. In particular, one needs assumptions on their thermodynamical properties, which suggest equations of state that consistently link the matter fields to each other and to the remaining fluid variables. Ideally, the equations of state should be the outcome of a rigorous treatment of the physics of the matter fields in terms of a kinetic theory description ( see, e.g., Ehlers (1971) [38] ). However, in general one resorts to the phenomenological fluid description of the matter content as presented here (which still proves to be sufficiently complicated). A first framework for a relativistic treatment of the thermodynamics of fluid matter was presented by Eckart in 1940 [35], which, however, suffers from the deficiency that thermal interactions propagate at infinite velocities. Eckart expanded (small) deviations from thermal equilibrium only to first order. This fundamental shortcoming was later bypassed by the suggestion of a new framework of relativistic thermodynamics, initiated by Müller (1967) [115], and later fully developed by Israel (1976) [79] and Israel and Stewart (1979) [80]. Here (small) deviations from thermal equilibrium are expanded to second order. That the second-order framework has indeed the required properties to supersede Eckart’s treatment, was shown by Hiscock and Lindblom (1983) [72]. However, so far the second-order framework has experienced only rather little practice in relativistic cosmology, which might change once numerical methods of analysis become more feasible and widespread. A series of investigations in this field are continued, among others by Pavón and collaborators ( see, e.g., Refs. [123] and [132] ). Another problematic aspect of the 1 + 3 covariant dynamical equations, as reviewed in the previous two sections, is that it is not entirely clear how a well-defined initial value problem could arise, if the


18

preferred timelike reference congruence u/c has non-zero vorticity, (ω/c) 6= 0 ( see, e.g., Jantzen et al [82] ). In that case there do not exist extended spacelike 3-surfaces everywhere orthogonal to u/c.

2.5 Vanishing vorticity If, however, the preferred timelike reference congruence u/c is irrotational, (ω/c) = 0, then the local rest 3-spaces orthogonal to every individual flow line of the preferred timelike reference congruence u/c mesh together to constitute spacelike 3-surfaces everywhere orthogonal to u/c. In that case, the Gauß equation ( see, e.g., Stephani (1991) [144] ), which describes how these spacelike 3-surfaces are embedded in ( M, g, u/c ), provides expressions encoding their intrinsic curvature properties in terms of the 3-Ricci tensor 3Rµν . By use of Eq. (2.21) one obtains 3

4πG c4

Sµν

=

(Eµν +

3

=

4 4πG c4 µ −

R

2 3

πµν ) −

1 3

(Θ/c) σµν /c + σµρ /c σ ρν /c −

2 3

(σ/c)2 hµν

2

(Θ/c) + 2 (σ/c)2 + 2 Λ ,

(2.37)

(2.38)

where the symmetric tracefree part of the 3-Ricci curvature tensor is defined by 3Sµν := 3Rµν − 1 3 3 R hµν , and represents the anisotropic part of the 3-Ricci curvature.

On the other hand, the 3-scalar 3R

represents its isotropic part. The conformal curvature properties of the spacelike 3-surfaces orthogonal to u/c are encoded in the symmetric tracefree 3-Cotton–York tensor (density) [33, 164], which has zero spatial divergence. In a form due to Jantzen et al [82] it is defined as the spatial rotation of the anisotropic 3-Ricci curvature tensor, i.e., 3

Cµν

:= − h1/3 hρ(µ hσν) ρτ κλ (∇τ 3S κσ ) uλ /c (2.39) = − h1/3 hρµ hσν (Hρσ )˙/c + 34 (Θ/c) Hµν − 3 σ ρ(µ /c Hν)ρ + σ ρσ /c H σρ hµν κ − h1/3 hρ(µ hσν) ρτ κλ ∇τ (σ κξ /c σ ξσ /c + 2 4πG c4 π σ ) − 13 Z τ σ κσ /c − 2 u˙ τ /c2 E κσ λ τ κ + 4πG c4 σ σ /c q /c u /c ,

(2.40)

where h is the determinant of the orthogonal projection tensor h, and where in Eq. (2.40) the relations (2.30) and (2.32) have been used to make substitutions. It follows from the twice-contracted second Bianchi identities that the 3-Einstein tensor of the spacelike 3-surfaces, given by 3Gµν := 3Sµν −

1 3 6 R hµν ,

also is spatial divergence-free, i.e.,

0 = hµρ hνσ (∇ν 3Gρσ ) .

(2.41)

For an irrotational perfect fluid, where 0 = q µ /c = πµν and Hµν = Kµν , using Eqs. (2.28), (2.35), (2.37) and (2.38) it can be shown that this corresponds to the identity 0

= − µνρσ σντ /c K τρ uσ /c +

1 2

σ µρ /c hνσ (∇ν σ ρσ /c)

+ hµρ σ νσ /c (∇ν σ ρσ /c) − S µ . This expression is just identity (B.25) in Appendix B as applied to Aµν = Bµν = σµν /c.

(2.42)

2.6. DERIVATION OF NEW EVOLUTION EQUATIONS

2.6

19

Derivation of new evolution equations for barotropic perfect fluid spacetime geometries

One of the nice aspects of the 1+3 covariant approach to GR is that the underlying dynamical equations have a stronger appeal from a physical point of view, as compared to the quasi-linear, second-order partial differential equation form which the EFE take in the metric based approach. That is, for a given initial configuration, one can directly read off from the dynamical equations what sort of interaction terms would generate what sort of physical fields in the subsequent evolution, and hence, the gravitational phenomena representative of a particular spacetime geometry. Moreover, most of the dynamical variables involved in this formulation in principle correspond to directly measurable physical fields. However, the dynamical equations also contain fully orthogonally projected covariant derivatives. This means that, implicitly, they contain spatial connection components, which are, by far, the most difficult variables to determine. This becomes more obvious in the 1 + 3 orthonormal frame (ONF) formulation discussed in Chapter 4 below. One also resorts to the 1 + 3 ONF formulation to reconstruct the metric tensor field g on ( M, g, u/c ), if one successfully manages to solve the 1 + 3 covariant dynamical equations. In view of the problems related to the description of the fluid matter source fields, briefly touched upon in Section 2.4, in 1 + 3 covariant discussions of fluid spacetime geometries, especially in relativistic cosmology, one (very) often specialises to the case of a perfect fluid matter source comoving with u/c. This situation is mathematically characterised by imposing the conditions 0 = q µ /c = πµν on the stress-energy-momentum tensor, as given in Eq. (2.13). Additionally, it is common to assume that the fluid matter flow be isentropic ( i.e., physical interactions between the matter source fields do not produce entropy ), such that its total energy density, µ, and isotropic pressure, p, can be related by a barotropic equation of state p = p(µ) (with

∂p ∂µ

6= 0, unless otherwise stated), and that (µ + p) > 0.

However, as in the physical world shearing motions of fluids (as they will occur in most of the subsequent examples) are closely tied to the induction of dissipative, i.e., entropy generating effects such as viscosity, the barotropic perfect fluid assumption is frequently criticised as rather being of practical appeal than of physical one. On the other hand, if, like Misner (1968) [113], one assumes shearing and viscous effects to be of importance at least during the early stages of the evolution of the Universe, then it is not immediately clear how these features can be smoothly accommodated within (any form of) Penrose’s WCH, as the 1 + 3 covariant dynamical equations of Sections 2.2 and 2.3 indicate that generically both temporal and spatial changes in the fluid rate of shear and also the viscosity terms generate non-zero Weyl curvature. In the cosmological context, (Θ/c) 6= 0, a further simplification often made is to employ a pressurefree fluid, p = 0, a so-called “dust” matter source. It is assumed that this reduction provides a viable approximation to the dynamically important fluid matter source fields present in the Universe after the thermal radiation, which nowadays constitutes the CMBR, stopped interacting (“decoupled”) with all forms of baryonic and leptonic (hence, fermionic) matter. That is, one assumes that the CMBR no longer makes an important contribution to the total energy density µ. Given the barotropic perfect fluid assumption, the non-zero geometrical quantities in the generic case of the 1 + 3 covariant approach to GR will be the 3-scalar field variables µ, p, (Θ/c), the 3vector field variables ω µ /c, u˙ µ /c2 , and the symmetric tracefree 3-tensor field variables σµν /c, Eµν


20

and Hµν . In the rest of this chapter the cosmological constant is assumed to be non-existent, i.e., Λ = 0. The interesting geometrical quantities in the resultant reduced form of the 1 + 3 covariant dynamical equations (2.25) - (2.36) are the spatial derivative terms in both the constraint equations and, acting as source terms, on the righthand side of some of the evolution equations.6 One would therefore like to know how these terms themselves evolve along the integral curves of the preferred timelike reference congruence u/c, especially in light of the question of whether the 1 + 3 covariant dynamical equations for barotropic perfect fluid spacetime geometries are integrable. On the basis of the twice-contracted second Bianchi identities ( cf. Eq. (A.32) in Appendix A ) integrability in this case is generally expected ( see, e.g., Wald (1984) [161] ), and this question is explicitly addressed in Section 2.7 below.

In the barotropic perfect fluid case, it turns out that the evolution equation (2.33) for the energy current density q µ /c reduces to an algebraic expression for the fluid acceleration in terms of the total energy density 3-gradient: −1

u˙ µ /c2 = − (µ + p)

−1 ∂p ∂µ

hµν ∇ν p = − (µ + p)

Xµ .

(2.43)

For the spatial divergence and the spatial rotation of the fluid acceleration one obtains, respectively, hµν ∇µ u˙ ν /c2

=

−2

(µ + p)

−1

∂p −1 [ ( ∂µ ) + 1 ] (hνµ ∇ν p) (hµρ ∇ρ p) − (µ + p)

hνµ ∇ν (hµρ ∇ρ p)

−1

∂p µ ν = − (µ + p) ∂µ h ν ∇µ X h i −1 ∂ 2 p −1 ∂p 2 − (µ + p) − (µ + p) ( ) Xµ X µ , 2 ∂µ ∂µ

(2.44)

and −1 ∂p µνρσ ∂µ

− µνρσ (∇ν u˙ ρ /c2 ) uσ /c = (µ + p)

∂p (∇ν Xρ ) uσ /c = 2 ∂µ (Θ/c) ω µ /c ,

(2.45)

where Eq. (B.2) of Appendix B is used to derive the second part of Eq. (2.45). Hence, u˙ µ /c2 will automatically have zero spatial rotation, if (ω/c) = 0. A 1+3 covariant evolution equation for the fluid acceleration vector field follows from investigating the preservation of the spatial constraint equations, including Eq. (2.43), along u/c, as will be discussed in Section 2.7 below ( cf. Eq. (2.79) ).

2.6.1

Evolution of spatial 3-gradients

In their 1 + 3 covariant and gauge-invariant treatment of linearised matter and curvature perturbations to FLRW barotropic perfect fluid spacetime geometries, Ellis and Bruni (1989) [50] derived exact propagation equations for the spatial 3-gradients of the total energy density, X µ , and the rate of expansion, Z µ . Here, this set of equations is extended to include new evolution equations for the spatial 3-gradients of some other 1 + 3 invariantly defined 3-scalars of interest, listed in Eq. (2.23). The derivation of this 6 These

terms are equally interesting in the generic, imperfect fluid case.

21

2.6. DERIVATION OF NEW EVOLUTION EQUATIONS

set of equations rests on repeated application of Eq. (B.1) in Appendix B, and use of the evolution equations (2.25) - (2.27), (2.31), (2.32), (2.34) and (2.79). One obtains: hµν (X ν )˙/c = − 43 (Θ/c) X µ − ( σ µν /c − µνρσ ωρ /c uσ /c ) Xν − (µ + p) Z µ

(2.46)

hµν (Z ν )˙/c = − (Θ/c) Z µ − ( σ µν /c − µνρσ ωρ /c uσ /c ) Zν − u˙ µ /c2 [

2

(Θ/c) − hνρ ∇ν u˙ ρ /c2 − (u/c ˙ 2 )2 + 2 (σ/c)2

1 3

− 2 (ω/c)2 − 2 4πG c4 µ ] + hµν ∇ν [ hρσ ∇ρ u˙ σ /c2 ] + Aµ − 2 S µ + 2 W µ −

4πG c4

Xµ

(2.47)

hµν (W ν )˙/c = − 53 (Θ/c) W µ − ( σ µν /c − µνρσ ωρ /c uσ /c ) Wν − u˙ µ /c2 [

(Θ/c) (ω/c)2 − 2 σνρ /c ω ν /c ω ρ /c

4 3

+ ων /c νρστ (∇ρ u˙ σ /c2 ) uτ /c ] −

4 3

(ω/c)2 Z µ + 2 hµν ∇ν [ σρσ /c ω ρ /c ω σ /c ]

− hµν ∇ν [ ωρ /c ρστ κ (∇σ u˙ τ /c2 ) uκ /c ] hµν (Aν )˙/c =

∂p 2 ( ∂µ −

1 2

(2.48)

) (Θ/c) Aµ − ( σ µν /c − µνρσ ωρ /c uσ /c ) Aν

+ 2 u˙ µ /c2 [ u˙ ν /c2 ∇ν [

∂p ∂µ

∂p (Θ/c) ] + ( ∂µ −

1 3

) (Θ/c) (u/c ˙ 2 )2

− σνρ /c u˙ ν /c2 u˙ ρ /c2 ] ∂p + 2 ( ∂µ −

1 3

) (u/c ˙ 2 )2 Z µ + 2 hµν ∇ν [ u˙ ρ /c2 ∇ρ [ 2 2

+ 2 (Θ/c) (u/c ˙ )

∂p hµν ∇ν ( ∂µ )

−

2 hµν ∇ν [

∂p ∂µ

(Θ/c) ] ]

ρ

σρσ /c u˙ /c2 u˙ σ /c2 ]

(2.49)

hµν (S ν )˙/c = − 53 (Θ/c) S µ − ( σ µν /c − µνρσ ωρ /c uσ /c ) Sν − u˙ µ /c2 [

4 3

(Θ/c) (σ/c)2 − σνσ /c (∇ν u˙ σ /c2 ) − σνσ /c u˙ ν /c2 u˙ σ /c2 + σνσ /c ω ν /c ω σ /c + σ νσ /c σ στ /c σ τν /c + σ νσ /c E σν ]

−

4 3

(σ/c)2 Z µ − hµν ∇ν [ σ ρσ /c σ στ /c σ τν /c ] − SE µ

+ hµν ∇ν [ σρσ /c (∇ρ u˙ σ /c2 ) + σρσ /c u˙ ρ /c2 u˙ σ /c2 − σρσ /c ω ρ /c ω σ /c ]

(2.50)

hµν (E ν )˙/c = − 73 (Θ/c) E µ − ( σ µν /c − µνρσ ωρ /c uσ /c ) Eν − u˙ µ /c2 [

4πG c4

(µ + p) σ νρ /c E ρν + 2 (Θ/c) E 2 − 3 σ νρ /c E ρσ E σν

− E νρ J ρν + 2 Eνκ νρστ u˙ ρ /c2 H κσ uτ /c ] ∂p ν ρ µ ∂µ ) (σ ρ /c E ν ) X

− 2 E2 Zµ

−

4πG c4

(1 +

−

4πG c4

(µ + p) SE µ + hµν ∇ν [ 3 σ ρσ /c E στ E τρ + E ρσ J σρ ]

− 2 hµν ∇ν [ Eρτ ρσκλ u˙ σ /c2 H τκ uλ /c ] hµν (H ν )˙/c = − 73 (Θ/c) H µ − ( σ µν /c − µνρσ ωρ /c uσ /c ) Hν

(2.51)


22

− u˙ µ /c2 [ 2 (Θ/c) H 2 − 3 σ νρ /c H ρσ H σν + H νρ I ρν − 2 Hνκ νρστ u˙ ρ /c2 E κσ uτ /c ] − 2 H 2 Z µ + hµν ∇ν [ 3 σ ρσ /c H στ H τρ − H ρσ I σρ ] + 2 hµν ∇ν [ Hρτ ρσκλ u˙ σ /c2 E τκ uλ /c ] .

2.6.2

(2.52)

Evolution of further geometrical quantities

Next, a derivation of new evolution equations for the spatial divergence terms of the fluid vorticity, the fluid rate of shear, and the “electric” and “magnetic” parts of the Weyl curvature is given. These terms occur in Eqs. (2.29), (2.28), (2.35) and (2.36), respectively. On (repeated) application of Eqs. (B.4) and (B.13) in Appendix B, and use of the evolution equations (2.26), (2.27), (2.31) and (2.32), one obtains: [ hµν ∇µ ω ν /c ]˙/c = − 32 (Θ/c) hµν ∇µ ω ν /c + σ µν /c ωµ /c u˙ ν /c2 −

1 2

u˙ µ /c2 µνρσ (∇ν u˙ ρ /c2 ) uσ /c + ωµ /c (u˙ µ /c2 )˙/c

hµν [ hνσ hρτ (∇ρ σ στ /c) ]˙/c = − (Θ/c) hµρ hνσ (∇ν σ ρσ /c) − σ µρ /c hνσ (∇ν σ ρσ /c) −

(2.53) 4 3

Sµ

− hµρ hνσ (∇ν E ρσ ) − hµκ ων /c νρστ (∇ρ σ κσ /c) uτ /c + σ µν /c νρστ (∇ρ ωσ /c) uτ /c − hµρ ων /c (∇ν ω ρ /c) − ω µ /c hνρ ∇ν ω ρ /c +

1 3

Wµ

+ µνρσ σντ /c [ H τρ − 2 K τρ ] uσ /c − u˙ ν /c2 [

1 6

(Θ/c) σ µν /c −

1 6

(Θ/c) µνρσ ωρ /c uσ /c

+ σ µρ /c σ νρ /c + σ µρ /c νρστ ωσ /c uτ /c + −

1 3

u˙ µ /c2 [

1 2

1 2

E µν ] 2

(Θ/c) − 2 hνρ ∇ν u˙ ρ /c2 − 2 (u/c ˙ 2 )2 + 4 (σ/c)2 + 2 (ω/c)2 − 2 4πG c4 µ ]

+

1 2

hµρ hνσ (∇ν ∇ρ u˙ σ /c2 ) +

+

3 2

hµρ u˙ ν /c2 (∇ν u˙ ρ /c2 ) −

+

1 2

(Θ/c) hµν (u˙ ν /c2 )˙/c

1 6

1 12

hµρ ∇ρ [ hνσ ∇ν u˙ σ /c2 ] Aµ

− µνρσ ων /c uσ /c (u˙ ρ /c2 )˙/c hµν [ hνσ hρτ (∇ρ E στ ) ]˙/c = − 34 (Θ/c) hµρ hνσ (∇ν E ρσ ) + + hµρ E νσ (∇ρ σ σν /c) −

5 4

4πG c4

σ µρ /c hνσ (∇ν E ρσ ) −

(2.54)

1 2

SE µ

(µ + p) hµρ hνσ (∇ν σ ρσ /c)

+

3 4

E µρ hνσ (∇ν σ ρσ /c) + hµρ hνσ (∇ν J ρσ )

−

1 2

hµκ ων /c νρστ (∇ρ E κσ ) uτ /c

−

1 µνρσ 2

ων /c hκλ (∇κ E λρ ) uσ /c

+

1 µνρσ 2

E τν (∇τ ωρ /c) uσ /c

−

1 2

−

4πG c4

+

1 µνρσ 2

E µν νρστ (∇ρ ωσ /c) uτ /c − E µν Z ν (1 +

∂p µ ν ∂µ ) σ ν /c X

[ σντ /c I τρ + 3 Eντ K τρ + 2 Eντ H τρ ] uσ /c

23

2.6. DERIVATION OF NEW EVOLUTION EQUATIONS − u˙ ν /c2 [ µνρσ hκλ (∇κ H λρ ) uσ /c − hµκ νρστ (∇ρ H κσ ) uτ /c +

4πG c4

−

5 2

(µ + p) σ µν /c

σ µρ /c E νρ −

1 2

E µρ σ νρ /c − J µν

− 3 E (µρ ν)ρστ ωσ /c uτ /c − µρστ H νρ u˙ σ /c2 uτ /c ] − 2 u˙ µ /c2 σ νρ /c E ρν + µνρσ H τν (∇τ u˙ ρ /c2 ) uσ /c − H µν νρστ (∇ρ u˙ σ /c2 ) uτ /c hµν [ hνσ hρτ (∇ρ H στ ) ]˙/c = − 34 (Θ/c) hµρ hνσ (∇ν H ρσ ) + + hµρ H νσ (∇ρ σ σν /c) + − hµρ hνσ (∇ν I ρσ ) −

1 2

3 4

5 4

(2.55)

σ µρ /c hνσ (∇ν H ρσ ) −

1 2

SH µ

H µρ hνσ (∇ν σ ρσ /c)

hµκ ων /c νρστ (∇ρ H κσ ) uτ /c

−

1 µνρσ 2

ων /c hκλ (∇κ H λρ ) uσ /c

+

1 µνρσ 2

H τν (∇τ ωρ /c) uσ /c

−

1 2

+

1 µνρσ 2

H µν νρστ (∇ρ ωσ /c) uτ /c − H µν Z ν [ σντ /c J τρ + 3 Hντ K τρ ] uσ /c

+ u˙ ν /c2 [ µνρσ hκλ (∇κ E λρ ) uσ /c − hµκ νρστ (∇ρ E κσ ) uτ /c +

5 2

σ µρ /c H νρ +

1 2

H µρ σ νρ /c − I µν

+ 3 H (µρ ν)ρστ ωσ /c uτ /c − µρστ E νρ u˙ σ /c2 uτ /c ] − 2 u˙ µ /c2 σ νρ /c H ρν − µνρσ E τν (∇τ u˙ ρ /c2 ) uσ /c + E µν νρστ (∇ρ u˙ σ /c2 ) uτ /c .

(2.56)

Here, in deriving Eqs. (2.54) - (2.56) the identity (B.25) of Appendix B was employed. The spatial divergence terms of Jµν and Iµν , that occur in Eqs. (2.55) and (2.56), respectively, are given in the following, and for sake of completeness also the spatial divergence of Kµν is included. This derivation rests on Eq. (B.15) in Appendix B, the constraints (2.35), (2.36) and (2.28), and the evolution equations (2.31), (2.32) and (2.27). It leads to: hµρ hνσ (∇ν I ρσ )

= − 21 SH µ + hµρ H νσ (∇ρ σ σν /c) + H µρ hνσ (∇ν σ ρσ /c) −

3 4

σ µρ /c hνσ (∇ν H ρσ )

3 µνρσ 2

(∇ν H τρ ) uσ /c ωτ /c

−

3 4

+

3 µνρσ 2

H τν (∇ρ ωτ /c) uσ /c + 2 ων /c J µν

−

1 µνρσ 2

[ σντ /c J τρ − Hντ K τρ ] uσ /c

+

1 3

(Θ/c) µνρσ σντ /c E τρ uσ /c − µνρσ σνκ /c Eρλ uσ /c σ κλ /c

− (Θ/c) ων /c E µν + 3 σ µν /c E νρ ω ρ /c + 6 E µν σ νρ /c ω ρ /c − 2 ω µ /c σ νρ /c E ρν + 2 µνρσ Eντ ωρ /c uσ /c ω τ /c −

2 4πG 3 c4

µ ν (µ + p) (Θ/c) ω µ /c − 2 4πG c4 (µ + p) σ ν /c ω /c


24

+ 2 µνρσ Hντ u˙ ρ /c2 uσ /c ω τ /c + 2 H µν νρστ u˙ ρ /c2 ωσ /c uτ /c hµρ hνσ (∇ν J ρσ )

=

1 2

SE µ − hµρ E νσ (∇ρ σ σν /c) − E µρ hνσ (∇ν σ ρσ /c) +

3 4

(2.57)

σ µρ /c hνσ (∇ν E ρσ )

3 µνρσ 2

(∇ν E τρ ) uσ /c ωτ /c

+

3 4

−

3 µνρσ 2

+

4πG c4

(µ + p) µνρσ (∇ν ωρ /c) uσ /c

+

4πG c4

(1 +

+

1 µνρσ 2

+

1 3

E τν (∇ρ ωτ /c) uσ /c − 2 ων /c I µν ∂p µνρσ ∂µ )

Xν ωρ /c uσ /c

[ σντ /c I τρ − 2 Eντ H τρ − Eντ K τρ ] uσ /c

(Θ/c) µνρσ σντ /c H τρ uσ /c − µνρσ σνκ /c Hρλ uσ /c σ κλ /c

− (Θ/c) ων /c H µν + 3 σ µν /c H νρ ω ρ /c + 6 H µν σ νρ /c ω ρ /c − 2 ω µ /c σ νρ /c H ρν + 2 µνρσ Hντ ωρ /c uσ /c ω τ /c − 2 µνρσ Eντ u˙ ρ /c2 uσ /c ω τ /c − 2 E µν νρστ u˙ ρ /c2 ωσ /c uτ /c hµρ hνσ (∇ν K ρσ )

= − µνρσ Eντ σ τρ /c uσ /c +

1 2

(2.58)

hµρ hνσ (∇ν ∇σ ω ρ /c)

−

1 2

hµρ ∇ρ [ ων /c u˙ ν /c2 ] + hµρ u˙ ν /c2 (∇ν ω ρ /c)

+

1 2

hµρ ων /c (∇ρ u˙ ν /c2 ) +

+ ω µ /c hνρ ∇ν u˙ ρ /c2 −

1 4

1 2

hµρ ων /c (∇ν u˙ ρ /c2 )

(Θ/c) µνρσ (∇ν u˙ ρ /c2 ) uσ /c

2

−

1 2

(Θ/c) ω µ /c + u˙ µ /c2 ων /c u˙ ν /c2

−

5 2

ων /c E µν −

2 4πG 3 c4

(2 µ + 3 p) ω µ /c .

(2.59)

Here, identity (B.25) of Appendix B was employed again. Additionally, Eq. (B.2) was used in Eqs. (2.57) and (2.59). Note that if the preferred timelike reference congruence u/c is geodesic and irrotational, 0 = (u/c ˙ 2 ) = (ω/c), then in Eq. (2.59) only the first term on the righthand side is non-zero. Thus, combined with Eq. (2.30), it reduces to a special subcase of Eq. (2.36). A symmetrised, fully orthogonally projected covariant derivative of the fluid vorticity occurs in the H-constraint (2.30). Its propagation along the integral curves of u/c can be derived from Eq. (B.4) in Appendix B, in conjunction with the vorticity evolution equation (2.26). This gives: h(µρ hν)σ

hρκ hσλ (∇κ ω λ /c) ˙/c = − (Θ/c) h(µρ hν)σ (∇ρ ω σ /c) − 2 σ (µ[ρ /c hν)σ] (∇ρ ω σ /c) − h(µκ ν)ρστ ωρ /c (∇σ ω κ /c) uτ /c + h(µρ hν)σ (∇ρ σ στ /c) ω τ /c −

2 3

ω (µ /c Z ν)

−

1 6

(Θ/c) σ (µρ /c ν)ρστ u˙ σ /c2 uτ /c

−

7 6

(Θ/c) ω (µ /c u˙ ν) /c2 +

−

1 2

σ (µρ /c ν)ρστ σσκ /c uτ /c u˙ κ /c2

1 2

hµν (Θ/c) ωρ /c u˙ ρ /c2

+ σ µν /c ωρ /c u˙ ρ /c2 − ω (µ /c σ ν)ρ /c u˙ ρ /c2 +

1 2

u˙ (µ /c2 σ ν)ρ /c ω ρ /c +

1 2

hµν σ ρσ /c ωρ /c u˙ σ /c2

25

2.6. DERIVATION OF NEW EVOLUTION EQUATIONS +

1 2

ω (µ /c ν)ρστ u˙ ρ /c2 ωσ /c uτ /c

− H (µρ ν)ρστ ωσ /c uτ /c +

1 2

E (µρ ν)ρστ u˙ σ /c2 uτ /c

−

1 2

h(µκ ν)ρστ (∇ρ ∇κ u˙ σ /c2 ) uτ /c

−

1 2

u˙ (µ /c2 ν)ρστ (∇ρ u˙ σ /c2 ) uτ /c

−

1 2

σ (µρ /c ν)ρστ uτ /c (u˙ σ /c2 )˙/c

−

1 2

ω (µ /c hν)ρ (u˙ ρ /c2 )˙/c

+

1 2

hµν ωρ /c (u˙ ρ /c2 )˙/c .

(2.60)

Finally, there are spatial rotation terms of the fluid vorticity and the fluid rate of shear, respectively, in the divergence equation for the fluid rate of shear, Eq. (2.28), and the H-constraint, Eq. (2.30), while spatial rotation terms of the “electric” and “magnetic” parts of the Weyl curvature alternatingly occur as source terms in the evolution equations of one another. Propagation equations along u/c for these geometrical quantities, which are new, are obtained from Eqs. (B.5) and (B.14) in Appendix B, by using the evolution equations (2.26), (2.27), (2.31) and (2.32). These are: hµν [ − νρστ (∇ρ ωσ /c) uτ /c ]˙/c =

(Θ/c) µνρσ (∇ν ωρ /c) uσ /c + µνρσ σ τν /c (∇ρ ωτ /c) uσ /c Wµ

+ µνρσ σ τν /c (∇τ ωρ /c) uσ /c +

1 2

− µνρσ (∇ν σ τρ /c) uσ /c ωτ /c +

2 µνρσ 3

+ ων /c H

µν

+ u˙ ν /c2 [

1 6

(Θ/c) σ µν /c +

1 2

Zν ωρ /c uσ /c

(Θ/c) µνρσ ωρ /c uσ /c

+ σ µρ /c σ νρ /c + σ µρ /c νρστ ωσ /c uτ /c − 2 µνρσ σρτ /c uσ /c ω τ /c + ω µ /c ω ν /c + − u˙ µ /c2 [

1 18

hνσ

1 2

E µν ] 2

(Θ/c) −

−

1 2

hµρ

+

1 4

Aµ −

−

1 2

(Θ/c) hµν (u˙ ν /c2 )˙/c

1 2

σ ρ

2 4πG 3 c4 2

(∇ν ∇ u˙ /c ) +

µ] 1 2

hµρ ∇ρ [ hνσ ∇ν u˙ σ /c2 ]

hµρ u˙ ν /c2 (∇ν u˙ ρ /c2 )

− µνρσ ων /c uσ /c (u˙ ρ /c2 )˙/c hµρ hνσ (K ρσ )˙/c = − (Θ/c) K µν − I µν + 3 H (µρ σ ν)ρ /c − H ρσ σ σρ /c hµν + h(µρ hν)σ (∇ρ σ στ /c) ω τ /c − 2 σ (µ[ρ /c hν)σ] (∇ρ ω σ /c) − h(µκ ν)ρστ ωρ /c (∇σ ω κ /c) uτ /c −

2 3

ω (µ /c Z ν)

−

1 6

(Θ/c) σ (µρ /c ν)ρστ u˙ σ /c2 uτ /c

−

1 2

(Θ/c) ω (µ /c u˙ ν) /c2 +

−

1 2

σ (µρ /c ν)ρστ σσκ /c uτ /c u˙ κ /c2 + σ µν /c ωρ /c u˙ ρ /c2

− ω (µ /c σ ν)ρ /c u˙ ρ /c2 + −

1 2

5 2

1 6

hµν (Θ/c) ωρ /c u˙ ρ /c2

u˙ (µ /c2 σ ν)ρ /c ω ρ /c

hµν σ ρσ /c ωρ /c u˙ σ /c2 +

1 2

ω (µ /c ν)ρστ u˙ ρ /c2 ωσ /c uτ /c

(2.61)


26

−

3 2

E (µρ ν)ρστ u˙ σ /c2 uτ /c

−

1 2

h(µκ ν)ρστ (∇ρ ∇κ u˙ σ /c2 ) uτ /c

−

3 2

u˙ (µ /c2 ν)ρστ (∇ρ u˙ σ /c2 ) uτ /c

+

1 2

hµν u˙ ρ /c2 ρσκλ (∇σ u˙ κ /c2 ) uλ /c

−

1 2

σ (µρ /c ν)ρστ uτ /c (u˙ σ /c2 )˙/c

+

3 2

ω (µ /c hν)ρ (u˙ ρ /c2 )˙/c −

hµρ hνσ (I ρσ )˙/c = − 43 (Θ/c) I µν −

4πG c4

1 2

hµν ωρ /c (u˙ ρ /c2 )˙/c

(2.62)

(µ + p) K µν + 3 H (µρ E ν)ρ − H ρσ E σρ hµν

+ h(µρ hν)σ ρτ κλ σ ξτ /c (∇ξ Eκσ ) uλ /c − 3 h(µρ hν)σ ρτ κλ ∇τ [ σ ξ(κ /c Eσ)ξ ] uλ /c − ω (µ /c hν)ρ hτσ (∇τ E ρσ ) + h(µρ hν)σ (∇ρ E στ ) ω τ /c −[

4πG c4

(1 +

∂p (µ ∂µ ) σ ρ /c Xσ

+ E (µρ Zσ +

4πG c4

(µ + p) σ (µρ /c u˙ σ /c2

+

2 3

(Θ/c) E (µρ u˙ σ /c2 −

−

5 2

E κ(µ σκρ /c u˙ σ /c2 + u˙ κ /c2 σ (µρ /c Eκσ

3 2

σ κ(µ /c Eκρ u˙ σ /c2

− u˙ (µ /c2 σρκ /c E |κ| σ ] ν)ρστ uτ /c + [ 2 E (µ[ρ hν)σ] + E µν hρσ ] ω ρ /c u˙ σ /c2 + 3 ω (µ /c u˙ ρ /c2 E ν)ρ − E ρσ ωρ /c u˙ σ /c2 hµν − [ 2 h(µ[ρ hν)[κ hλ] σ] − h(µ[ρ hν)λ hκσ] − hµν hκ[ρ hλσ] ] × [ ∇κ + u˙ κ /c2 ] [ ∇ρ H σλ + 2 u˙ ρ /c2 H σλ + ω ρ /c E σλ ]

(2.63)

hµρ hνσ (J ρσ )˙/c = − 43 (Θ/c) J µν + 3 H µρ H νρ − 2 H 2 hµν + h(µρ hν)σ ρτ κλ σ ξτ /c (∇ξ Hκσ ) uλ /c − 3 h(µρ hν)σ ρτ κλ ∇τ [ σ ξ(κ /c Hσ)ξ ] uλ /c − ω (µ /c hν)ρ hτσ (∇τ H ρσ ) + h(µρ hν)σ (∇ρ H στ ) ω τ /c − [ H (µρ Zσ +

2 3

−

(Θ/c) H (µρ u˙ σ /c2 − 5 2

3 2

σ κ(µ /c Hκρ u˙ σ /c2

H κ(µ σκρ /c u˙ σ /c2 + u˙ κ /c2 σ (µρ /c Hκσ

− u˙ (µ /c2 σρκ /c H |κ| σ ] ν)ρστ uτ /c + [ 2 H (µ[ρ hν)σ] + H µν hρσ ] ω ρ /c u˙ σ /c2 + 3 ω (µ /c u˙ ρ /c2 H ν)ρ − H ρσ ωρ /c u˙ σ /c2 hµν + [ 2 h(µ[ρ hν)[κ hλ] σ] − h(µ[ρ hν)λ hκσ] − hµν hκ[ρ hλσ] ] × [ ∇κ + u˙ κ /c2 ] [ ∇ρ E σλ + 2 u˙ ρ /c2 E σλ − ω ρ /c H σλ ] .

(2.64)

Here, in deriving Eq. (2.62), the relation7 − h(µρ hν)σ ρτ κλ ∇τ [ σκξ /c σ σξ /c ] uλ /c = − h(µρ hν)σ ρτ κλ hξη ∇ξ [ σ στ /c σ ηκ /c ] uλ /c (2.65) 7 Thanks

are due to Roy Maartens for pointing out this identity to me. See Maartens (1996) [101].

27

2.7. PRESERVATION OF CONSTRAINTS

was used, as well as the identity (B.22) in Appendix B. The tracefree property of Eq. (2.62) follows by use of Eq. (2.28). Note that the evolution equation for the spatial rotation of the “electric” part of the Weyl curvature tensor, Eq. (2.63), contains a double-spatial rotation of the “magnetic” part, while the same situation applies in an opposite constellation in Eq. (2.64).

2.7

Preservation of constraints for barotropic perfect fluid spacetime geometries

With all required evolution equations for the spatial derivative terms at one’s disposal, one can now explicitly address the question of integrability of the 1+3 covariant dynamical equations for (comoving) barotropic perfect fluid spacetime geometries, i.e., the question of preservation of the spatial constraints along the integral curves of the preferred timelike reference congruence u/c. Recall that the non-zero geometrical dynamical quantities are the fluid matter variables µ, p, (Θ/c), σµν /c, ω µ /c and u˙ µ /c2 , and the Weyl curvature variables Eµν and Hµν , and µ and p are assumed to be related by an equation of state of the form p = p(µ). As mentioned before, the spatial constraints that need to be examined in the barotropic perfect fluid case are the four equations containing spatial divergence terms of each of the fluid rate of shear, Eq. (2.28), the fluid vorticity, Eq. (2.29), and the “electric” and “magnetic” parts of the Weyl curvature, given by Eqs. (2.35) and (2.36). Furthermore, one needs to take into account the H-constraint (2.30), and finally, as a perfect fluid has 0 = q µ /c = πµν , the new constraint Eq. (2.43), which the evolution equation (2.33) converts into. To this end, it proves to be convenient to introduce specific symbols denoting the righthand sides of each of these six spatial constraints.8 It is suggested to use, respectively, (C1 )µ

(C2 )

(C3 )µν

:= hµρ hνσ (∇ν σ ρσ /c) −

Z µ − µνρσ [ ∇ν ωρ /c + 2 u˙ ν /c2 ωρ /c ] uσ /c

:= hµν ∇µ ω ν /c − u˙ µ /c2 ω µ /c

(2.67)

1 3

[ 2 u˙ ρ /c2 ω ρ /c + hρσ ∇ρ ω σ /c ] hµν

:= hµρ hνσ (∇ν E ρσ ) −

2 4πG 3 c4

(2.69)

µ := hµρ hνσ (∇ν H ρσ ) + 2 4πG c4 (µ + p) ω /c

+ 3 ων /c E µν − µνρσ σντ /c E τρ uσ /c

(C6 )µ

(2.68)

Xµ

− 3 ων /c H µν + µνρσ σντ /c H τρ uσ /c

(C5 )µ

(2.66)

:= H µν + 2 u˙ (µ /c2 ω ν) /c + h(µρ hν)σ (∇ρ ω σ /c) − K µν −

(C4 )µ

2 3

:= hµν ∇ν p + (µ + p) u˙ µ /c2 =

∂p ∂µ

X µ + (µ + p) u˙ µ /c2 .

(2.70)

(2.71)

8 An unknown referee has to be acknowledged for suggesting the (C )-format of the constraints and their propagations along i u/c ( see Maartens (1996) [101] ).

28


Now one assumes that at an initial instant along every individual flow line of the invariantly defined preferred timelike reference congruence u/c all of the (Ci ) vanish, i.e., 0 = (C1 )µ = (C2 ) = (C3 )µν

0 = (C4 )µ = (C5 )µ =

4πG c4

(C6 )µ .

(2.72)

Then, propagating the set of equations (2.66) - (2.70) along the integral curves of u/c, and using Eqs. (2.46), (2.47), (2.53) - (2.60) and (2.62) of Section 2.6 above, one obtains the set hµν [ (C1 )ν ]˙/c = − (Θ/c) (C1 )µ − σ µν /c (C1 )ν −

5 3

ω µ /c (C2 )

+ 2 µνρσ σντ /c (C3 )τρ uσ /c − 2 ων /c (C3 )µν − (C4 )µ

[ (C2 ) ]˙/c =

0

(2.73)

(2.74)

hµρ hνσ [ (C3 )ρσ ]˙/c = − (Θ/c) (C3 )µν − h(µρ hν)σ ρτ κλ σ στ /c (C1 )κ uλ /c hµν [ (C4 )ν ]˙/c = − 34 (Θ/c) (C4 )µ +

1 2

σ µν /c (C4 )ν +

hµν [ (C5 )ν ]˙/c = − 34 (Θ/c) (C5 )µ +

1 2

σ µν /c (C5 )ν +

(2.75)

1 µνρσ ων /c (C4 )ρ uσ /c 2 µ µ ν µνρσ 3 − 4πG Eντ (C3 )τρ uσ /c c4 (µ + p) (C1 ) + 2 E ν (C1 ) − µ ν − 32 µνρσ u˙ ν /c2 (C5 )ρ uσ /c − 4πG (2.76) c4 σ ν /c (C6 )

H µν (C1 )ν −

1 µνρσ ων /c (C5 )ρ 2 µνρσ Hντ (C3 )τρ uσ /c

+

3 2

+

3 µνρσ u˙ ν /c2 (C4 )ρ uσ /c 2 ∂p −1 4πG µνρσ ( ∂µ ) u˙ ν /c2 (C6 )ρ c4

+

uσ /c .

uσ /c

(2.77)

Here, in order to eliminate terms on the righthand sides of the set (2.73) - (2.77), identities listed in Appendix B have been applied in the derivation. Thus: (B.20) in (2.73); (B.20), (B.21), (B.23) and (B.24) in (2.76); and (B.2), (B.20), (B.21), (B.23) and (B.24) in (2.77). Finally, demanding that hµν [ (C6 )ν ]˙/c = 0 ,

(2.78)

leads to an evolution equation for the fluid acceleration vector field, ∂p ∂µ

∂p (Θ/c) ] + ( ∂µ − 13 ) (Θ/c) u˙ µ /c2 − ( σ µν /c − µνρσ ωρ /c uσ /c ) u˙ ν /c2 , (2.79) which was given before, e.g., in Ellis and Bruni (1989) [50].

hµν u˙ ν /c2 ˙/c = hµν ∇ν [

Hence, it follows from Eqs. (2.73) - (2.78) that, given all of the conditions (2.72) are satisfied at an initial instant (in the local rest 3-spaces) along any individual flow line of the invariantly defined preferred timelike reference congruence u/c and if Eq. (2.79) holds, then they are preserved throughout the entire evolution of a relativistic cosmological model based on a barotropic perfect fluid matter source, as long as no spacetime singularities arise as a consequence of the fluid-spacetime dynamics. At this point it should be pointed out though, that the set of spatial constraints given by Eqs. (2.66) - (2.71) is highly likely to be overdetermined, i.e., in terms of numbers it contains more restrictive conditions than one would expect to arise by imposing the barotropic perfect fluid condition on the specific

2.8. CONCLUSION

29

form of the stress-energy-momentum tensor of the matter sources. This is so, as the twice-contracted second Bianchi identities ensure that the EFE constitute a consistent set of dynamical relations for a general (un-restricted) matter source ( see, e.g., Wald (1984) [161] ), and the assumed barotropic perfect fluid form does not contain as many assumptions as (apparent) constraints follow from Eqs. (2.66) - (2.71). In this investigation, the target to establish a necessary and sufficient set of spatial constraints restricting spacetime geometries with a barotropic perfect fluid matter source in 1 + 3 covariant formulation was not pursued, and so this specific question continues to remain to be solved. Leaving the problem just touched upon aside, note that all the conclusions drawn from the analysis of the conditions (2.66) - (2.71) given above apply in particular in the following special cases of • a rotating dust matter source, p = 0 ⇒ (u/c ˙ 2) = 0 , • an irrotational dust matter source, (ω/c) = 0, and p = 0 ⇒ (u/c ˙ 2 ) = 0, as recently demonstrated by Maartens (1996) [101]; additionally he showed that, given these assumptions and, furthermore, assuming vanishing spatial divergence of the “magnetic” part of the Weyl curvature, the spatial constraints are preserved as well , • orthogonally spatially homogeneous (OSH) perfect fluids ( see Ellis and MacCallum (1969) [49] ); all spatial 3-gradients vanish, Fµ = 0, and consequently, by Eqs. (2.43) and (B.2), 0 = (u/c ˙ 2 ) = (ω/c); this case is considered in more detail in Section 3.2 of Chapter 3 below , • an irrotational dust matter source under the further assumption that the resulting spacetime geometry has zero “magnetic” part of the Weyl curvature, i.e., (ω/c) = 0, p = 0 ⇒ (u/c ˙ 2 ) = 0, and H = 0. These configurations have been dubbed “silent” models of the Universe by Matarrese et al (1994/95) [111, 21]. However, the “silent” assumption generates a new spatial constraint, the preservation along u/c of which will be investigated in Section 5.1 of Chapter 5 below . It should be mentioned that only a few exact solutions to the EFE are known even in these simpler subcases, most of them for the OSH perfect fluid family.

2.8

Conclusion

In this chapter a detailed review was given of the 1 + 3 covariant approach to GR, resting on the existence of an invariantly defined preferred timelike reference congruence u/c on the spacetime manifold ( M, g ). This review contained the introduction of some useful notation. Then new evolution equations for the spatial derivative terms of geometrical quantities, defined in the local rest 3-spaces orthogonal to u/c, were derived for the case of barotropic perfect fluid spacetime geometries. It was subsequently demonstrated that for these spacetime geometries the spatial constraints are preserved along the integral curves of u/c. The next chapter contains a presentation of specific examples of 1 + 3 covariant formulations of well-known barotropic perfect fluid spacetime geometries.

30


Chapter 3

Special Perfect Fluid Spacetime Geometries. I. 3.1

Locally rotationally symmetric spacetime geometries

The covariant 1 + 3 decomposition of ( M, g ), together with its associated geometrical fields and dynamical equations, as discussed in Chapter 2, rested entirely on the assumption of the existence of an invariantly defined (normalised) preferred timelike reference congruence, u/c. In addition to that assumption, one can consider the situation where ( M, g, u/c ) allows for a second invariantly defined (normalised) preferred reference congruence, e, which is spacelike and orthogonal to u/c, i.e., eµ eµ = 1 ⇒ eν (∇µ eν ) = 0 and uµ /c eµ = 0. Then, analogous to Eq. (2.11) for the covariant derivative of u/c, one can decompose the covariant derivative of e into its irreducible parts. Using Eqs. (2.7) - (2.10) of Chapter 2, this yields ∇µ eν

= − uµ /c [ uν /c u˙ ρ /c2 eρ + pρν (eρ )˙/c ] + [ σµρ /c eρ +

1 3

(Θ/c) eµ − sµρ ω ρ /c ] uν /c

+ eµ pρν (eρ )0 + dµν +

1 2

a pµν +

1 2

k sµν .

(3.1)

Here, the following shorthand notation is introduced: (i) a tensor p projecting orthogonal both to u/c and e, defined by pµν := hµν − eµ eν , with the properties pµρ pρν = pµν , 0 = pµν uν /c = pµν eν , pµµ = 2, and (ii) a 2-surface element s, again orthogonal both to u/c and e, defined by sµν := − µνρσ eρ uσ /c = s[µν] , and satisfying 0 = sµν uν /c = sµν eν . Also, analogous to Eqs. (2.7) (2.10), the irreducible geometrical quantities pµν (eν )0 dµν

:= pµν eρ (∇ρ eν ) :=

[

pρ(µ

pσν)

−

1 2

(3.2) pµν p

ρσ

] (∇ρ eσ ) = d(µν)

a := pµν ∇µ eν k eµ

(3.3) (3.4)

:= − µνρσ (∇ν eρ ) uσ /c + sµν (eν )0 ,

(3.5)

associated with the preferred spacelike reference congruence e may be defined, where pµν (eν )0 is the non-geodesity (“acceleration”) of e, dµν its symmetric tracefree intrinsic distortion (“shear”), a the magnitude of its spatial divergence (“expansion”), and k the magnitude − eµ µνρσ (∇ν eρ ) uσ /c of the 31

32

CHAPTER 3. SPECIAL PERFECT FLUID SPACETIME GEOMETRIES. I.

projection of the spatial rotation (“twist”) of e along e itself. Local rotational symmetry (LRS) is imposed by demanding that the Riemann curvature tensor of the spacetime manifold ( M, g, u/c ), and all its covariant derivatives, be locally invariant under spatial rotations about the direction defined by the preferred spacelike reference congruence e. In particular, this requires that the preferred timelike reference congruence u/c, the fluid matter source variables, and all their covariant derivatives be left invariant under the action of these local spatial rotations ( see Ellis (1967) [44] ). In this case ( M, g, u/c ), then extended to a setting ( M, g, u/c, e ), gives rise to a covariant 1 + 1 + 2 decomposition — a subcase of the covariant 1 + 3 splitting — of all its geometrical fields and dynamical equations. This goal is pursued in the following. In differential geometrical language the existence of LRS implies that locally there exists either a 1-D or 3-D continuous isotropy group and, consequently, a multiply-transitive isometry group acting on ( M, g, u/c, e ). This isometry group is at least a G3 acting on spacelike 2-surfaces orthogonal to u/c and e, and it turns out that in the non-vacuum LRS case of highest symmetry the isometry group is a G7 , multiply-transitive on ( M, g, u/c, e ) ( see Kramer et al (1980) [88] ). In order for the spatial direction given by e to constitute a local axis of symmetry (and thus the spacetime to be LRS) there are no physical features allowed that could single out any other preferred spatial direction; two out of three local spatial directions have to be physically equivalent. Hence, the tangents to the preferred spacelike reference congruence e need to be Fermi-transported along the integral curves of the preferred timelike congruence u/c, and e has to be geodesic and shearfree; in mathematical terms pµν (eν )˙/c = pµν (eν )0 dµν

0

(3.6)

= 0

(3.7)

= 0.

(3.8)

From an observational point of view, in conjunction with the dynamical equations of GR, a LRS symmetry of ( M, g, u/c ) would manifest itself (and thus be invariantly defined) as, e.g., a degeneracy in the eigenvalues of the rate of shear tensor field and the “electric” part of the Weyl curvature tensor field, or a non-zero vorticity vector field, or else a non-vanishing acceleration vector field, with the latter two having fixed spatial orientation. A unique rank 2 symmetric tracefree spacelike tensor field eµν orthogonal to u/c is defined from e by eµν :=

1 2

( 3 eµ eν − hµν ) = hµν −

3 2

pµν = e(µν) ,

(3.9)

which has the properties eµµ = 0, eµρ eρν = eµν + 43 pµν , eµν eνµ = 32 , eµν eν = eµ and eµν uν /c = 0. For its propagation along the integral curves of u/c, its spatial divergence and its spatial rotation one obtains, respectively, hµρ hνσ (eρσ )˙/c = 0 hµρ

hνσ

ρσ

(3.10)

(∇ν e ) =

3 2

µ

ae

(3.11)

− h(µρ hν)σ ρτ κλ (∇τ eσκ ) uλ /c =

3 2

k eµν .

(3.12)

33

3.1. LOCALLY ROTATIONALLY SYMMETRIC SPACETIME GEOMETRIES

In presence of LRS, any spacelike vector field V orthogonal to u/c, any spatial 3-gradient of a 3-scalar f , and any rank 2 symmetric tracefree spacelike tensor field A orthogonal to u/c, which can be 1 + 3 ( 1 + 1 + 2 ) invariantly defined on ( M, g, u/c, e ), takes one of the forms Fµ = f 0 eµ

V µ = V eµ 1 2

where V 2 := Vµ V µ ≥ 0, A2 :=

Aµν =

√2 3

A eµν ,

(3.13)

Aµν Aνµ ≥ 0, and f 0 := eµ ∇µ f . Note that the magnitudes V

and A themselves can take either positive or negative values. For a non-zero Weyl curvature tensor, Eq. (3.13) implies that all LRS solutions are of algebraic Petrov type D ( see Wainwright (1970) [157] and Kramer et al (1980) [88] ). Furthermore, it automatically follows from Eq. (3.13) that 0 = − µνρσ Vν Wρ uσ /c

0 = − µνρσ Aντ B τρ uσ /c ,

(3.14)

where W and B denote further arbitrary spacelike vector and rank 2 symmetric tracefree tensor fields orthogonal to u/c. Under the LRS assumption, Eq. (2.11) of Chapter 2 specialises to ∇µ uν /c = − (u/c ˙ 2 ) uµ /c eν +

√2 3

(σ/c) eµν +

1 3

(Θ/c) hµν + (ω/c) sµν ,

(3.15)

while Eq. (3.1) reduces to ∇µ eν = − (u/c ˙ 2 ) uµ /c uν /c +

1 3

√ [ (Θ/c) + 2 3 (σ/c) ] eµ uν /c +

1 2

a pµν +

1 2

k sµν .

(3.16)

Fully orthogonally projected propagation terms along u/c as well as spatial divergence and spatial rotation terms of any V are given by hµν (V ν )˙/c = V˙ /c eµ hµν ∇µ V ν µνρσ

−

(3.17)

= V 0 + aV

(3.18)

µ

(∇ν Vρ ) uσ /c = k V e ,

(3.19)

˙ √2 eµν hµρ hνσ (Aµν )˙/c = A/c 3

(3.20)

and those of any A by

hµρ hνσ (∇ν Aρσ ) −

h(µρ

hν)σ ρτ κλ

(∇τ Aσκ ) uλ /c

= [ A0 + =

[

3 2

3 2

kA]

a A ] √23 eµ

(3.21)

µν

(3.22)

√2 3

e

.

In the following discussion the matter content of a spacetime geometry with LRS symmetry is assumed to be a comoving barotropic perfect fluid, and the cosmological constant is set to zero, i.e., 0 = q µ /c = πµν , p = p(µ), and Λ = 0. The LRS perfect fluid models have been rigorously investigated from the isometry point of view, in terms of a connection component oriented 1 + 3 ONF treatment, by Stewart and Ellis in 1968 [147]. In the exposition presented here the emphasis will be on their 1 + 3 ( 1 + 1 + 2 ) covariant properties, which, so far, have not been systematically examined. It is clear from the preliminary considerations pursued up to now, that, with LRS spacetime symmetry imposed, the 1 + 3 covariant dynamical equations (2.25) - (2.36) of Chapter 2 reduce to a set of differential relations between 1 + 3 ( 1 + 1 + 2 ) invariantly defined 3-scalar fields f , since all directional features have been factored out. In the barotropic perfect fluid case the non-zero dynamical 3-scalar fields will be the fluid matter variables µ, p, (Θ/c), (σ/c), (ω/c) and (u/c ˙ 2 ), the spatial derivative magnitudes a and k, and

34


the Weyl curvature variables E and H. From the point of view of the Cartan–Karlhede equivalence problem formalism for invariantly classifying different spacetime geometries ( see Cartan (1946) [27] and Karlhede (1980) [84] ), the existence of spacelike rotational isometries in the 1 + 3 ( 1 + 1 + 2 ) covariant approach to LRS spacetime geometries is condensed into the existence of the invariantly defined preferred spacelike reference congruence e. To cover translational isometries, one can specify a maximal set of four generalised essential coordinates, which can then be investigated for the existence of functional dependencies ( see Bradley and Karlhede (1990) [16] ). For (non-vacuum) models with non-vanishing rate of expansion, it is proposed to choose this set to be constituted by the four 1 + 3 invariantly defined 3-scalar fields S4 := { µ, (Θ/c), (σ/c), E }. It remains to be determined if this is a good choice or not. LRS perfect fluid spacetime geometries have found frequent application in the literature in, e.g., the modelling of the dynamical processes surrounding the formation of relativistic stars or galaxies ( see, e.g., Refs. [99], [144], [114] and [138] ). Irrespective of the large degree of idealisation of the physics underlying their evolution, they have proved to be a valuable test ground for more complex astrophysical scenarios. Finally, using Eqs. (3.13), (3.14) and (3.17) - (3.22) in simplifying the set (2.25) - (2.36), the resulting reduced set of 1 + 3 ( 1 + 1 + 2 ) covariant dynamical equations for barotropic perfect fluid models with LRS spacetime symmetry imposed is constituted by:

3.1.1

The Ricci identities

Time derivative equations 2

(Θ/c)˙/c = − 31 (Θ/c) + u/c ˙ 2 −

4πG c4

(σ/c)˙/c = − 23 (Θ/c) (σ/c) + −

2 2 2 + a u/c ˙ 2 + u/c ˙ 2 − 2 (σ/c) + 2 (ω/c)

(µ + 3p)

(ω/c)˙/c = − 32 (Θ/c) (ω/c) + √1 3

0

2

(σ/c) −

(3.23) √2 3 √1 3 √1 3

2

(σ/c) (ω/c) + 12 k u/c ˙ 0 1 u/c ˙ 2 − 2√ a u/c ˙ 2 + 3

(3.24) √1 3

u/c ˙

2

2 2

(ω/c) − E .

(3.25)


0

√

(Θ/c) + 23 k (ω/c) 0 0 = (ω/c) + a (ω/c) − u/c ˙ 2 (ω/c) √ √ 0 = H + 3 u/c ˙ 2 (ω/c) − 23 a (ω/c) − 32 k (σ/c) .

0

3.1.2

=

(σ/c) +

3 2

a (σ/c) −

√1 3

(3.26) (3.27) (3.28)

The second Bianchi identities

Time derivative equations ˙ E/c = − 4πG c4 (µ + p) (σ/c) − (Θ/c) E + √ ˙ H/c = − (Θ/c) H + 3 (σ/c) H − 23 k E µ/c ˙ = − (µ + p) (Θ/c) .

√

3 (σ/c) E +

3 2

kH

(3.29) (3.30) (3.31)

35

3.1. LOCALLY ROTATIONALLY SYMMETRIC SPACETIME GEOMETRIES Constraint equations

3.1.3

4πG 0 √ µ − 3 (ω/c) H 3c4 √ 0 3 4πG H + 2 a H + 3 c4 (µ + p) (ω/c) + 3 (ω/c) E ∂p 0 µ + (µ + p) u/c ˙ 2 . p0 + (µ + p) u/c ˙ 2 = ∂µ

0

= E0 +

0

=

0

=

3 2

aE −

(3.32) (3.33) (3.34)

Vanishing vorticity

With LRS spacetime symmetry imposed, Eqs. (2.37) and (2.38) of Chapter 2 expressing the intrinsic 3-Ricci curvature of spacelike 3-surfaces orthogonal to an irrotational timelike reference congruence u/c, (ω/c) = 0, become 3

Sµν

=

3

=

R

2 √2 [ E − 1 (Θ/c) (σ/c) + √1 (σ/c) 3 3 3 2 2 2 4 4πG c4 µ − 3 (Θ/c) + 2 (σ/c) .

] eµν

(3.35) (3.36)

The specific form the 3-Cotton–York tensor can take under LRS spacetime symmetry will be discussed in the subsequent subsections.

3.1.4

The equations for e

Using the Ricci identities for e in the contracted form 0 = eν ∇µ (∇ν eµ ) − (∇µ eµ )0 − Rµν eµ eν , leads to 0 = a0 +

1 2

a2 −

2 9

2

(Θ/c) −

2 √ 3 3

(Θ/c) (σ/c) +

4 3

(σ/c)2 −

1 2

k2 +

√2 3

E+

4 4πG 3 c4

µ.

(3.37)

An evolution equation for a along u/c can be derived from Eq. (B.4) in Appendix B, setting V µ = eµ . This yields a/c ˙ = − 13 [ (Θ/c) −

√

3 (σ/c) ] [ a − 2 (u/c ˙ 2 ) ] + k (ω/c) .

(3.38)

By an analogous substitution one obtains from Eq. (B.6) that 0 = k0 + a k −

2 3

√ [ (Θ/c) + 2 3 (σ/c) ] (ω/c) ,

(3.39)

and from Eq. (B.5), using Eq. (3.28), the evolution equation for k along u/c becomes √ ˙ = − 1 [ (Θ/c) − 4 3 (σ/c) ] k . k/c 3

3.1.5

(3.40)

The consistency equations and their different cases

The LRS spacetime symmetry assumption, as reflected in Eqs. (3.13), (3.14) and (3.17) - (3.22), imposes strong restrictions on the form of the general barotropic perfect fluid 1 + 3 covariant dynamical equations, the integrability of which was demonstrated in Section 2.7 of Chapter 2. Therefore, one cannot expect that the associated LRS-reduced set (3.23) - (3.34), combined with Eqs. (3.37) - (3.40), will automatically be consistent. The necessary consistency analysis will be addressed in this subsection. First, by the LRS symmetry, the spacetime gradient of any 1 + 3 ( 1 + 1 + 2 ) covariant 3-scalar is given by ∇µ f = −f˙/c uµ /c + f 0 eµ .

(3.41)

36


Taking a second covariant derivative then leads to ∇ν ∇µ f = − (∇ν f˙/c) uµ /c − f˙/c (∇ν uµ /c) + (∇ν f 0 ) eµ + f 0 (∇ν eµ ) ,

(3.42)

which, on contraction with − νµρσ eρ uσ /c = sνµ , yields the important result 0 = 2 (ω/c) f˙/c − k f 0 .

(3.43)

Applying this first to (ω/c) and using Eqs. (3.24) and (3.27), and then to k and using Eqs. (3.40) and (3.39), one finds 0 = (ω/c) D = k D , where D :=

4 3

[ (Θ/c) −

√

(3.44)

3 (σ/c) ] (ω/c) − a k .

(3.45)

Now Eq. (3.44) implies D = 0, unless 0 = k = (ω/c), but then D is zero in any case, from its definition. Hence, one always has D=0

⇔

0 = ak −

4 3

[ (Θ/c) −

√

3 (σ/c) ] (ω/c) .

(3.46)

Also, applying Eq. (3.43) to p leads to ∂p 2 ∂µ (Θ/c) (ω/c) = k (u/c ˙ 2) .

(3.47)

To check the consistency of the LRS-reduced set (3.23) - (3.34), together with Eqs. (3.37) - (3.40), again the constraints (3.26), (3.27), (3.28), (3.32) and (3.33), and additionally Eqs. (3.37) and (3.39), are propagated along the integral curves of the preferred timelike reference congruence u/c, as was outlined in detail in Section 2.7 of Chapter 2. Within this procedure frequent use is made of the relation (f 0 )˙/c = (f˙/c)0 + (u/c ˙ 2 ) f˙/c −

1 3

√ [ (Θ/c) + 2 3 (σ/c) ] f 0 ,

(3.48)

to commute fully orthogonally projected covariant spatial and temporal derivatives acting on a 1 + 3 ( 1 + 1 + 2 ) covariant 3-scalar f . This is just the LRS-reduced version of Eq. (B.1) in Appendix B. It is a further spin-off of the LRS-reduction of the 1 + 3 dynamical variables to covariant 3-scalars, that the Riemann curvature tensor of ( M, g, u/c, e ) does not contribute to the constraint analysis via the Ricci identities. Propagation along u/c of Eq. (3.32), which contains the magnitude of the spatial divergence of the “electric” part of the Weyl curvature tensor, after suitable re-substitutions yields the new condition 0

=

√

3 4πG c4 (µ + p) k (ω/c) ,

(3.49)

and since it is assumed that (µ + p) > 0, this implies 0 = k (ω/c) .

(3.50)

Now one can multiply Eq. (3.43) first by k and then by (ω/c) to attain 0 = (ω/c)2 f˙/c = k 2 f 0 ,

(3.51)


37

which is always valid. Similarly, from Eq. (3.47) one gets 0 = k 2 (u/c ˙ 2) =

∂p ∂µ

(Θ/c) (ω/c)2 .

(3.52)

Applying Eq. (3.51) first to (ω/c) and then to k one finds that 0 = (ω/c) (ω/c)˙/c and 0 = k k 0 , which implies that one always has 0 = (ω/c)˙/c = k 0 . (3.53) Putting the first parts of Eqs. (3.52) and Eq. (3.53) into the LRS-reduced vorticity evolution equation (3.24) and then using Eq. (3.46) leads to 0 = ak . (3.54) The invariantly defined preferred spacelike reference congruence e in LRS spacetime geometries is thus either (i) orthogonal to spacelike 2-surfaces (k = 0), and can be derived from a scalar potential, or (ii) divergence-free (a = 0), and can be derived from a vector potential, or (iii) covariantly constant (0 = a = k), when considered in the local rest 3-spaces orthogonal to u/c. Finally, putting the second of Eq. (3.53) in Eq. (3.39) and using Eq. (3.54), and employing the first of Eq. (3.52) and the first of Eq. (3.53) in the LRS-reduced vorticity evolution equation (3.24), gives 0 = (Θ/c) (ω/c) = (σ/c) (ω/c) .

(3.55)

Using these results in the remaining propagations along u/c of the constraints referred to above, one can show that, on appropriate re-substitution, they are preserved. The evolution equation for the magnitude of the fluid acceleration is obtained as the LRS-reduced version of Eq. (2.79) in Chapter 2. It reads (u/c ˙ 2 )˙/c =

[

∂p ∂µ

∂p (Θ/c) ] 0 + ( ∂µ −

1 3

) (Θ/c) (u/c ˙ 2) −

√2 3

(σ/c) (u/c ˙ 2) ,

(3.56)

and corresponds to the condition that the constraint (3.34) be preserved along u/c. Hence, the set of 1+3 ( 1+1+2 ) covariant evolution and constraint equations describing barotropic perfect fluid LRS spacetime geometries with (µ + p) > 0, Eqs. (3.23) - (3.34) and (3.37) - (3.40), is consistent provided that the conditions 0 = k (ω/c)

0 = a k = (Θ/c) (ω/c) = (σ/c) (ω/c)

0 = (ω/c) f˙/c = k f 0 ,

(3.57)

are true simultaneously.1 This ensures especially that also the propagation of the constraints (3.37) and (3.39) along u/c vanishes identically, if they were satisfied initially. Assuming (µ + p) > 0, there are thus three cases that can occur, which were classified by Ellis (1967) [44] and Stewart and Ellis (1968) [147] as follows: • LRS class I: (ω/c) 6= 0 ⇒ 0 = k = (Θ/c) = (σ/c) , f˙/c = 0 . e is hypersurface orthogonal, u/c is twisting. • LRS class II: 0 = k = (ω/c) . e and u/c are both hypersurface orthogonal. • LRS class III: k 6= 0 ⇒ 0 = (ω/c) = a = (u/c ˙ 2) , f 0 = 0 . e is twisting, u/c is hypersurface orthogonal. 1 The case f˙/c = (u/c ˙ 2 ) is not included in this notation, as (u/c ˙ 2 ) is not the covariant time derivative of a 1 + 3 invariantly defined 3-scalar.

38


These three cases will be discussed in turn, in each case considering first the generic situation and then the subcases that occur. It remains to check the consistency of the algebraic expression (3.28) for the magnitude H with the constraint (3.33). When this is done and the conditions (3.57) have been imposed, one obtains √

0=[

3 (u/c ˙ 2 )0 +

√

3 (u/c ˙ 2 )2 −

4πG √ 3c4

(µ + 3p) − 2 E ] (ω/c) .

(3.58)

This condition is only of interest when (ω/c) 6= 0 (see the following subsection).

3.1.6

(ω/c) 6= 0: Rotating solutions (LRS class I)

When (ω/c) 6= 0, one immediately finds that 0 = k = (Θ/c) = (σ/c), so the fluid matter source of the models within this LRS class of solutions can neither intrinsically expand nor distort. As k = 0, the non-zero vorticity vector field has vanishing spatial rotation ( cf. Eq. (3.19) ). Furthermore, the consistency conditions (3.57) show that f˙/c = 0 for all 1 + 3 ( 1 + 1 + 2 ) invariantly defined 3-scalars f , and so there exists an additional timelike Killing vector field such that all spacetimes within this LRS class will also be stationary. Thus, in this case there exists a G4 multiply-transitive on timelike 3-surfaces. Non-zero quantities, besides (ω/c), are in general µ, p, (u/c ˙ 2 ), a, E and H. The set of equations one needs to solve in this LRS class is given by u/c ˙ 2

0 0

(ω/c)

a0 (u/c ˙ 2)

2 2 = − a u/c ˙ 2 − u/c ˙ 2 − 2 (ω/c) + = − a (ω/c) + u/c ˙ 2 (ω/c)

4πG c4

(µ + 3p)

= − 12 a2 + a (u/c ˙ 2 ) + 2 (ω/c)2 − 2 4πG c4 (µ + p) = − (µ +

−1 ∂p p) ∂µ

µ0 ,

(3.59) (3.60) (3.61) (3.62)

which follow from (3.23), (3.27), (3.37) and (3.34), respectively. The magnitudes of the “electric” and “magnetic” parts of the Weyl curvature tensor in these models are given algebraically by a combination of Eqs. (3.23) and (3.25), and Eq. (3.28), respectively, which are E H

= −

√

3 2

√

a (u/c ˙ 2) − 2

= − 3 [ (u/c ˙ )−

√ 1 2

3 (ω/c)2 +

4πG √ 3c4

(µ + 3p)

a ] (ω/c) .

(3.63) (3.64)

With Eqs. (3.59) and (3.63), Eq. (3.58) is identically satisfied, as are the constraint equations (3.32) and (3.33) on substitution of Eqs. (3.63) and (3.64). Algorithm: The boundary data, which can be specified freely on a timelike 3-surface orthogonal to e, are the values of µ, its spatial derivative µ0 , (ω/c) and a. An equation of state of the form p = p(µ) is assumed. (u/c ˙ 2 ) is then given by Eq. (3.62), while E and H follow from Eqs. (3.63) and (3.64). For a treatment of this LRS class in terms of local coordinates and a metric tensor field refer to Stewart and Ellis (1968) [147], Eq. (2.8), and Kramer et al (1980) [88], Eq. (11.4).

39

3.1. LOCALLY ROTATIONALLY SYMMETRIC SPACETIME GEOMETRIES Solutions with a = 0

A subclass of solutions with a = 0 exists. However, for consistency they demand an equation of state of the form p(µ) = − 13 µ + const, which is usually dismissed as unphysical. One ordinary differential equation describing the spatial distribution of the total energy density µ remains to be solved, which follows from Eq. (3.59) and is given by 2 µ00 − 13 µ0 / (µ + p) + 3 4πG µ2 − p2 = 0 . (3.65) c4 One then obtains the following expressions for the remaining non-zero quantities: 1/2 4πG 1/2 (ω/c) = (µ + p) u/c ˙ 2 = 13 µ0 / (µ + p) c4 1/2 1/2 E = − √23 4πG H = − 4πG µ0 / (µ + p) . c4 µ 3c4

(3.66)

Models of this kind can be regarded as (non-physical) generalisations of the Gödel LRS case (see below). Solutions with p = 0 In the subclass of dust models, p = 0 ⇒ u/c ˙ 2 = 0, from Eq. (3.61) one obtains again one ordinary differential equation for µ which remains to be solved: µ00 −

5 4

2

2 µ0 /µ − 2 4πG c4 µ = 0 .

(3.67)

The other non-zero quantities are: a

= − 12 µ0 /µ

E

=

(ω/c)

2πG −√ µ 3c4

H

= = −

√

3 4

2πG 1/2 c4 2πG 1/2 c4

µ1/2

(3.68)

µ0 /µ1/2 .

Models with these properties can also be regarded as generalisations of the Gödel LRS case (see below). If one were to impose the dynamical restriction (u/c ˙ 2 ) = 0 instead of p = 0, Eq. (3.62) would be solved 0 0 by either µ = 0, which leads to f = 0 (see below), or ∂p/∂µ = 0 ⇒ p = const, which leads to a slight generalisation of Eq. (3.67). Solutions with H = 0 The dynamical restriction H = 0 implies from Eq. (3.64) that a = 2 u/c ˙ 2 . Then from Eqs. (3.59) and (3.61) one finds that 2 2 (ω/c) = − u/c ˙ 2 + 23 4πG (3.69) c4 (µ + 2p) . Finally, consistency of this expression with Eq. (3.60), on using Eq. (3.62), demands that ∂p/∂µ = 1 ⇒ p(µ) = µ + const. Solutions with E = 0 The dynamical restriction E = 0 implies from Eq. (3.63) that 2

(ω/c) = − 12 a (u/c ˙ 2) +

4πG 3c4

(µ + 3p) .

(3.70)

Consistency of this expression with Eq. (3.60) then requires that i 2 h 2 1 ∂p −1 4πG u/c ˙ 2 − 12 a + 34a 4πG p + [ ( ) + 1 ] (µ + p) u/c ˙ 2 c4 3 a 3 ∂µ c4 + 4πG 3c4 (µ + 3p) = in order to obtain a “pure magnetic” Weyl curvature solution.

0,

(3.71)

40


f 0 = 0: Gödel’s rotating model of the Universe If one imposes the additional condition that f 0 = 0, the symmetry group becomes a G5 multiplytransitive on the full spacetime manifold ( M, g, u/c, e ), which consequently is homogeneous. As the spatial derivatives of all non-zero 1 + 3 ( 1 + 1 + 2 ) covariant 3-scalars f vanish, one immediately finds from Eq. (3.62) that the matter moves geodesically: u/c ˙ 2 = 0. However, since no spacelike 3-surfaces of constant f exist, which are the group orbits of a simply-transitive G3 , models of this LRS

subclass are not spatially homogeneous (see Section 3.1.7 below). Equation (3.60) shows that a = 0, and from the algebraic equation (3.64) one obtains that the magnitude of the “magnetic” part of the Weyl curvature tensor is zero, H = 0. All 1 + 3 ( 1 + 1 + 2 ) covariant 3-scalars f are constant on ( M, g, u/c, e ), so that the remaining differential relations reduce to purely algebraic ones. From Eqs. (3.59), (3.61) and (3.63) one finds 2

2 (ω/c)

E

=

4πG c4

√

(µ + 3p) = 2 4πG c4 (µ + p) 2

= − 3 (ω/c) +

4πG √ 3c4

(µ + 3p) .

(3.72) (3.73)

It can easily be seen that this system of algebraic equations is consistent, provided the equation of state is p(µ) = µ, that is for “stiff matter” only. An equivalent configuration would be given by a combination of dust matter, p = 0, and a negative cosmological constant, Λ < 0 ( see Ref. [144] ), which gives Gödel’s 1949 rotating model of the Universe in its original form [61]. In summary, one observes that LRS class I of barotropic perfect fluid spacetime geometries contains a variety of stationary solutions, that, apart from the Gödel model, are differentially rotating. Most of them, however, are of minor interest for astrophysical and cosmological purposes, as it is difficult to see how, given non-zero vorticity, e.g., the geometry of a star model could correspond to exact rotational symmetry about every point.

3.1.7

k 6= 0: Homogeneous orthogonal models with twist (LRS class III)

When k 6= 0, the consistency conditions (3.57) demand that f 0 = 0 and 0 = (ω/c) = a. Thus, all spatial 3-gradients vanish and it follows immediately from Eq. (3.34) that the matter in these models moves on timelike geodesics, i.e., u/c ˙ 2 = 0. As a result the preferred timelike reference congruence

u/c is normal and geodesic, all 1 + 3 ( 1 + 1 + 2 ) covariant 3-scalars f are spatially homogeneous and there exists a G4 of isometries multiply-transitive on spacelike 3-surfaces orthogonal to u/c. That is,

the spacetimes themselves are orthogonally spatially homogeneous (OSH) ( see Ellis and MacCallum (1969) [49] ). Note that, apart from the preferred spacelike reference congruence e (and its classdefining spatial rotation), all 1 + 3 invariantly defined spacelike vector fields V orthogonal to u/c vanish. Also, since f 0 = 0 and a = 0, it follows from Eq. (3.21) that the fluid rate of shear and the “electric” and “magnetic” parts of the Weyl curvature are 1 + 3 ( 1 + 1 + 2 ) covariantly spatial divergence-free. Hence, the constraints (3.26), (3.32) and (3.33) are trivially satisfied. The non-zero geometrical quantities in the generic case, besides k, are µ, p, (Θ/c), (σ/c), E and H. From the 3Ricci identities for e and the Gauß embedding equation ( see, e.g., Ref. [144] ), or the LRS-reduced

41


form of Eq. (2.37), the intrinsic 3-Ricci curvature of the spacelike 3-surfaces orthogonal to u/c can be determined to be 3

E − 13 (Θ/c) (σ/c) + √13 (σ/c)2 ] eµν √ − 2 3 E + √23 (Θ/c) (σ/c) − 2 (σ/c)2

Sµν

=

3

=

√2 [ 3 3 2 2 k

=

4 4πG c4 µ −

R

2 3

(3.74)

2

(Θ/c) + 2 (σ/c)2 ,

(3.75)

where the expression for the 3-Ricci curvature scalar yields a generalised Friedmann equation. The magnitudes of the “electric” and “magnetic” parts of the Weyl curvature tensor are determined algebraically from Eqs. (3.75) or (3.37) and Eq. (3.28), respectively, to be E

=

H

=

2 1 √ (Θ/c) 3 3 3 2 k (σ/c) .

+

1 3

(Θ/c) (σ/c) −

√2 3

(σ/c)2 +

√

3 4

4πG k2 − 2 √ µ 3c4

(3.76) (3.77)

The time evolution equation for the dynamical variable k follows from Eq. (3.40). Note that, since k 6= 0 defines this LRS class, it follows from Eq. (3.77) that (σ/c) 6= 0 ⇒ H 6= 0. Using Eqs. (2.40) and imposing the LRS class III reductions, one can show that the conformal 3-Cotton–York tensor in this LRS class is given by ( see also Wainwright (1979) [158] ) ˙ + Cµν = − √23 h1/3 [ H/c

3

4 3

(Θ/c) H −

√4 3

(σ/c) H ] eµν .

(3.78)

The set of dynamical equations describing this class of expanding OSH LRS models reads 2

(Θ/c)˙/c = − 13 (Θ/c) − 2 (σ/c)2 − (σ/c)˙/c = − 3

1 √

4πG c4

(µ + 3p)

2

3

(Θ/c) − (Θ/c) (σ/c) +

√1 3

(3.79) 2

(σ/c) −

√

3 4

2

k +2

4πG √ 3c4

µ/c ˙ = − (µ + p) (Θ/c) ˙ k/c = − (Θ/c) k + 1 3

µ

(3.80) (3.81)

√4 3

(σ/c) k .

(3.82)

Consistency of this set of equations with the evolution equation (3.29) was tested for, which, with Eq. (3.76), is identically satisfied. An evolution equation for the 3-Ricci scalar, Eq. (3.75), can be derived: √ √ 3 R ˙/c = − 23 [ (Θ/c) − 3 (σ/c) ] 3R − 3 k 2 (σ/c) .

(3.83)

Alternatively, given the aim of using the set S4 = { µ, (Θ/c), (σ/c), E } as the (maximum set of) four generalised essential coordinates for the investigation of functional dependencies in models with non-zero rate of expansion, one can solve Eq. (3.37) for k 2 instead, obtaining 2

k 2 = − 94 (Θ/c) −

4 √ 3 3

(Θ/c) (σ/c) +

8 3

(σ/c)2 +

√4 3

E+

8 4πG 3 c4

µ.

(3.84)

The reality condition on k, k 2 > 0, leads to an algebraic restriction on the values of µ, (Θ/c), (σ/c) and E. Then the set of relevant 1 + 3 ( 1 + 1 + 2 ) covariant dynamical equations is given by 2

(Θ/c)˙/c = − 31 (Θ/c) − 2 (σ/c)2 − (σ/c)˙/c = − 23 (Θ/c) (σ/c) − ˙ E/c =

√1 3

4πG c4 2

(µ + 3p)

(σ/c) − E

√ (5µ − p) (σ/c) − (Θ/c) E + 4 3 (σ/c) E √ 2 − (Θ/c) (σ/c) − 3 (Θ/c) (σ/c)2 + 6 (σ/c)3

(3.85) (3.86)

4πG c4

µ/c ˙ = − (µ + p) (Θ/c) .

(3.87) (3.88)

42


The maximal dimension of the physical state space of the dynamical systems constituted by either Eqs. (3.79) - (3.82) or Eqs. (3.85) - (3.88) is four. Consistency of the latter set of equations with the evolution equation (3.82) was tested for, which, with Eq. (3.84), is identically satisfied. Algorithm: The initial data, which can be specified freely on a spacelike 3-surface orthogonal to u/c, are the values of µ, (Θ/c), (σ/c) and E. Given the equation of state p(µ), all covariant time derivatives are determined, together with k given by Eq. (3.84) and H by Eq. (3.77). If local comoving coordinates are chosen, the line element can be cast into the following form ( cf. Eq. (2.8) of Stewart and Ellis (1968) [147] and Eq. (11.4) of Kramer et al (1980) [88] ): ds2 = − d(ct)2 + X 2 (ct) [ dx − h(y) dz ]2 + Y 2 (ct) [ dy 2 + Σ2 (y) dz 2 ] .

(3.89)

Then one immediately obtains relations for the following 1 + 3 ( 1 + 1 + 2 ) covariant 3-scalars: (Θ/c)

=

(σ/c)

=

˙ X/c Y˙ /c +2 X Y ! ˙ X/c Y˙ /c 1 √ − . X Y 3

(3.90) (3.91)

This enables one to integrate Eq. (3.82), which gives X , Y2 where C1 denotes an integration constant. The solution to Eq. (3.83) is given by k = C1

(3.92)

C2 (C1 )2 X 2 − , (3.93) Y2 2 Y4 where C2 denotes a further integration constant, and thus Eq. (3.75) constitutes a first integral. Consequently, one is left with only three essential dynamical equations, these being Eqs. (3.75), (3.80) and (3.81), or equivalently, the set of equations !2 Y¨ /c2 Y˙ /c C2 3 (C1 )2 X 2 + + 2− = − 2 4πG (3.94) 2 c4 p Y Y Y 4 Y4 !2 ˙ X/c Y˙ /c Y˙ /c C2 (C1 )2 X 2 2 + + 2− = 2 4πG (3.95) c4 µ X Y Y Y 4 Y4 3

R=2

¨ 2 ˙ X/c X/c Y˙ /c Y¨ /c2 (C1 )2 X 2 (3.96) + + + = − 2 4πG c4 p . X X Y Y 4 Y4 This set of equations was given by Ellis (1967) [44] for p = 0. It should be pointed out that, in contrast to C1 , no 1 + 3 ( 1 + 1 + 2 ) covariant geometrical meaning can be associated with the constant C2 . In fact, C2 is just a constant of integration, resulting from the fact that Eq. (3.95) is a first integral of Eqs. (3.94) and (3.96); so any two of these equations implies the third. One obtains a form of the field equations independent of C2 from Eq. (3.96) and the difference between Eqs. (3.94) and (3.95). Exact solutions to the dynamical equations within this LRS class have been discussed in Kramer et al (1980), chapter 12.3 [88]. The different underlying simply-transitive subgroups G3 of the possible G4 isometry groups can be of Bianchi Type–II, Type–VIII/III and Type–IX ( see Ellis and MacCallum (1969) [49] ). For Type–IX one can have 3R > 0, which means 3R can change its sign in this class of models. Type–II and Type–VIII/III have 3R < 0 throughout their entire evolution and are only distinguished by different values of the sectional 3-curvature. Dynamically they are identical.


43

Solutions with E = 0 (“Pure magnetic”) Imposing the dynamical restriction E = 0 in LRS class III, its covariant time derivative, Eq. (3.87), requires that for (σ/c) 6= 0 the algebraic condition 2

(Θ/c) =

4πG c4

(5µ − p) −

√

2

3 (Θ/c) (σ/c) + 6 (σ/c) ,

(3.97)

must hold, which, in order to satisfy the rate of expansion evolution equation (3.85), restricts the functional form of the equation of state through an algebraic condition for ∂p/∂µ, with ∂p/∂µ 6= 0. The value of k 2 , as determined from Eq. (3.84), is k2 =

4 4πG 9 c4

(µ + p) ,

(3.98)

which is clearly consistent with the condition k 2 > 0. Hence, given that (σ/c) 6= 0, one has from Eq. (3.77) that H 6= 0. Alternatively, this condition arises from the LRS-reduced version of the relation 0=

4πG c4

(µ + p) σµν /c − Jµν ,

(3.99)

which follows from setting Eµν = 0 in the OSH perfect fluid form of Eq. (2.31) in Chapter 2 ( see also subsection 3.2.2 below ). Solutions of this kind are called “pure magnetic” in view of the form of their Weyl curvature tensor. The character of these solutions is very peculiar, since, although the fluid matter flow is expanding and, especially, shearing, no tidal distortions are being generated in the resultant spacetime geometry. The latter are commonly associated with the non-radiative part of Eµν , the physical origin of which is linked to temporal variations in the fluid rate of shear ( cf. Eq. (3.86) ).2 Here, the solutions seem to arise from an interplay between the OSH LRS condition and a strong restriction on the barotropic equation of state. Note that p 6= 0. It might even be that self-similar evolution of the spacetime geometry is a necessary requirement (see subsection 3.1.10 below). An example of a “pure magnetic” solution is given by a special subcase of the self-similar Bianchi Type–II OSH LRS solutions of Collins and Stewart (1971) [30], which have a linear barotropic equation of state of the form p(µ) = (γ − 1)µ. Here, one obtains E = 0 for γ = 6/5 ( cf. Eq.(3.241) below ). The FLRW subcase If one demands that for (σ/c) 6= 0 the spacelike 3-surfaces orthogonal to u/c are of constant curvature, i.e., from Eq. (3.74) that √ (3.100) E = 31 [ (Θ/c) − 3 (σ/c) ] (σ/c) , 2

one obtains from Eq. (3.87) an algebraic expression for (Θ/c) , which, when substituted into Eq. (3.84), gives k 2 = 0 and thus violates the definition of this specific LRS class. Thus there do not exist any shearing solutions in this LRS class with isotropic 3Rµν ; the 3R > 0 - FLRW models where (σ/c) = 0 are the only LRS models with k 6= 0 and spacelike 3-surfaces of constant curvature, as follows from Eq. (3.75). These are invariant under the action of a G6 of isometries, which is multiply-transitive on the spacelike 3-surfaces orthogonal to u/c. In this case there exists a 3-D family of rotational symmetries rather than one. This is the exceptional case where the preferred spacelike reference congruence e is not uniquely defined, and there is no 1 + 3 covariant feature that picks out a preferred spatial direction. All 1 + 3 covariant vectorial and tensorial physical quantities that could give e an invariant meaning are zero on the spacelike 3-surfaces orthogonal to u/c, while only 1 + 3 2 This

is not entirely rigorous, as it is difficult to invariantly distinguish between the radiative and non-radiative parts.

44


covariant 3-scalars like µ and (Θ/c) have non-trivial values. Nevertheless, one can still find local coordinate or orthonormal frame bases as before. In the local comoving coordinates of Eq. (3.89) one now has Y = X. Einstein’s static model of the Universe, the first relativistic model in the history of cosmology put forward in 1917 [41], is the special FLRW case with vanishing expansion, (Θ/c) = 0, and it follows 8 3

from Eq. (3.84) that k 2 =

(4πG/c4 ) µ > 0. The spacetime manifold ( M, g, u/c, e ) is then

invariant under a multiply-transitive G7 of isometries and therefore homogeneous. From Eq. (3.85) one finds that for consistency an equation of state of the form p(µ) = − 31 µ = const is required, which can be interpreted as a dust model, p = 0, with an additional positive cosmological constant, Λ > 0, balancing the tendency towards gravitational collapse within the fluid matter source. This is the form of matter content that was assumed in Einstein’s original version of the model. In summary, the general OSH LRS class III perfect fluid spacetime geometries with equation of state p = p(µ) are characterised by the existence of a G4 isometry group multiply-transitive on spacelike 3-surfaces, with a simply-transitive subgroup G3 which belongs to one of the various Bianchi types. Special cases are the 3R > 0 - FLRW models. Of particular interest is the possibility of constructing simple cosmological models with purely “magnetic” Weyl curvature. This offers a step to studying the underlying physical mechanisms which could generate peculiar solutions of this kind.

3.1.8

0 = k = (ω/c): The inhomogeneous orthogonal family (LRS class II)

When 0 = k = (ω/c), there exist spacelike 3-surfaces orthogonal to u/c in which there acts a G3 multiply-transitive on spacelike 2-surfaces orthogonal to e. These are the spherically symmetric solutions and their generalisations with plane and hyperbolic 2-spaces. All members within this class of expanding (or in the time-reversed case contracting), spatially inhomogeneous LRS models have vanishing “magnetic” part of the Weyl curvature tensor, as follows directly from the H-constraint (3.28), i.e., H =0,

(3.101)

and since k = 0 the fluid acceleration has zero spatial rotation ( cf. Eq. (3.19) ). The non-zero quantities in the generic case are µ, p, (Θ/c), (σ/c), u/c ˙ 2 , a and E. Depending on which of these dynamical variables might be zero, a broad variety of different special cases arise, which will be discussed in detail in this and the following subsection. From the 3-Ricci identities and the Gauß equation for e ( see, e.g., Ref. [144] ), the intrinsic 3-Ricci curvature of the spacelike 3-surfaces orthogonal to u/c can be determined to be 3

Sµν 3

R

= − 31 [ a0 + 2 K ] eµν 0

= −[ 2a + =

4 4πG c4 µ −

3 2 2 3

(3.102)

2

a − 2K ] 2

(Θ/c) + 2 (σ/c)2 ,

(3.103)

where K denotes the (constant) Gaußian curvature 2R := 2 K of the 2-D spacelike symmetry group orbits orthogonal both to u/c and e. The generalised Friedmann equation (3.103) provides a further constraint. From this one obtains for a0 the expression a0 = − 43 a2 + K − 2 4πG c4 µ +

1 3

2

(Θ/c) − (σ/c)2 .

(3.104)

45


This relation can be combined with Eq. (3.37) to give a purely algebraic expression for E in the form E=−

√

3 2

K−

1 √

6 3

[ (Θ/c) −

√

3 (σ/c) ]2 +

√

3 8

a2 +

4πG √ 3c4

µ.

(3.105)

Then the set of equations describing all LRS models within this class, the metric tensor of which can always be diagonalised ( see, e.g., Ref. [88], and subsection 5.2.1 of Chapter 5 ), is 2

(Θ/c)˙/c = − 13 (Θ/c) + (u/c ˙ 2 )0 + a (u/c ˙ 2 ) + (u/c ˙ 2 )2 − 2 (σ/c)2 −

4πG c4

(µ + 3p)

(σ/c)˙/c = − (Θ/c) (σ/c) + −

1 √ 2 3

(3.106)

1 √ 6 3

a (u/c ˙ 2) +

a/c ˙ = − 13 (Θ/c) a +

√1 3

˙ K/c = − 32 (Θ/c) K +

2

(Θ/c) − √1 3

(u/c ˙ 2 )2 −

(σ/c) a +

√2 3

1 √ 2 3

2 3

(σ/c)2 + √

3 8

a2 +

√

√1 (u/c ˙ 2 )0 3

3 2

(Θ/c) (u/c ˙ 2) −

K−

√2 3

4πG √ 3c4

(σ/c) (u/c ˙ 2)

(σ/c) K

(σ/c)0 +

3 2

=

0

= a0 +

0

= K0 + a K

0

=

∂p ∂µ

3 4

1 3

(3.108)

(3.110)

a (σ/c) −

a2 −

(3.107)

(3.109)

µ/c ˙ = − (µ + p) (Θ/c) 0

µ

√1 3

0

(Θ/c)

(3.111)

2

(Θ/c) + (σ/c)2 − K + 2 4πG c4 µ

(3.112) (3.113)

µ0 + (µ + p) (u/c ˙ 2) .

(3.114)

The evolution of the fluid acceleration is determined by Eq. (3.56). Consistency of this set of equations with the evolution equation (3.29) was tested for, which, on using Eq. (3.105), is identically satisfied. The evolution equation (3.109) derives from demanding preservation of the constraint (3.104) along the integral curves of u/c, whereas the constraint (3.113) is a consequence of the constraint (3.32). The covariant time derivative of the constraint (3.113) vanishes identically. Using Eqs. (2.40) and imposing the LRS class II reductions, one can show that the conformal 3Cotton–York tensor for all models within this LRS class vanishes ( see also Wainwright (1979) [158] ), 3

Cµν = 0 ,

(3.115)

i.e., the spacelike 3-surfaces orthogonal to u/c are conformally flat. The evolution equation for the 3-Ricci scalar (3.103) is √ √ 3 R ˙/c = − 31 [ 2 (Θ/c) + 3 (σ/c) ] [ 3R + 2 a (u/c ˙ 2) ] + 2 3 [ K − √ − 43 [ (Θ/c) − 3 (σ/c) ] [ (u/c ˙ 2 )0 + (u/c ˙ 2 )2 ] .

1 4

a2 ] (σ/c) (3.116)

Alternatively, given the aim of using the set S4 = { µ, (Θ/c), (σ/c), E } as the (maximum set of) four generalised essential coordinates for the investigation of functional dependencies in models with nonzero rate of expansion, one can solve the combined Eqs. (3.37) and (3.104) for K instead, obtaining K = − √23 E −

1 9

[ (Θ/c) −

√

3 (σ/c) ]2 +

1 4

a2 +

2 4πG 3 c4

µ.

(3.117)

46


The set of relevant 1 + 3 ( 1 + 1 + 2 ) covariant dynamical equations then becomes 2

(Θ/c)˙/c = − 13 (Θ/c) + (u/c ˙ 2 )0 + a (u/c ˙ 2 ) + (u/c ˙ 2 )2 − 2 (σ/c)2 4πG c4

−

(µ + 3p)

2 3

(σ/c)˙/c = − (Θ/c) (σ/c) + √1 3

−

(3.118) √1 3

2 0

(u/c ˙ ) −

1 √

2 3

2

a (u/c ˙ )+

√1 3

2 2

(u/c ˙ )

(σ/c)2 − E

1 3

√1 3

(3.119) 2

2 3

(Θ/c) (u/c ˙ )− √ ˙ E/c = − 4πG 3 (σ/c) E c4 (µ + p) (σ/c) − (Θ/c) E + a/c ˙ = − (Θ/c) a +

(σ/c) a +

√2 3

2

(σ/c) (u/c ˙ )

(3.121)

µ/c ˙ = − (µ + p) (Θ/c) 0 0

=

(σ/c)0 + 0

+ 0 0

= E0 + =

∂p ∂µ

a −

2 9

4 4πG 3 c4

µ

3 2

(3.122)

a (σ/c) −

2

1 2

= a +

3 2

aE −

(3.120)

√1 3

2

0

(Θ/c) 2 √

(Θ/c) −

3 3

(Θ/c) (σ/c) +

(3.123) 4 3

2

(σ/c) +

√2 3

E (3.124)

4πG √ 3c4

0

µ0

(3.125)

2

(3.126)

µ + (µ + p) (u/c ˙ ).

The evolution of the fluid acceleration is determined by Eq. (3.56). Consistency of this set of equations with the evolution equation (3.109) and the constraint equation (3.113) was tested for, which, with Eq. (3.117), are identically satisfied. Algorithm: The initial data, which can be specified freely on a spacelike 3-surface orthogonal to u/c, are the values of µ and (Θ/c), while (σ/c), E and a can be given at a point and their spatial distribution is determined by Eqs. (3.123), (3.125) and (3.124), respectively. Given a choice of the equation of state p(µ), (u/c ˙ 2 ) then follows from Eq. (3.126), all covariant time derivatives are known and K is given by Eq. (3.117). If local comoving coordinates are chosen, the line element can be cast into the following form (cf. Eq. (2.8) of Stewart and Ellis (1968) [147] and Eq. (13.2) of Kramer et al (1980) [88]): ds2 = − F −2 (x, ct) d(ct)2 + X 2 (x, ct) dx2 + Y 2 (x, ct) [ dy 2 + Σ2 (y) dz 2 ] .

(3.127)

One then immediately obtains relations for the following 1 + 3 ( 1 + 1 + 2 ) covariant 3-scalars: (u/c ˙ 2) (Θ/c)

(σ/c)

1 F0 X F ! ˙ X/c Y˙ /c = F +2 X Y ! ˙ F X/c Y˙ /c = √ − X Y 3 = −

a =

2 Y0 . X Y

(3.128) (3.129) (3.130) (3.131)

Eqs. (3.109) and (3.113) can then be integrated to give K=

C1 , Y2

(3.132)

47


with C1 an integration constant. Next, from Eq. (3.103), one obtains the differential expression " 0 2 # 2 X0 Y 0 Y C1 Y 00 3 R=− 2 2 −2 + +2 2 . (3.133) X Y X Y Y Y In integrating the field equations, Eq. (3.103) is commonly used in place of Eq. (3.106). Exact solutions to the dynamical equations within this LRS class with the assumption of spherical symmetry (K > 0) have been discussed by Kramer et al (1980), chapter 14.2 [88]. Employing the identity (3.48), Eqs. (3.118) - (3.124) allow the following set of useful relations to be derived, which show the non-linear growth of spatial inhomogeneities within this LRS class ( cf. Refs. [50] and [19] ): √ 0 [ µ0 ]˙/c = − 23 [ 2 (Θ/c) + 3 (σ/c) ] µ0 − (µ + p) (Θ/c) √ 00 0 0 (Θ/c) ˙/c = − [ (Θ/c) + 2 3 (σ/c) ] (Θ/c) + u/c ˙ 2 + a (u/c ˙ 2 )0 h 0 2 2 1 + 3 u/c ˙ 2 u/c ˙ 2 + 6 a (σ/c) − u/c ˙ 2 9 (Θ/c) −

(3.134)

2 √ 3 3

(Θ/c) (σ/c)

2

∂p −1 4πG (σ/c) + √23 E + 12 a2 − 23 4πG c4 µ − c4 (µ + p) ( ∂µ ) 2 i − a u/c ˙ 2 − u/c ˙ 2 (3.135) √ 00 0 0 1 (σ/c) ˙/c = − √13 [ (Θ/c) + 2 3 (σ/c) ] (Θ/c) + √13 u/c ˙ 2 − 2√ a (u/c ˙ 2 )0 3 √ √ 0 2 + 3 u/c ˙ 2 u/c ˙ 2 + 2 3 a (σ/c) + 32 a (Θ/c) (σ/c) + 32 a E h 1 2 2 7 1 √ √2 − √13 u/c ˙ 2 9 (Θ/c) + 3 3 (Θ/c) (σ/c) + 3 (σ/c) + 3 E

+

10 3

[ E 0 ]˙/c = − 23 [ 2 (Θ/c) + [ a0 ]˙/c =

2 3

√

−

1 4

a2 −

+

1 2

a (u/c ˙ 2 ) − u/c ˙

3 (σ/c) ]

2 4πG 3 c4

4πG √ 3c4

µ−

µ0 −

∂p −1 (µ + p) ( ∂µ ) i 2 2 4πG c4

4πG √ 3c4

(3.136)

0

(µ + p) (Θ/c) √ + 32 4πG ˙ 2 ] [ (Θ/c) − 3 (σ/c) ] E (3.137) c4 (µ + p) a (σ/c) + [ 2 a − u/c √ 0 3 2 4 2 [ (Θ/c) − 3 (σ/c) ] u/c ˙ 2 − 27 (Θ/c) − 3√ (Θ/c) (σ/c) 3 √ 3 2 4 (σ/c) + 13 a2 [ (Θ/c) − 3 (σ/c) ] + 23 (Θ/c) (σ/c) + 3√ 3 i h √ 2 4πG √ + 3√ [ 2 (Θ/c) + 3 (σ/c) ] E + 2 µ 4 3 3c h 2 1 4 2 − (u/c ˙ ) 3 a (Θ/c) − √3 a (σ/c) − 3 (Θ/c) u/c ˙ 2 i + √23 (σ/c) u/c ˙ 2 . (3.138)

In cosmological structure formation scenarios it is a well-established practice to study the evolution of linearised spatially inhomogeneous perturbations around an assumed FLRW background spacetime geometry. In their FLRW-linearised form the set of equations (3.134) - (3.138) simplifies significantly and could be used, e.g., to investigate covariant and gauge-invariant spherically symmetric perturbations of a FLRW spacetime geometry ( cf. Refs. [50] and [19] ). The description of scalar (total energy density) perturbations, which are the only ones contributing to the clumping of matter at linear order, would be entirely covered by only two equations from this set, e.g., Eqs. (3.134) and (3.135), and the equations underlying the evolution of the FLRW background spacetime geometry (see below).

48


Spatially inhomogeneous LRS dust models With the condition p = 0 ⇔ u/c ˙ 2 = 0 imposed, the cosmological models within this LRS subclass are the Lemaˆıtre–Tolman–Bondi spherically symmetric solutions (K > 0) ( see Refs. [94], [154] and [15] ) and their generalisations to spacelike 2-surfaces with vanishing or negative Gaußian curvature scalar K ( see Ellis (1967) [44] ). These spatially inhomogeneous dust models belong to the so-called “silent” class, recently discussed by Matarrese et al (1994/95) [111, 21] ( see also Section 5.1 of Chapter 5 below ). The relevant equations, which one obtains from specialisation of the set (3.118) - (3.124), are 2

(Θ/c)˙/c = − 13 (Θ/c) − 2 (σ/c)2 − (σ/c)˙/c = − 23 (Θ/c) (σ/c) − 1 3

√1 3

a/c ˙ = − (Θ/c) a +

√1 3

4πG c4 2

µ

(3.139)

(σ/c) − E

(3.140)

(σ/c) a

˙ E/c = − 4πG c4 µ (σ/c) − (Θ/c) E +

(3.141) √

3 (σ/c) E

(3.142)

µ/c ˙ = − (Θ/c) µ 0 0

=

(σ/c)0 + 0

= a +

0

0

= E +

a (σ/c) −

2

1 2

+

3 2

(3.143)

a −

2 9

4 4πG 3 c4

µ

3 2

aE −

√1 3

2

(Θ/c) −

0

(Θ/c) 2 √

3 3

(3.144)

(Θ/c) (σ/c) +

4 3

2

(σ/c) +

√2 3

E (3.145)

4πG √ 3c4

0

µ .

(3.146)

Algorithm: The initial data, which can be specified freely on a spacelike 3-surface orthogonal to u/c, are the values of µ and (Θ/c), while (σ/c), E and a can be given at a point and their spatial distribution is determined by Eqs. (3.144), (3.146) and (3.145), respectively. Then all covariant time derivatives are known and K is given by Eq. (3.117). Exact solutions to the dynamical equations within this LRS subclass have been discussed by Lemaˆıtre (1933) [94], Tolman (1934) [154], Bondi (1947) [15], and Kramer et al (1980), chapter 13.5 [88], for the spherically symmetric case (K > 0). Ellis (1967) [44] discussed the more general cases. For dust, in the local comoving coordinates of Eq. (3.127), one has from Eq. (3.128) that F = 1. The set of equations (3.134) - (3.138) describing the non-linear growth of spatial inhomogeneities simplifies considerably and reads: √ 0 [ µ0 ]˙/c = − 23 [ 2 (Θ/c) + 3 (σ/c) ] µ0 − µ (Θ/c) (3.147) √ 0 0 2 0 (Θ/c) ˙/c = − 4πG (3.148) c4 µ − [ (Θ/c) + 2 3 (σ/c) ] (Θ/c) + 6 a (σ/c) √ √ 0 2 0 0 4πG 1 (σ/c) ˙/c = − √3c4 µ − √3 [ (Θ/c) + 2 3 (σ/c) ] (Θ/c) + 2 3 a (σ/c) 3 2

a (Θ/c) (σ/c) + 32 a E √ 4πG 0 [ E 0 ]˙/c = − 23 [ 2 (Θ/c) + 3 (σ/c) ] √ µ − 3c4 √ + 2 a E [ (Θ/c) − 3 (σ/c) ] +

0

[ a ]˙/c = −

4 27

3

(Θ/c) −

2 √

2

(3.149) 4πG √ 3c4

2 3

0

µ (Θ/c) +

3 4πG 2 c4

µ a (σ/c) (3.150)

2

4 √

3

(Θ/c) (σ/c) + (Θ/c) (σ/c) + 3 3 (σ/c) √ √ 2 + 31 a2 [ (Θ/c) − 3 (σ/c) ] + 3√ [ 2 (Θ/c) + 3 (σ/c) ] 3 3 3

4πG µ]. ×[ E + 2 √ 3c4

The linearised set of equations corresponding to Ellis and Bruni (1989) [50] easily follows.

(3.151)

49

3.1. LOCALLY ROTATIONALLY SYMMETRIC SPACETIME GEOMETRIES (σ/c) = 0: The shearfree subcase

An interesting subcase within LRS class II arises, if one demands that (σ/c) = 0. Then one immediately obtains from Eq. (3.119) that the magnitude of the “electric” part of the Weyl curvature tensor is given by E=

√1 3

(u/c ˙ 2 )0 −

1 √ 2 3

a (u/c ˙ 2) +

√1 3

(u/c ˙ 2 )2 .

(3.152)

Hence, in order to obtain a solution with E 6= 0, the fluid acceleration has to be non-zero, and here also a non-zero rate of expansion of the fluid matter source is assumed (the case with (Θ/c) = 0 will be treated below). Next, from the constraint (3.123) it follows that 0

(Θ/c) = 0 ,

(3.153)

i.e., the spatial distribution of the rate of expansion is homogeneous and thus constant on the spacelike 3-surfaces orthogonal to u/c. With Eqs. (3.152) and (3.126) the conditions deriving from Eqs. (3.135) and (3.136) for (σ/c) = 0 are equivalent: they provide the expression u/c ˙ 2

00

=

0 0 µ0 − a u/c ˙ 2 − 73 u/c ˙ 2 u/c ˙ 2 h1 2 1 2 2 4πG + u/c ˙ 2 9 (Θ/c) + 2 a − 3 c4 µ −

4πG c4

(3.154) 4 3

a u/c ˙

2

−

1 3

u/c ˙

2 2

i

,

which then ensures that the constraint (3.125) is solved identically. Using relation (3.126), this is a constraint equation for the spatial distribution of the total energy density µ of third order, which is coupled to the constraint equation (3.124), giving the spatial distribution of the spatial divergence a. In the local comoving coordinates of Eq. (3.127) one has that (σ/c) = 0

⇒

F

˙ X/c Y˙ /c =F X Y

⇒

(Θ/c) = 3 F

˙ X/c . X

(3.155)

Then one can immediately integrate Eq. (3.134) to find µ0 =

C2 , X4

(3.156)

where C2 denotes an integration constant. Using this result and Eqs. (3.126) and (3.128), an expression for ∂p/∂µ is established, constraining the functional form of the equation of state. Exact solutions to the dynamical equations within this LRS subclass have been discussed in Kramer et al (1980), chapter 14.2.3 [88], for the spherically symmetric case (K > 0) and p = p(µ). Recently, Stephani and Wolf (1996) [145] discussed the spherically symmetric case within this class of shearfree solutions from a Lie point symmetry point of view. It would be interesting to investigate, whether a solution with (Θ/c)˙/c = 0 ⇒ (Θ/c) = const, (µ + 3p) > 0 and negative deceleration parameter q ( as defined in Eq. (7.20) of Chapter 7 ) exists within this subclass of LRS class II, which is consistent. A solution of this kind would provide a very special spatially inhomogeneous inflationary cosmological model.

50


a = 0: The non-diverging subcase If a = 0, the normals to the spacelike 2-surfaces spanned by the isometry group are non-diverging. The conditions for this dynamical restriction to be consistent are the following: for a = 0, Eq. (3.120) demands that √ 0 = [ (Θ/c) − 3 (σ/c) ] u/c ˙ 2 , (3.157) √ while Eq. (3.124) gives E algebraically. If (i) (Θ/c) = 3 (σ/c), inserting E into the constraint (3.125) requires that µ0 = 0, while Eqs. (3.119) and (3.121) give (µ + p) = 0 and (µ + p) (Θ/c) = 0, respectively. Consequently, this case can be discarded. If on the other hand (ii) u/c ˙ 2 = 0, then for f 0 6= 0 one needs to have p = 0. Equation (3.146) then results in the condition √ 0 0 (Θ/c) = 3 [ (Θ/c) − 3 (σ/c) ]−1 4πG (3.158) c4 µ , and the covariant time derivatives along the fluid matter flow lines of the constraints (3.144) - (3.145) vanish identically. That is, only dust provides an appropriate matter source for spatially inhomogeneous models of LRS class II with non-diverging integral curves of the preferred spacelike congruence e. These models are thus a further specialisation of the “silent” class discussed above. In conclusion, the generically spatially inhomogeneous LRS class II constitutes the largest class of solutions of perfect fluid spacetime geometries with equation of state p = p(µ). The models are in general time-dependent, and in the past they have been prime candidates for the theoretical and numerical description of star and galaxy formation processes as well as supernova explosions ( see, e.g., Refs. [114], [138], [99] and [144] ). Due to their highly idealised spacetime symmetry properties the relevant 1 + 3 ( 1 + 1 + 2 ) covariant dynamical equations become relatively simple and easily tractable. However, as outlined in the introduction of this section, an isentropic fluid matter flow, resulting from a barotropic equation of state, will in general be too restrictive to realistically model, e.g., explosions of stars during the late stages of their evolution. In the following subsection those subcases of cosmological models within LRS class II are discussed, which contain an additional translational (timelike or spacelike) Killing symmetry, apart from the multiply-transitive G3 isometry group.

3.1.9

Higher symmetry subcases: Hypersurface homogeneous models in LRS class II

f˙/c = 0: The static subcase Imposing the geometrical condition f˙/c = 0 on the set of equations (3.118) - (3.124), i.e., assuming the existence of an additional timelike Killing vector field such that the group of isometries is a G4 multiply-transitive on timelike 3-surfaces, for (µ + p) > 0 and E 6= 0 one immediately obtains from Eqs. (3.122) and (3.121) that 0 = (Θ/c) = (σ/c). Then, by combining Eqs. (3.118) and (3.119) one obtains for the magnitude E √ 4πG (µ + 3p) − 23 a u/c ˙ 2 , (3.159) E=√ 3c4 which, when substituted into Eq. (3.117), yields for the (constant) Gaußian curvature of the spacelike 2-D symmetry orbits of the G3 subgroup K = 14 a2 + a u/c ˙ 2 − 2 4πG (3.160) c4 p .


51

In principle K can be positive, negative, or zero. Finally, the remaining equations of the set (3.118) (3.124) lead to the set 0 2 (3.161) u/c ˙ 2 = − a u/c ˙ 2 − u/c ˙ 2 + 4πG c4 (µ + 3p) 2 0 4πG 1 2 ˙ − 2 c4 (µ + p) (3.162) a = − 2 a + a u/c µ0

∂p −1 ∂p −1 0 = − (µ + p) ( ∂µ ) (u/c ˙ 2 ) = ( ∂µ ) p .

(3.163)

The constraint (3.125) including the spatial divergence of the “electric” part is identically satisfied, upon using Eq. (3.159). In the local comoving coordinates of Eq. (3.127) one has F = F (x), X = X(x), Y = Y (x). In modelling static, spherically symmetric matter configurations (where K > 0), however, it is more convenient to use Schwarzschild ( say: “Shvarts-shilt” ) coordinates instead, such that the line element assumes the form ( see, e.g., Stephani (1991) [144] ) ds2 = − eν(r) d(ct)2 + eλ(r) dr2 + r2 [ dϑ2 + sin2 ϑ dϕ2 ] ,

(3.164)

and the fluid 4-velocity is given by uµ /c = e−ν/2 δ µ0 . Upon combination with the EFE this leads to ( see Ref. [144] ) i−1 1 h 3 , (3.165) u/c ˙ 2 = 1 − 2 m(r) m(r) + 4πG 4 r p r c 2 r where Z r m(r) := 4πG µ(x) x2 dx . (3.166) c4 0

Inserting expression (3.165) into Eq. (3.163) then yields the Tolman–Oppenheimer–Volkoff structure equation, which describes the interior spacetime geometry of static, spherically symmetric relativistic stars ( see, e.g., Refs. [114] and [138] ). If one uses expression (3.165) in Eqs. (3.161) and (3.162), one has to solve a coupled set of two ordinary differential equations for the two functions µ(r) and a(r), provided an equation of state p = p(µ) has been specified. The well-known conformally flat interior Schwarzschild solution published in 1916 [137] arises by imposing E = 0. Then, from Eq. (3.159) one obtains the condition 0 = a (u/c ˙ 2) −

2 4πG 3 c4

(µ + 3p) ,

(3.167)

which has to be preserved along the integral curves of the preferred spacelike reference congruence e. Taking the prime derivative of Eq. (3.167) and re-substituting where appropriate, one finds ∂p −1 0 = ( ∂µ )

2 4πG 3 c4

(µ + p) (u/c ˙ 2) .

(3.168)

Hence, for (µ + p) > 0 and non-zero acceleration between the individual fluid matter elements (which is required to sustain them against attractive gravitational forces) one obtains ∂p −1 0 = ( ∂µ ) ,

(3.169)

which by Eq. (3.163) implies µ = const. Furthermore, Eq. (3.160) now shows that the Gaußian 2-curvature of the symmetry group orbits necessarily needs to be positive, K=

1 4

a2 +

2 4πG 3 c4

µ > 0,

(3.170)

i.e., spherical symmetry of the configuration follows naturally. In the remaining two paragraphs the spatially homogeneous subcases within LRS class II are addressed, which generally occur in two different ways.

52


f 0 = 0: Orthogonally spatially homogeneous models In the OSH subcase of LRS class II there exists an extra translational symmetry acting within the spacelike 3-surfaces orthogonal to u/c. Since now f 0 = 0 for all 1 + 3 ( 1 + 1 + 2 ) covariant 3-scalars ( which implies (u/c ˙ 2 ) = 0 ), and a = 0 in order to satisfy the constraints (3.123) and (3.125), and also k = 0 (by the definition of LRS class II), it follows from Eqs. (3.21) and (3.22) that both the fluid rate of shear and the “electric” part of the Weyl curvature are 1 + 3 ( 1 + 1 + 2 ) covariantly spatial divergence-free and spatial rotation-free. The anisotropic part of the 3-Ricci curvature tensor for OSH models of this subclass is given from Eq. (3.102) by 3

Sµν = − 23 K eµν ,

(3.171)

while the generalised Friedmann equation is 3

R = 2 K = 4 4πG c4 µ −

2 3

2

(Θ/c) + 2 (σ/c)2 .

(3.172)

The evolution of 3R is governed by the equation √ √ 3 R ˙/c = − 31 [ 2 (Θ/c) + 3 (σ/c) ] 3R + 2 3 K (σ/c) ,

(3.173)

and from Eq. (3.124) it follows that E=

1 √ 3 3

2

(Θ/c) +

1 3

(Θ/c) (σ/c) −

√2 3

2

4πG (σ/c) − 2 √ µ. 3c4

(3.174)

The equations (3.118) - (3.124) reduce to the dynamical system 2

2

(Θ/c)˙/c = − 31 (Θ/c) − 2 (σ/c) −

4πG c4

2

(µ + 3p)

1 (σ/c)˙/c = − 3√ (Θ/c) − (Θ/c) (σ/c) + 3

µ/c ˙ = − (µ + p) (Θ/c) ,

√1 3

(3.175) 2

4πG (σ/c) + 2 √ µ 3c4

(3.176) (3.177)

whose physical state space has at most dimension three. The isometry group underlying OSH models within this subclass can be either of Bianchi Type–I/VII0 (where the group orbits have isotropic 3Ricci curvature with 3R = 0 = 2 K, i.e., they are flat), or of a special case of Bianchi Type-III (with K < 0) ( see Ellis and MacCallum (1969) [49] ), or instead it can be the spherically symmetric case of Kantowski–Sachs (KS) from 1966 [83] (with K > 0), which is the unique case where the group of isometries G4 multiply-transitive on spacelike 3-surfaces orthogonal to u/c does not contain a simplytransitive G3 subgroup ( see, e.g., Ref. [105] ). In the local comoving coordinates of Eq. (3.127) one has that F = 1, X = X(ct), Y = Y (ct), and the solution to Eq. (3.173) is given by 3

R=2

C1 C2 + , Y2 XY

(3.178)

where C1 and C2 are integration constants. Thus, Eq. (3.172) constitutes a first integral and is generally used in place of Eq. (3.175).

53

3.1. LOCALLY ROTATIONALLY SYMMETRIC SPACETIME GEOMETRIES FLRW

In this OSH subcase of LRS class II one has that (u/c ˙ 2 ) = 0 and (σ/c) = 0 ⇔ E = 0. This is a further specialisation of the previous OSH subcases, but it no longer follows that necessarily a = 0 and a0 = 0, as this is the exceptional case where the preferred spacelike reference congruence e is not uniquely defined. As mentioned in subsection 3.1.7 there is no 1 + 3 covariant feature that picks out a preferred spatial direction. With the condition a0 = − 2 K required to obtain spacelike 3-surfaces orthogonal to u/c which have isotropic curvature ( cf. Eq. (3.102) ), the Friedmann equation is given by 3

R=6 K−

1 4

a2 = 4 4πG c4 µ −

2 3

2

(Θ/c) .

(3.179)

The evolution and constraint equations for the non-zero variables are 2

(Θ/c)˙/c = − 13 (Θ/c) −

4πG c4

(µ + 3p)

(3.180)

µ/c ˙ = − (µ + p) (Θ/c)

(3.181)

1 3

(3.182)

a/c ˙ = − (Θ/c) a ˙ K/c = − 23 (Θ/c) K 3 R ˙/c = − 23 (Θ/c) 3R 0

(3.183) (3.184)

= K0 + a K .

(3.185)

However, once the sign of 3R was fixed, where either 3R > 0, 3R = 0 or 3R < 0, only two of these equations are essential, which are commonly chosen to be Eq. (3.179) and Eq. (3.181). In the local comoving coordinates of Eq. (3.127) one has that F = 1, X = X(ct), Y = X(ct) Z(x), and Eq. (3.184) can be integrated to give 3

R=

C , X2

(3.186)

where C is an integration constant. For a pressure-free fluid matter source, p = 0, and flat spacelike 3-surfaces orthogonal to u/c, 3R = 0 ⇒ K = a2 /4, one obtains the well-known Einstein–de Sitter (1932) [42] model. The Einstein static model is the special FLRW case when (Θ/c) = 0 and thus

1 3 6 R

= K − a2 /4 =

2 3

(4πG/c4 ) µ > 0.

From Eq. (3.180) one then finds that for consistency an equation of state of the form p(µ) = − 13 µ = const is required, which can be interpreted as a dust model with a positive cosmological constant, Λ > 0, as was already discussed at the end of subsection 3.1.7. Tilted spatially homogeneous LRS dust models Here there exists an extra translational symmetry, but not (as was the case in the previous paragraph) in the 3-surfaces orthogonal to u/c. Instead, the 3-surfaces of homogeneity are tilted with respect to u/c, i.e., u/c does not provide the unit normal congruence to these 3-surfaces ( see King and Ellis (1973) [86] ), and in general one will have Fµ 6= 0 for the 3-gradients of any 1 + 3 invariantly defined 3-scalars f . In particular, in the tilted case, with the cosmological conditions (Θ/c) 6= 0 and (µ + p) > 0 imposed (which are assumed), one will have Xµ 6= 0. From a mathematical point of view spatial homogeneity (SH) can be treated in terms of functional dependencies between the dynamical variables of a particular model ( see, e.g., Refs. [43] and [135] ).

54


Using, e.g., orthonormal frames that are adapted to the 3-D spacelike orbits of the isometry group, it is known that in the SH situation all dynamical scalar invariants are functions of one independent variable only, which in general is a timelike coordinate ( see, e.g., Ellis and MacCallum (1969) [49] or King and Ellis (1973) [86] ). In line with an approach to the equivalence problem of spacetime geometries as discussed by Bradley and Karlhede (1990) [16], in a 1 + 3 covariant treatment of (tilted) SH one can choose one of the non-trivial 1 + 3 covariant 3-scalars as an essential generalised coordinate — µ, say — and consider the remaining 1 + 3 covariant 3-scalars f as functions thereof. That is, f = f (µ) , and so ∇µ f =

(3.187)

df (µ) ∇µ µ , dµ

(3.188)

df (µ) df (µ) ν h µ ∇ν µ = Xµ dµ dµ

(3.189)

which implies the two relations Fµ

= hνµ ∇ν f =

f˙/c =

df (µ) µ/c ˙ . dµ

(3.190)

Similar relations will hold for any other pair of 1 + 3 covariant 3-scalars, like, e.g., { (Θ/c) , f }. Equation (3.189) implies that the spatial directions of Xµ and Fµ are parallel, i.e., 0 = − µνρσ Xν Fρ uσ /c .

(3.191)

In the tilted SH situation to be investigated, one has from the LRS condition Xµ = µ0 eµ and Fµ = f 0 eµ , so that Eq. (3.191) will be trivially satisfied. Thus, one also needs the magnitude information. In general f (µ) is not known, but it is clear that it must be the same in both Eqs. (3.189) and (3.190), so one gets a non-trivial magnitude relation by eliminating df (µ)/dµ between the two, leading to µ/c ˙ Fµ = f˙/c Xµ

⇔

µ/c ˙ f 0 = f˙/c µ0 .

(3.192)

This is another way of saying that the surfaces of constant µ and f must be the same. The set of equations corresponding to Eqs. (3.192) will hold for every pair of 1 + 3 covariant 3-scalars f , defined from the fluid matter and Weyl curvature variables, and all their covariant derivatives. It is known that the perfect fluid cases of tilted SH models in LRS class II are of Bianchi Type– V/VIIh ( see King and Ellis (1973) [86] ), studied in some detail in Ref. [32]. However, for reasons of simplicity, the following discussion focuses on the situation where the equation of state of the fluid matter source is that of dust, p = 0 ⇔ (u/c ˙ 2 ) = 0, only. The exact solution to the EFE for this nonrotating tilted SH LRS dust model was given by Farnsworth (1967) [56]. It belongs to the “silent” class ( see Refs. [111] and [21] ) mentioned in subsection 3.1.8 above. Now it is assumed that the functional dependency relations (Θ/c) = (Θ/c)[µ], (σ/c) = (σ/c)[µ], E = E[µ], a = a[µ] and K = K[µ] hold, and also that all of these dynamical variables are non-zero. Using the set of 1 + 3 ( 1 + 1 + 2 ) covariant dynamical equations (3.139) - (3.145) of subsection 3.1.8, it follows from Eq. (3.192) that the following five consistency equations need to be satisfied: First, 0 0 = µ/c ˙ (Θ/c) − (Θ/c)˙/c µ0 , (3.193)

55

3.1. LOCALLY ROTATIONALLY SYMMETRIC SPACETIME GEOMETRIES which can be solved to give 2

0

(Θ/c) =

2

(4πG/c4 ) µ + 13 (Θ/c) + 2 (σ/c) 0 µ . µ (Θ/c)

(3.194)

Second, 0

0 = µ/c ˙ (σ/c) − (σ/c)˙/c µ0 ,

(3.195)

which, on using Eq. (3.194), gives µ0 = − 32

E − (4πG/

√

a µ (Θ/c) (σ/c) √ . 1 − 3√ [ (Θ/c) − 3 (σ/c) ]2 3

3c4 ) µ

(3.196)

Third, ˙ µ0 , 0 = µ/c ˙ E 0 − E/c

(3.197)

which with Eq. (3.196) leads to the algebraic relation √ √ 1 [ (Θ/c) − 3 (σ/c) ] [ (Θ/c) + 2 3 (σ/c) ] √ E − 3√ 3 4πG √ 3 E. c4 µ = E − [ (Θ/c) − 3 (σ/c) ] (σ/c)

(3.198)

Fourth, the condition 0 = µ/c ˙ a0 − a/c ˙ µ0 ,

(3.199)

which, after inserting from Eqs. (3.194), (3.196) and (3.198), leads to a higher-order multivariate polynomial equation in the variables (Θ/c), (σ/c), E and a, which is quadratic in a, and finally the relation ˙ µ0 , 0 = µ/c ˙ K 0 − K/c

(3.200)

which, after inserting from Eqs. (3.194), (3.196) and (3.198), yields a second higher-order multivariate polynomial equation in the variables (Θ/c), (σ/c), E, that however does not contain the variable a. The next step to be taken is to consider the covariant time derivatives along the integral curves of the preferred timelike reference congruence u/c of the constraints (3.193), (3.195), (3.197), (3.199) and (3.200), and work out the conditions which establish that they will be preserved. Those of the relations for the pairs { µ, (Θ/c) }, Eq. (3.193), { µ, (σ/c) }, Eq. (3.195), and { µ, E }, (3.197), vanish identically, when Eqs. (3.194), (3.196) and (3.198) are employed, while the relations for the pairs { µ, a }, Eq. (3.199), and { µ, K }, Eq. (3.200), give two further higher-order multivariate polynomial equations, where again the first is quadratic in a, and the second again does not contain a. Combining these equations with the two obtained from the functional dependency relations (3.199) and (3.200), one has to solve a set of four simultaneous non-linear algebraic equations in the variables (Θ/c), (σ/c), ¨BNER package of the algebraic E and a. This problem is tackled by means of application of the GRO computing system REDUCE. The solutions obtained in this way are (Θ/c) = 0 E=

2 √ 3 3

2

(σ/c) =

2

(σ/c) = − √13 (Θ/c) ,

2

(σ/c) =

2

(σ/c) = − √16 (Θ/c) .

(Θ/c)

2 (Θ/c) E = − 3√ 3

E=

1 √ 3 6

(σ/c) = 0 ,

(Θ/c)

1 E = − 3√ (Θ/c) 6

√1 3

√1 6

(Θ/c) ,

(Θ/c) ,

(3.201) (3.202) (3.203) (3.204) (3.205)

56


However, all of them have to be discarded, as they lead to consequences for µ and µ0 which violate the initial assumptions. The failure of solution (3.201) is obvious, while Eq. (3.202) with Eq. (3.198) gives a zero denominator in Eq. (3.196), and solutions (3.203) - (3.205) lead to µ = 0 from Eq. (3.198). Thus, as one can show that the alternative assumptions a = 0, K 6= 0 and a = 0, K = 0 also lead to contradictions, it is concluded that for consistency Eqs. (3.193) - (3.200) have to be solved for the case K = 0, a 6= 0 instead ( see also Ref. [105] ). That is, from Eq. (3.105) the magnitude of the “electric” part of the Weyl curvature tensor is algebraically given by 1 E = − 6√ [ (Θ/c) − 3

√

3 (σ/c) ]2 +

√

3 8

a2 +

4πG √ 3c4

µ,

(3.206)

which simplifies the relevant calculations significantly. With Eq. (3.206) the evolution and constraint equations (3.142) and (3.146) are identically satisfied. Now Eq. (3.194) is still valid, whereas from Eq. (3.195) one gets µ0 =

√ 3 3 a µ (Θ/c) (σ/c) √ , [ (Θ/c) − 3 (σ/c) ]2 − 34 a2

(3.207)

which shows that for a = 0 one necessarily obtains µ0 = 0, and then successively f 0 = 0 for all 1 + 3 ( 1 + 1 + 2 ) covariant 3-scalars f , i.e., one arrives at OSH conditions. With Eqs. (3.194) and (3.207) inserted, Eq. (3.199) yields the algebraic expression 4πG c4

µ=

1 6

h

2

2

(Θ/c) − 3 (σ/c) −

3 4

i (Θ/c) − √3 (σ/c) 2 − a √ 2 (Θ/c) − 3 (σ/c) − 2

9 4 3 4

a2

,

(3.208)

a2

while Eq. (3.197) is identically satisfied. In principle, one can invert this (quartic) algebraic equation to determine the dynamical variable a in terms of µ, (Θ/c) and (σ/c). Note that for a = 0, Eq. (3.208) reduces to Eq. (3.172) with K = 0. Here, as a → 0, the tilted SH LRS dust model of Bianchi Type–V with non-zero spatial 3-Ricci curvature ( cf. Eq. (3.103) ) continuously reduces to the spatially flat OSH √ LRS dust model of Bianchi Type–I/VII0 . Note also that for a2 = 43 [ (Θ/c) − 3 (σ/c) ]2 , Eq. (3.208) shows that there occurs a singularity in the Ricci curvature tensor, and from Eq. (3.206) simultaneously in the Weyl curvature tensor. Equation (3.208) is compatible with Eq. (3.207), when Eqs. (3.144), (3.145), (3.194) and (3.206) are employed. It remains to investigate the covariant time derivatives along the integral curves of the preferred timelike reference congruence u/c of the constraints (3.193), (3.195), (3.197) and (3.199). One finds that they vanish identically, after expressions (3.194), (3.207) and (3.208) have been substituted, and thus for the tilted SH dust subcase within the generically spatially inhomogeneous LRS class II consistency of the initial assumptions and the results obtained from them has been established. Algorithm: In the tilted SH dust subcase of LRS class II the initial data, which can be specified freely on a spacelike 3-surface orthogonal to u/c, are the values of µ, (Θ/c) and (σ/c) at a point. The values of a and E at that point follow from Eqs. (3.208) and (3.206), respectively. The spatial derivatives on the initial spacelike 3-surface orthogonal to u/c of the non-zero dynamical variables are determined from Eqs. (3.207), (3.194), (3.144), (3.145) and (3.146), while their covariant time derivatives come from Eqs. (3.139) - (3.141). K = 0, as demonstrated. Applying the steps outlined in this subsection to the case with non-vanishing pressure, and hence, non-zero acceleration should also lead to similar results.


3.1.10

57

Self-similar OSH models in LRS classes II and III

Concluding the discussion of a 1 + 3 ( 1 + 1 + 2 ) covariant formulation of barotropic perfect fluid LRS spacetime geometries, a brief exposition of the existence of so-called self-similar OSH subcases within LRS classes II and III will now be given. In GR the physical dimension of any geometrical field of importance can be expressed in terms of an (integer) power q of the dimension [ length ].3 The physical dimension of most of the geometrical quantities occurring in this work is given at the end of Appendix A. Note that, except for scalars, the physical dimension of fields is dependent on the basis fields one chooses in the local description of the spacetime manifold ( M, g ). Following Eardley (1974) [34], with respect to a local coordinate basis, a homothety (self-similarity) exists, if under a constant rescaling of the unit of length l by l −→ eλ l

λ = const (dimensionless) ,

(3.209)

the metric tensor field g ( q = 2 ) transforms like gµν −→ e2 λ gµν ,

(3.210)

while a physical field Φ of dimension [ length ]q will transform like Φ −→ eq λ Φ .

(3.211)

In infinitesimal form Eqs. (3.210) and (3.211) correspond to the Lie-dragging of the geometrical fields along the integral curves of a homothety generating vector field ξ (which might be either timelike or spacelike), i.e., one obtains £ξ gµν = 2 λ gµν

(3.212)

£ξ Φ = q λ Φ

(3.213)

for the metric, and

for any physical field. Note that all dimensionless quantities ( q = 0 ) constructed from physical (geometrical) fields remain constant along ξ. It is straightforward to prove that for comoving perfect fluid spacetime geometries with barotropic equation of state, the homothety condition imposes the specific linear form ( see, e.g., Wainwright (1985) [159] ) p(µ) = (γ − 1) µ ,

0≤γ ≤2,

γ = const .

(3.214)

In the barotropic perfect fluid case it also follows that if ξ is orthogonal to the preferred timelike reference congruence u/c, then γ = 2, while if ξ is colinear with u/c, then γ = 2/3 ( see Ref. [159] ). The literature on self-similar spacetime geometries is vast. Ground-breaking work was done by Cahill and Taub (1971) [23], who set up the equations for expanding spherically symmetric ( hence, LRS class II with K > 0 ), self-similar cosmological models and discussed various special subcases thereof. Eardley (1974) [34] gave a systematic discussion of the concept of self-similarity in GR. Hsu and Wainwright (1986) [75] later discussed all self-similar OSH vacuum and barotropic perfect fluid solutions to the EFE. As the consideration here confines itself to the OSH situation, the reader is referred to recent work by Nilsson and Uggla (1996) [118] on spatially inhomogeneous self-similar 3 The dimension parameter q here should not be confused with the deceleration parameter q, later defined in Eq. (7.20) of Chapter 7.

58


LRS solutions, where dynamical systems methods have been applied. Due to the LRS property of ( M, g, u/c, e ), and implicitly using Eardley’s result that any homothety group Hr contains an isometry subgroup Gr−1 [34], one can make the Ansatz ξ µ = ξ1 uµ /c + ξ2 eµ

ξµ ξ µ = − (ξ1 )2 + (ξ2 )2

⇒

(3.215)

for the homothety generating field. The OSH situation is characterised by f 0 = 0 ⇒ (u/c ˙ 2 ) = 0 and a = 0, and, furthermore, k = 0 in LRS class II, and k 6= 0 in LRS class III. The homothety group H5 is multiply-transitive on ( M, g, u/c, e ). With these conditions taken into account the homothety condition for g is expressed by £ξ gµν

= ∇µ ξ ν + ∇ν ξ µ = − 2 (ξ1 )˙/c uµ /c uν /c h + (ξ1 )0 − (ξ2 )˙/c + + 2 (ξ2 )0 eµ eν + 2 ξ1 =

1 3

(Θ/c) ξ2 +

h

√2 3

√2 3

(σ/c) eµν +

(σ/c) ξ2 1 3

i

( uµ /c eν + uν /c eµ ) i

(Θ/c) hµν

2 λ gµν .

(3.216)

The homothety condition for any 1 + 3 ( 1 + 1 + 2 ) invariantly defined 3-scalar f of dimension q is given by £ξ f = ξ µ ∇µ f = ξ1 f˙/c = q λ f .

(3.217)

Contracting Eq. (3.216) with pµν one obtains λ=

1 3

[ (Θ/c) −

√

3 (σ/c) ] ξ1 .

(3.218)

On the other hand one has from £ξ uµ /c = ξ ν (∇ν uµ /c) − uν /c (∇ν ξ µ ) = − λ uµ /c £ξ eµ

(3.219)

= ξ ν (∇ν eµ ) − eν (∇ν ξ µ ) = − λ eµ ,

(3.220)

( in a local coordinate basis both uµ /c and eµ have q = − 1 ) after successive contraction with uµ /c and eµ , respectively, and using Eq. (3.218), that (ξ1 )˙/c =

1 3

(ξ2 )˙/c =

1 3

[ (Θ/c) −

√

3 (σ/c) ] ξ1 √ [ (Θ/c) + 2 3 (σ/c) ] ξ2

(ξ1 )0

=

(ξ2 )0

= − 3 (σ/c) ξ1 .

(3.221) (3.222)

0

(3.223) √

(3.224)

Decomposing the covariant derivative of ξ one obtains ∇µ ξ ν

= − (ξ1 )˙/c uµ /c uν /c − (ξ2 )˙/c uµ /c eν √ + 31 ξ2 [ (Θ/c) + 2 3 (σ/c) ] eµ uν /c + (ξ2 )0 eµ eν + ξ1 [

√2 3

(σ/c) eµν +

1 3

(Θ/c) hµν + (ω/c) sµν ] +

1 2

ξ2 k sµν .

(3.225)


59

It follows, after substituting from Eqs. (3.221) - (3.224), that ∇µ ξ ν

=

1 3

[ (Θ/c) − +

1 2

√

3 (σ/c) ] ξ1 gµν

k ξ2 sµν −

1 3

√ [ (Θ/c) + 2 3 (σ/c) ] ξ2 ( uµ /c eν − uν /c eµ ) . (3.226)

If ξ1 = 0, one can see that in that case ξ is a spacelike Killing vector field orthogonal to u/c. Now Eq. (3.217) is successively applied to each 1 + 3 covariant 3-scalar f in the set 4πG c4 µ, (Θ/c) , (σ/c), E , and substitutions from the respective evolution equations are made, leading to purely algebraic conditions to be solved. The self-similar OSH cases in both LRS class II and LRS class III are separately discussed in the following. LRS class II First, the spatially flat ( 3R = 0 = K ) OSH subcase within LRS class II is considered. Here Eqs. (3.172) and (3.174) give 4πG c4

µ =

1 6

E

1 3

=

2

(Θ/c) −

2

1 2

(σ/c) √ [ (Θ/c) − 3 (σ/c) ] (σ/c) ,

(3.227) (3.228)

and the homothety condition (3.217), applied to (Θ/c) and (σ/c), demands that for (Θ/c) 6= 0 the set of equations 2

2

ξ1

=

2 λ (Θ/c) / [ γ (Θ/c) + 3 (2 − γ) (σ/c) ]

(3.229)

0

=

(2 − γ) (σ/c)

(3.230)

has to be satisfied. So from Eq. (3.230) either (σ/c) = 0, which gives the 3R = 0 - FLRW model with arbitrary γ, or γ = 2, leading to a Bianchi Type–I/VII0 LRS model with “stiff matter” ( see, e.g., Ref. [159] ). For the dimensionless cosmological density parameter ( defined in Eq. (7.1) of Chapter 7 below ) one obtains (σ/c)2 Ω=1−3 (3.231) 2 . (Θ/c) Finally, Eqs. (3.180) and (3.175) can be integrated to give, respectively, (Θ/c) = 2 C / (γ cτ ) and (Θ/c) = C / cτ , where C is an integration constant and τ denotes cosmic proper time along u/c. For OSH models in LRS class II with K 6= 0, self-similar solutions with (σ/c) 6= 0 are determined from Eq. (3.217) by ξ1

=

2 λ / ( γ (Θ/c) )

(3.232)

1 (3γ − 2) (Θ/c) (σ/c) = − 2√ 3 4πG c4

µ = E

=

1 2

(3.233)

2

(1 − γ) (Θ/c)

1 √ (3γ 2 3 1 4 (3γ −

(3.234) 2

− 2) (1 − γ) (Θ/c) 2

2) (γ − 2) (Θ/c)

K

=

Ω

= 3 (1 − γ) .

(3.235) (3.236) (3.237)

They are rather peculiar in that they require γ < 1 in order that µ > 0. A self-similar Kantowski– Sachs model (with K > 0) only exists for matter with γ < 2/3, which thus violates the strong energy condition ( see Carr and Koutras (1993) [26] ). The self-similar OSH solution obtained for 2/3 < γ
0 - FLRW 0 0 •= 6 0 • 0, “magnetic” •= 6 0 • 0, 3R > 0 - FLRW

Table 3.1: Survey of the existing LRS perfect fluid spacetime geometries with barotropic equation of state p = p(µ) and (µ + p) > 0. where f again, as in the previous sections and chapters, denotes any 1 + 3 covariant 3-scalar. Hence, all 1 + 3 covariant spatial 3-gradients defined in subsection 2.1.2 of Chapter 2 are zero. In particular, by Eq. (2.43) and Eq. (B.2) of Appendix B it follows from the vanishing of the spatial 3-gradient of the total energy density that Xµ = 0 ⇒ 0 = (u/c ˙ 2 ) = (ω/c). That integrability is maintained when imposing these specialisations on the set (2.25) - (2.36) of 1 + 3 covariant evolution and constraint equations of Chapter 2 was demonstrated in Section 2.7. The resulting OSH-reduced set of dynamical equations for zero cosmological constant, Λ = 0, is given in the following: Time derivative equations 2

2

(Θ/c)˙/c = − 13 (Θ/c) − 2 (σ/c) −

4πG c4

(µ + 3 p)

hµρ hνσ (σ ρσ /c)˙/c = − 23 (Θ/c) σ µν /c − σ µρ /c σ νρ /c − E µν +

(3.246) 2 3

(σ/c)2 hµν

(3.247)

µν µν hµρ hνσ (E ρσ )˙/c = − 4πG + 3 σ (µρ /c E ν)ρ + J µν c4 (µ + p) σ /c − (Θ/c) E

− hµν σ ρσ /c E σρ µ/c ˙ = − (µ + p) (Θ/c) .

(3.248) (3.249)

Constraint equation 0 = hµρ hνσ (∇ν σ ρσ /c) .

(3.250)

Hence, in all OSH barotropic perfect fluid spacetime geometries the fluid rate of shear is 1 + 3 covariantly spatial divergence-free, i.e., it has no sources. Also, under the OSH assumptions, it follows from Eq. (2.30) that the “magnetic” part of the Weyl curvature is completely determined by the spatial rotation of the fluid rate of shear, Hµν = Kµν ,

(3.251)

62


such that in Eq. (3.248) one has Jµν = − hρ(µ hσν) ρτ κλ (∇τ K κσ ) uλ /c ,

(3.252)

which is just the spatial rotation of the spatial rotation of the fluid rate of shear ( or its second spatial rotation, for that matter ). The spatial divergence terms of the “electric” and “magnetic” parts of the Weyl curvature are obtained algebraically from the OSH-reduced forms of Eqs. (2.35) and (2.36), leading to hµρ hνσ (∇ν E ρσ ) hµρ

hνσ

(∇ν H

ρσ

)

= − µνρσ σντ /c K τρ uσ /c µνρσ

=

σντ /c E τρ

uσ /c .

(3.253) (3.254)

By use of the twice-contracted 3-Bianchi identity, Eq. (2.42), one can show that hµρ hνσ (∇ν E ρσ ) = − hµρ σ νσ /c (∇ν σ ρσ /c) ,

(3.255)

which relates the spatial divergence of the “electric” part of the Weyl curvature to spatial derivatives of the fluid rate of shear, and so does not occur as an additional constraint. It is known from the 1 + 3 ONF analysis of the OSH perfect fluid spacetime geometries by Ellis and MacCallum (1969) [49] that in OSH class A all 1 + 3 invariantly defined rank 2 symmetric tracefree spacelike tensor fields A orthogonal to u/c can be simultaneously diagonalised.4 Hence, for OSH models of class A the righthand sides of Eqs. (3.253) and (3.254) vanish, thus 1 + 3 covariantly characterising them as having “electric” and “magnetic” parts of the Weyl curvature that are spatial divergence-free. In that respect, except for Bianchi Type–I where H = 0,5 the OSH class A models may lend themselves to an interpretation as exact non-linear 1 + 3 covariant gravitational wave spacetime geometries, in analogy to Maxwell’s theory of electromagnetism, where propagating waves are associated with the sourcefree time-dependent parts of the fields. This interpretation is further motivated by the results obtained from the analysis of linearised spatially inhomogeneous perturbations to the matter content and spatial 3curvature of FLRW spacetime geometries ( see, e.g., Goode (1989) [62] and Bruni et al (1992) [19] ). Also King (1991) [87], by application of a particular geometrical decomposition method, reformulated OSH class A spacetime geometries of Bianchi Type–IX as an exact 3R > 0 - FLRW model on which gravitational waves of special character are superimposed. However, as the situation in the present case is spatially homogeneous (rather than spatially inhomogeneous, as one would expect in the case of a gravitational wave spacetime geometry), the gravitational wave interpretation is debatable, and also depends on whether the spatial rotation terms Kµν , Iµν and Jµν are non-zero or not. It had to be investigated whether the indicated characterisation of OSH class A is unique, or whether it would also allow for some models of OSH class B to fall into this category. As mentioned in Section 3.1, this spatial divergence-free feature of various OSH models follows automatically under LRS conditions ( where H = 0 in some cases ). The anisotropic and isotropic parts of the 3-Ricci curvature of the spacelike 3-surfaces orthogonal to u/c follow from Eqs. (2.37) and (2.38) as 3

Sµν 3

R

4 For

= Eµν − =

1 3


4 4πG c4 µ −

2 3

2

(Θ/c) + 2 (σ/c)2 ,

2 3

(σ/c)2 hµν

(3.256) (3.257)

the more general case see MacCallum et al (1970) [106]. 5 See, e.g., Ellis and MacCallum (1969) [49] for the definition of the different Bianchi types of OSH perfect fluid spacetime geometries.

63

3.2. ORTHOGONALLY SPATIALLY HOMOGENEOUS SPACETIME GEOMETRIES respectively, while their conformal 3-curvature properties relate to 3

Cµν

= − h1/3

hρµ hσν (Hρσ )˙/c +

4 3

(Θ/c) Hµν − 3 σ ρ(µ /c Hν)ρ + σ ρσ /c H σρ hµν

− h1/3 hρ(µ hσν) ρτ κλ ∇τ [ σ κξ /c σ ξσ /c ] uλ /c ,

(3.258)

which follows from OSH-reducing Eq. (2.40). Here, again Eq. (3.251) can be used to express Hµν . As was briefly hinted at, in the rate of shear and Weyl curvature propagation picture of the OSH models it is naturally the spatial rotation terms of these 3-tensor fields that are of interest. Their evolution along the integral curves of u/c follows from OSH-specialising Eqs. (2.62) - (2.64) of Chapter 2 and using Eq. (3.251). Thus, one arrives at, respectively, hµρ hνσ (K ρσ )˙/c = − (Θ/c) K µν − I µν + 3 σ (µρ /c K ν)ρ − σ ρσ /c K σρ hµν hµρ hνσ (I ρσ )˙/c = − 34 (Θ/c) I µν −

4πG c4

(3.259)

(µ + p) K µν + 3 E (µρ K ν)ρ − E ρσ K σρ hµν

+ h(µκ ν)ρστ σ λρ /c (∇λ E κσ ) uτ /c − 3 h(µρ hν)σ ρτ κλ ∇τ [ σ ξ(κ /c Eσ)ξ ] uλ /c − h(µρ hν)σ ρτ κλ (∇τ J σκ ) uλ /c

(3.260)

hµρ hνσ (J ρσ )˙/c = − 34 (Θ/c) J µν + 3 K µρ K νρ − 2 K 2 hµν + h(µκ ν)ρστ σ λρ /c (∇λ K κσ ) uτ /c − 3 h(µρ hν)σ ρτ κλ ∇τ [ σ ξ(κ /c Kσ)ξ ] uλ /c + h(µρ hν)σ ρτ κλ (∇τ I σκ ) uλ /c .

(3.261)

Imposition of the OSH barotropic perfect fluid conditions on the 1 + 3 covariant dynamical equations (2.25) - (2.36) reduces the dimension of their underlying physical state space from infinite to a finite number, as can be seen from the fact that by an appropriate choice of reference frame the OSH dynamical equations can be expressed as a closed set of coupled ordinary differential equations ( see, e.g., Ref. [160] ). For example, from the standard connection component oriented 1 + 3 ONF treatment of these spacetime geometries it is known that the maximal number of dimensions of their physical state space is six ( see MacCallum (1973) [104], and, on related issues, Siklos (1996) [139] ). Hence, in a complete 1 + 3 covariant formulation of these models, one would ideally like to obtain a closed set of six coupled ordinary differential equations relating the rates of change of six 1 + 3 invariantly defined 3-scalars to each other. Given an equation of state of the form p = p(µ), the variables µ and (Θ/c) are obvious as a choice in this target, identifying Eqs. (3.246) and (3.249) as two members in the set of dynamical equations aimed for. Then, Eq. (3.246) introduces (σ/c)2 as a new unknown, the evolution equation of which itself ( cf. Eq. (3.262) below ) introduces both the 1 + 3 covariant 3-scalars σ µν /c σ νρ /c σ ρµ /c and σ µν /c E νµ . In this way, again new 1 + 3 covariant 3-scalars arise from their respective evolution equations ( cf. Eqs. (3.277) and (3.267) below ), and so on. At first sight it is not entirely clear how this procedure could possibly terminate. However, one suspects that the functional dependency aspect for SH spacetime geometries mentioned in subsection 3.1.9 above will reveal itself in one form or another. For example, this may arise in an algebraic way from the fact that any (positive definite) rank 2 symmetric tracefree tensor A only defines two algebraically independent scalar invariants, namely A2 :=

1 2

Aµν Aνµ and Aµν Aνρ Aρµ . The fourth-order contraction of A with itself

64


is already given by Aµν Aνρ Aρσ Aσµ = 2 A4 . This simply corresponds to the obvious property that A possesses at most two independent eigenvalues. Similarly, there should exist a limited number of algebraically independent contractions between two (positive definite) rank 2 symmetric tracefree tensors A and B, such as Aµν B νµ and contractions of higher order, and analogously for any larger number of different tensors. In the OSH perfect fluid situation discussed so far, the non-zero rank 2 symmetric tracefree 3-tensor fields are the fluid rate of shear and the “electric” part of the Weyl curvature, σµν /c and Eµν , their spatial rotation terms, Kµν and Iµν , and the second spatial rotation of the fluid rate of shear, Jµν . An investigation of the 1 + 3 covariant 3-scalars constructed from them and their respective evolution equations should shed light on the problem outlined. It should be noted that this question is closely related to the implicit presence of purely spatial connection components in the aforementioned spatial rotation terms, and a possible 1 + 3 covariant way to express them explicitly and substitute for them in terms of other 1 + 3 covariant 3-scalars. Linear combinations of analogues of these spatial connection components are part of the dynamical variables chosen in the standard 1 + 3 ONF discussion of the OSH models ( cf. Chapter 4 ). In the OSH LRS class III of subsection 3.1.7 above, where all rank 2 symmetric tracefree 3-tensors are degenerate ( i.e., they have two equal eigenvalues ), there exists only one essential spatial connection component, k, which can be substituted for from Eq. (3.84). In the following, OSH-reduced evolution equations for all second-order, and for some third-order 1 + 3 covariant 3-scalars constructed from σµν , Eµν , Kµν , Iµν and Jµν are listed: Second-order 3-scalar evolution

(σ/c)2 ˙/c = − 43 (Θ/c) (σ/c)2 − σ µν /c σ νρ /c σ ρµ /c − σ µν /c E νµ E 2 ˙/c = − 2 (Θ/c) E 2 −

4πG c4

(µ + p) σ µν /c E νµ + 3 σ µν /c E νρ E ρµ

+ E µν J νµ

(3.263)

K 2 ˙/c = − 2 (Θ/c) K 2 + 3 σ µν /c K νρ K ρµ − K µν I νµ

I 2 ˙/c = − 83 (Θ/c) I 2 −

(3.262)

4πG c4

(3.264)

(µ + p) K µν I νµ + 3 E µν K νρ I ρµ

+ I µν νρστ σ κρ /c (∇κ Eσµ ) uτ /c

− 3 I µν νρστ ∇ρ [ σ κ(σ /c Eµ)κ ] uτ /c − I µν νρστ (∇ρ J νσ ) uτ /c

(3.265)

J 2 ˙/c = − 83 (Θ/c) J 2 + 3 J µν K νρ K ρµ

+ J µν νρστ σ κρ /c (∇κ Kσµ ) uτ /c − 3 J µν νρστ ∇ρ [ σ κ(σ /c Kµ)κ ] uτ /c

+ J µν νρστ (∇ρ I νσ ) uτ /c [ σ µν /c E νµ ]˙/c = − 53 (Θ/c) σ µν /c E νµ + 2 E µν σ νρ /c σ ρµ /c

(3.266)

65

3.2. ORTHOGONALLY SPATIALLY HOMOGENEOUS SPACETIME GEOMETRIES 2 2 µ ν − 2 4πG c4 (µ + p) (σ/c) − 2 E + σ ν /c J µ

(3.267)

[ σ µν /c K νµ ]˙/c = − 53 (Θ/c) σ µν /c K νµ + 2 K µν σ νρ /c σ ρµ /c − σ µν /c I νµ − E µν K νµ [ σ µν /c I νµ ]˙/c = − 2 (Θ/c) σ µν /c I νµ −

4πG c4

(3.268)

(µ + p) σ µν /c K νµ − I µν σ νρ /c σ ρµ /c

+ 3 σ µν /c E νρ K ρµ − E µν I νµ + σ µν /c νρστ σ κρ /c (∇κ Eσµ ) uτ /c − 3 σ µν /c νρστ ∇ρ [ σ κ(σ /c Eµ)κ ] uτ /c − σ µν /c νρστ (∇ρ J νσ ) uτ /c

(3.269)

[ σ µν /c J νµ ]˙/c = − 2 (Θ/c) σ µν /c J νµ − J µν σ νρ /c σ ρµ /c + 3 σ µν /c K νρ K ρµ − E µν J νρ + σ µν /c νρστ σ κρ /c (∇κ Kσµ ) uτ /c − 3 σ µν /c νρστ ∇ρ [ σ κ(σ /c Kµ)κ ] uτ /c + σ µν /c νρστ (∇ρ I νσ ) uτ /c [ E µν K νµ ]˙/c = − 2 (Θ/c) E µν K νµ −

4πG c4

(3.270)

(µ + p) σ µν /c K νµ + 6 σ µν /c E νρ K ρµ

− E µν I νµ + K µν J νµ [ E µν I νµ ]˙/c = − 73 (Θ/c) E µν I νµ −

4πG c4

(3.271)

(µ + p) σ µν /c I νµ −

4πG c4

(µ + p) E µν K νµ

+ 3 K µν E νρ E ρµ + 3 σ µν /c E νρ I ρµ + I µν J νµ + E µν νρστ σ κρ /c (∇κ Eσµ ) uτ /c − 3 E µν νρστ ∇ρ [ σ κ(σ /c Eµ)κ ] uτ /c − E µν νρστ (∇ρ J νσ ) uτ /c [ E µν J νµ ]˙/c = − 73 (Θ/c) E µν J νµ −

4πG c4

(3.272)

(µ + p) σ µν /c J νµ + 3 σ µν /c E νρ J ρµ

+ 2 J 2 + 3 E µν K νρ K ρµ + E µν νρστ σ κρ /c (∇κ Kσµ ) uτ /c − 3 E µν νρστ ∇ρ [ σ κ(σ /c Kµ)κ ] uτ /c + E µν νρστ (∇ρ I νσ ) uτ /c [ K µν I νµ ]˙/c = − 37 (Θ/c) K µν I νµ −

4πG c4

(µ + p) K 2 − 2 I 2 + 3 E µν K νρ K ρµ

+ 3 σ µν /c K νρ I ρµ + K µν νρστ σ κρ /c (∇κ Eσµ ) uτ /c − 3 K µν νρστ ∇ρ [ σ κ(σ /c Eµ)κ ] uτ /c

(3.273)

66

CHAPTER 3. SPECIAL PERFECT FLUID SPACETIME GEOMETRIES. I. − K µν νρστ (∇ρ J νσ ) uτ /c

(3.274)

[ K µν J νµ ]˙/c = − 37 (Θ/c) K µν J νµ + 3 K µν K νρ K ρµ + 3 σ µν /c K νρ J ρµ − I µν J νµ + K µν νρστ σ κρ /c (∇κ Kσµ ) uτ /c − 3 K µν νρστ ∇ρ [ σ κ(σ /c Kµ)κ ] uτ /c + K µν νρστ (∇ρ I νσ ) uτ /c

[ I µν J νµ ]˙/c = − 83 (Θ/c) I µν J νµ −

4πG c4

(3.275)

(µ + p) K µν J νµ + 3 I µν K νρ K ρµ

+ 3 E µν K νρ J ρµ + I µν νρστ σ κρ /c (∇κ Kσµ ) uτ /c − 3 I µν νρστ ∇ρ [ σ κ(σ /c Kµ)κ ] uτ /c + I µν νρστ (∇ρ I νσ ) uτ /c + J µν νρστ σ κρ /c (∇κ Eσµ ) uτ /c − 3 J µν νρστ ∇ρ [ σ κ(σ /c Eµ)κ ] uτ /c − J µν νρστ (∇ρ J νσ ) uτ /c .

(3.276)

Third-order 3-scalar evolution [ σ µν /c σ νρ /c σ ρµ /c ]˙/c = − 2 (Θ/c) σ µν /c σ νρ /c σ ρµ /c − 3 E µν σ νρ /c σ ρµ /c − 2 (σ/c)4

(3.277)

µ ν ρ [ E µν E νρ E ρµ ]˙/c = − 3 (Θ/c) E µν E νρ E ρµ − 3 4πG c4 (µ + p) σ ν /c E ρ E µ

− 6 E 2 (σ µν /c E νµ ) + 9 σ µν /c E νρ E ρσ E σµ + 3 J µν E νρ E ρµ

(3.278)

[ E µν σ νρ /c σ ρµ /c ]˙/c = − 73 (Θ/c) E µν σ νρ /c σ ρµ /c − − 2 σ µν /c E νρ E ρµ −

2 3

4πG c4

(µ + p) σ µν /c σ νρ /c σ ρµ /c

(σ/c)2 (σ µν /c E νµ )

+E µν σ νρ /c σ ρσ /c σ σµ /c + J µν σ νρ /c σ ρµ /c

(3.279)

ν ν ρ [ σ µν /c E νρ E ρµ ]˙/c = − 38 (Θ/c) σ µν /c E νρ E ρµ − 2 4πG c4 (µ + p) E µ σ ρ /c σ µ /c

+

4 3

(σ/c)2 E 2 − 2 (σ µν /c E νµ )2

+ 2 σ µρ /c σ ρν /c E σµ E νσ + 3 σ µρ /c E ρν σ νσ /c E σµ − E µν E νρ E ρµ + 2 σ µν /c E νρ J ρµ .

(3.280)

3.2. ORTHOGONALLY SPATIALLY HOMOGENEOUS SPACETIME GEOMETRIES

3.2.1

67

Solutions with isotropic 3-Ricci curvature

The question of obtaining a closed 1 + 3 covariant dynamical system describing (some of the) OSH perfect fluid spacetime geometries resolves itself in a straightforward manner, if the extra assumption is imposed that the 3-surfaces of homogeneity orthogonal to u/c be of isotropic 3-Ricci curvature, i.e., by Eq. (3.256) 3Sµν = 0, and 3R = const on 3-surfaces of constant affine parameter along the integral curves of u/c. From Ellis and MacCallum (1969) [49] it is well-known that OSH models with this property must be of either Bianchi Type–I ( where 3R = 0 ), or Bianchi Type–V ( where 3R < 0 ), or, in the spatially isotropic case with (σ/c) = 0, of Bianchi Type–IX ( where 3R > 0 ). It follows from Eq. (3.256) that Eµν =

(Θ/c) σµν /c − σµρ /c σ ρν /c +

1 3

2 3

(σ/c)2 hµν ,

(3.281)

which, when substituted into Eq. (3.247), leads to hµρ hνσ (σ ρσ /c)˙/c = − (Θ/c) σ µν /c .

(3.282)

On contraction with σµν /c this yields a simple evolution equation for (σ/c), which is identical to Eq. (3.262), when Eq. (3.281) is used. The resultant dynamical system is constituted by 2

2

(Θ/c)˙/c = − 13 (Θ/c) − 2 (σ/c) −

4πG c4

(µ + 3 p)

(σ/c)˙/c = − (Θ/c) (σ/c)

(3.283) (3.284)

µ/c ˙ = − (µ + p) (Θ/c) ,

(3.285)

which is further constrained by the generalised Friedmann equation 3

R = 4 4πG c4 µ −

2 3

2

(Θ/c) + 2 (σ/c)2 .

(3.286)

These two spacetime geometries will be used for various considerations on inflationary evolution of OSH cosmological models in Chapter 7 below. Note that by inserting Eq. (3.281) into Eq. (3.254), one immediately finds that for those OSH models with isotropic 3-Ricci curvature that have H 6= 0 ( which only applies to OSH models of Bianchi Type–V, cf. Ref. [49] ), the “magnetic” part of the Weyl curvature is 1 + 3 covariantly spatial divergence-free. This example was recently used by Maartens (1996) [101] to illustrate the existence of (known) exact solutions to the 1 + 3 covariant dynamical equations describing irrotational dust fluid spacetime geometries that satisfy the condition 0 = hµρ hνσ (∇ν H ρσ ).

3.2.2

Solutions with E = 0 (“Pure magnetic”)

Under the assumption of vanishing “electric” part of the Weyl curvature, E = 0 ⇒ I = 0, the remaining non-zero rank 2 symmetric tracefree 3-tensor fields are σµν /c, Kµν and Jµν , i.e., the fluid rate of shear and the first and second spatial rotation thereof; or, for that matter, the “magnetic” part of the Weyl curvature and its spatial rotation. The evolution equation (3.261) indicates that a nonzero Kµν in general will induce a non-zero Jµν . With the given assumptions imposed, the evolution equations (3.247) and (3.259) become involutive in character. It follows from Eqs. (3.248), (3.253),

68


(3.254), and Eqs. (2.57) and (2.58) of Chapter 2, respectively, that “pure magnetic” OSH spacetime geometries have to satisfy the constraints: 0

=

4πG c4 (µ + µνρσ

p) σµν /c − Jµν

σντ /c K τρ

0

= −

0

= hµρ hνσ (∇ν K ρσ ) µνρσ

σντ /c J τρ

0

= −

0

= hµρ hνσ (∇ν J ρσ ) .

uσ /c

(3.287) (3.288) (3.289)

uσ /c

(3.290) (3.291)

Here, Eq. (3.287) algebraically relates the second spatial rotation of the fluid rate of shear to the fluid rate of shear itself, and, hence, Eqs. (3.290) and (3.291) are identically satisfied, the latter since by Eq. (3.250) σµν /c is spatial divergence-free. Equations (3.288) and (3.289) demand that σµν /c and Kµν share a common eigenframe and that Kµν have zero spatial divergence, respectively. Hence, according to the discussion above, “pure magnetic” solutions should only exist in OSH class A. The constraints (3.288) and (3.289) are automatically preserved along the integral curves of u/c, as follows from the results obtained in Section 2.7 of Chapter 2. No new condition arises from Eq. (3.260), so, consequently, Eq. (3.287) provides the only new constraint to be satisfied. Its propagation along u/c, by use of Eqs. (3.247), (3.249) and (3.261), yields a restriction on the functional form of the barotropic equation of state, given by an algebraic condition on ∂p/∂µ, where ∂p/∂µ 6= 0. The situation here, where, under specific assumptions, a former evolution equation transforms into a new constraint, is similar to the consequences arising from the perfect fluid matter content assumption discussed in Section 2.6 of Chapter 2.

3.2.3

Conclusion

In view of the investigations made in this section and their related results, it seems possible to obtain a fully 1+3 covariant formulation of the well-known OSH barotropic perfect fluid spacetime geometries. A particular way to achieve this goal was suggested and outlined. Thereafter, special subcases as they arise from imposing geometrical and dynamical restrictions on the general OSH perfect fluid models were discussed in brief form.

Chapter 4

1 + 3 Decomposition of ( M, g, u/c ) by Orthonormal Frames One of the physical motivations behind the idea of introducing the notion of a preferred timelike reference congruence u/c into relativistic cosmological modelling is to provide an effective way of representing wo/man-kind’s Milky Way based status as spectators (observers) of dynamical processes on the cosmic stage. In the phenomenological fluid description of the matter content in the Universe, every individual curve in the congruence u/c can be viewed as the (timelike) worldline of a galaxy, or, more appropriately, a whole cluster thereof, on which (abstract) observers could be located. On the other hand, physical measurements — experiments and observations — are most conveniently referred to a special relativistic Minkowski reference frame. Combining these two aspects one naturally arrives at the concept of the 1 + 3 orthonormal frame (ONF) methods and their application to relativistic cosmology.1 Following the comprehensive description given by Pirani (1956) [129] and employing the notation set up in Appendix A the following framework arises. Within a family of hypothetical observer as represented by the preferred timelike reference congruence u/c, which in the cosmological context is generally identified with the average 4-velocity of the constituent particles of the fluid matter, everyone erects a Cartesian spatial coordinate system in the local (but extended) neighbourhood of each event along her/his individual worldline. The Cartesian spatial coordinate axes are spanned by three mutually orthogonal unit vector fields eα , which, furthermore, are also orthogonal to u/c, the tangent to the observers’ worldlines. Hence, identifying u/c with a timelike unit basis field, e0 = u/c, and joining it to the orthonormal spatial basis { eα }, a Minkowskian spacetime vector basis { ea } is obtained (here called a 1 + 3 ONF), which covers the neighbourhood of every single event on ( M, g, u/c ). It should be noted that any 1 + 3 ONF obtained in this way is linked to arbitrarily many others by a Lorentz transformation, which, however, will no longer be aligned with u/c, if this transformation was a timelike rotation ( i.e., a Lorentz boost ). Naturally, with respect to a single 1 + 3 ONF { ea } the spacetime metric tensor g has constant dimensionless physical components, given by ηab := g(ea , eb ) = diag [ − 1, 1, 1, 1 ]. 1 It should be emphasised that general ONF methods, not depending on the existence of a preferred timelike reference congruence on ( M, g ), have also been used in many areas of GR.

69

CHAPTER 4. 1 + 3 ONF DECOMPOSITION OF ( M, g, u/c )

70

Alternatively ( cf. Appendix A ), the 1 + 3 ONF can be defined in terms of the dual basis of 1-form fields { ω a } associated with the vector basis { ea } ( such that h ω a , eb i = δ a b ) by the requirement that the line element, which gives an invariant measure of infinitesimal 4-D physical distances on ( M, g, u/c ), is explicitly expressible in the particular form ds2 = ηab ω a ω b ,

(4.1)

where ηab is the same Minkowskian frame metric as above. Given this arrangement of a Minkowskian 1 + 3 ONF covering the nearby neighbourhood of every event on ( M, g, u/c ), the results of measurements of the components of any tensor field A defined on the spacetime manifold with respect to these frames represent their physical values at that particular event. The physical components of any tensor are spacetime scalars that are invariant under arbitrary local coordinate transformations. If ( M, g, u/c ) is not globally flat (for which in relativistic cosmology there does not seem to exist any evidence), then, due to spacetime curvature, the frame fields ea of a particular 1 + 3 ONF will soon lose their orthonormal arrangement and start twisting and distorting with respect to one another as one shifts the frame from one event to a nearby one. Mathematically this feature can be formulated in terms of either ( cf. Appendix A ) (i) the directional covariant derivatives of the frame fields ea along the integral curves of one another, given by ∇eb ea := Γcab ec ;

(4.2)

this defines the Ricci rotation coefficients Γabc ( where Γ(ab)c = 0 ), or (ii) the anti-symmetrised version thereof, a generalisation of Lie-transport of the frame fields ea along the integral curves of one another, yielding the commutator given by [ ea , eb ] := ∇ea eb − ∇eb ea := γ cab ec ;

(4.3)

this defines the commutation functions γ abc ( where γ a(bc) = 0 ). The individual frame fields ea in Eq. (4.3) are understood to act as differential operators, ea (A), on the frame components of any geometrical object A. It is clear from Eq. (4.3) though, that these frame derivatives are not (commuting) partial derivatives. Hence, any changes in the orthonormal arrangement of the frame fields ea as the 1+3 ONF is moved around on ( M, g, u/c ) is represented by either Γabc , or by γ abc , each of which has 24 algebraically independent components. By Eqs. (4.2) and (4.3) these geometrical objects are related by γ abc = − [ Γabc − Γacb ] ,

(4.4)

which can be inverted to give Γabc =

1 2

ηad γ dcb − ηbd γ dca + ηcd γ dab

.

(4.5)

In the subsequent part of this and the following chapter the commutation functions will be employed. By definition, the commutator satisfies the Jacobi identities, which imposes differential relations on the commutation functions. The frame components of the Riemann curvature tensor are defined via the

71

4.1. THE COMMUTATORS

Ricci identities, as applied to the frame fields ea . For more details the reader is referred to Appendix A ( cf. Eqs. (A.16), (A.25) and (A.27) ). With the 1 + 3 ONF structure set up on ( M, g, u/c ), analogous to the procedure described in Chapter 2, again one can invoke an irreducible 1 + 3 splitting of the geometrical fields defined on the spacetime manifold as well as of the dynamical equations tying them together. This involves the commutation functions, the fluid matter and the Weyl curvature variables, and their field equations, where the latter are given by the EFE, the Jacobi identities, and the second Bianchi identities. This goal will be pursued in the following. To the best of the author’s knowledge the second Bianchi identities have not been given previously in fully expanded 1 + 3 ONF decomposed form. In practical considerations of relativistic cosmology it is often helpful to have both the 1 + 3 covariant formulation as well as its 1 + 3 ONF counterpart available as complementary tools in the description of dynamical processes. By including the second Bianchi identities into the 1 + 3 ONF dynamical equations both frameworks are placed on an equal footing. In its conception the extended 1 + 3 ONF formalism is completely analogous to the decomposition method of ( M, g ) with respect to a set of frame spanning (complex) null fields, as developed by Newman and Penrose in 1962 [117]. Besides Pirani (1956) [129], the main features of the geometrical 1 + 3 ONF splitting were nicely reviewed by Ellis (1967) [44], MacCallum2 (1973) [104], and also, recently, in the book edited by Wainwright and Ellis (1996) [160]. Other references addressing general properties of ONF methods are, e.g., the book by Wald (1984) [161] and the paper by Edgar (1980) [36].

4.1 The commutators Aligning the timelike direction of the 1+3 ONF { ea } with the preferred timelike reference congruence, e0 = u/c ( ua /c = δ a0 , ua /c = − δ 0a ), leads to V a ⇒ V α and Aab ⇒ Aαβ , respectively, for any spacelike vector field V and any rank 2 symmetric tracefree spacelike tensor field A, which are both orthogonal to u/c ( see Appendix A for notation ). Now first the commutation functions in Eq. (4.3) with one or two indices equal to zero can be expressed in terms of the frame components of the irreducible dynamical variables associated with the covariant derivative of u/c, as defined in Eqs. (2.7) - (2.10) of Chapter 2, and the quantity Ωa /c := − 12 η abcd eb · e˙ c /c ud /c ,

(4.6)

where e˙ a /c := ∇u/c ea . The commutation functions Ωa /c ( ⇒ Ωα /c ) can be interpreted as the local angular velocity of the particular spatial frame { eα } an observer has set up at one event, as she/he propagates { eα } along her/his worldline within the preferred timelike reference congruence u/c. However, there exists no prescription for the way an observer should transport her/his spatial frame from one event on her/his worldline to another one (see below). This freedom could be used by the observer to choose, e.g., Ωα /c = 0, i.e., to Fermi-transport her/his spatial frame along u/c, which is the analogue of choosing a non-rotating reference frame in Newton’s theory of mechanics and gravitation. Physically a Fermi-transported spatial frame { eα } can be defined, e.g., by a set of three mutually orthogonal gyroscopes, which are carried along u/c. 2 Also:

MacCallum, Private Notes, Ref. [103]. Thanks are due to the author for making his notes available to me.


72

Secondly, the purely spatial commutation functions, γ αβγ , are decomposed, following the work of Schücking, Kundt and Behr ( see, e.g., Ref. [49] ), into a 1-index object aα and a symmetric 2-index object nαβ as follows: γ αβγ := 2 a[β δ αγ] + βγδ nδα .

(4.7)

Here, αβγ is the totally anti-symmetric 3-D permutation tensor with 123 = 1 = 123 , and spatial frame indices are raised and lowered with δ αβ and δαβ . The purely spatial commutation functions and the Fermi-rotation have no immediate counterparts in the 1 + 3 covariant formalism. In that framework they are implicitly contained in the fully orthogonally projected covariant derivatives ( see Appendix B ). The commutators in Eq. (4.3) are described by the expressions for γ abc . Using the variables just introduced, their 1 + 3 decomposition leads to [ e0 , eα ]

= u˙ α /c2 e0 −

[ e α , eβ ]

= 2 αβγ ω γ /c e0 + 2 a[α δ γβ] + αβδ nδγ eγ .

1 3

(Θ/c) δ βα + σ βα /c − βαγ (ω γ /c + Ωγ /c)

eβ ,

(4.8)

(4.9)

The resultant commutation functions, γ abc , and the associated Ricci rotation coefficients, Γabc , are explicitly listed in Appendix C. As an aside the conditions that a particular spatial frame vector eα be hypersurface orthogonal (HSO) are mentioned below ( see, e.g., Wald (1984) [161] ). HSO implies that there exist integrable timelike 3-surfaces orthogonal to eα at all non-singular events on ( M, g, u/c ). The HSO conditions are given by 0 = σαβ /c − αβγ ( ω γ /c + Ωγ /c ) α 6= β 6= γ 0 = nαα no summation .

4.2

(4.10)

The curvature

The local curvature of ( M, g, u/c ) is encoded in the components of its Riemann curvature tensor. By Eq. (A.25) of Appendix A its frame components are related via the Ricci rotation coefficients to the commutation functions and their frame derivatives. This relation arises in the following way: Rabcd := ec (Γabd ) − ed (Γabc ) + Γaec Γebd − Γaed Γebc − Γabe γ ecd .

(4.11)

The symmetry property Ra[bcd] = 0 of the frame components of the Riemann curvature tensor is ensured to hold by the fact that the commutation functions have to satisfy the differential Jacobi identities (A.27) of Appendix A. On the other hand, the Rabcd are given by the frame analogue of Eq. (2.21) in Chapter 2 in the form Rabcd

=

h

i 4 δ [ae δ b]f δ g[c δ hd] − η abef η ghcd E eg uf /c uh /c h i + 2 η abef δ g[c δ hd] + δ [ae δ b]f η ghcd H eg uf /c uh /c [a b] b] + 4 4πG c4 δ [a ( q /c ud] /c + u /c qd] /c ) [a b] + 2 4πG c4 (µ + 3 p) δ [c u /c ud] /c

−

2 4πG 3 c4

(µ − 3 p) δ a [c δ b d] +

2 3

Λ δ a [c δ b d] .

(4.12)

In this way they are related to both the Weyl curvature variables and, using the EFE (A.31) in Appendix A to substitute for the Ricci curvature variables, to the fluid matter variables. The frame components of

73

4.2. THE CURVATURE

the Riemann curvature tensor as given by Eq. (4.12) are explicitly listed in Appendix C. By inserting Eq. (4.11) for Rabcd into the lefthand side of Eq. (4.12) and adapting to the 1 + 3 ONF structure one can now obtain the EFE ( by contraction on the indices a and c ), the Jacobi identities ( from Ra[bcd] = 0 ), and also expressions for Eαβ and Hαβ in terms of the basic variables

(Θ/c) , σαβ /c, ωα /c, u˙ α /c2 , Ωα /c, aα , nαβ , µ, p, qα /c, παβ

,

(4.13)

and their e0 and eα frame derivatives. These equations and expressions are presented in the following subsections.

4.2.1

Einstein field equations e0 (Θ/c)

= − 13 (Θ/c)2 + (eα + u˙ α /c2 − 2 aα ) (u˙ α /c2 ) − 2 (σ/c)2 + 2 (ω/c)2 −

e0 (σ αβ /c)

4πG c4

(µ + 3p) + Λ ,

(4.14)

= − (Θ/c) σ αβ /c + (δ γ(α eγ + u˙ (α /c2 + a(α ) (u˙ β) /c2 ) − 2 ω (α /c Ωβ) /c αβ + 2 4πG − ∗S αβ c4 π − 13 δ αβ (eγ + u˙ γ /c2 + aγ ) (u˙ γ /c2 ) − 2 ωγ /c Ωγ /c i h + γδ(α 2 Ωγ /c σ β)δ /c − nβ)γ u˙ δ /c2 ,

(Θ/c)2 − (σ/c)2 + (ω/c)2 − 2 ωα /c Ωα /c − 2 4πG c4 µ +

1 ∗ 2 R

0

=

1 3

0

=

α (eβ − 3 aβ ) (σ αβ /c) − 23 δ αβ eβ (Θ/c) − nαβ ω β /c + 2 4πG c4 q + αβγ (eβ + 2 u˙ β /c2 − aβ ) (ωγ /c) − nβδ σ δγ /c ,

(4.15)

− Λ , (4.16)

(4.17)

where ∗

Sαβ

:= e(α (aβ) ) + bαβ −

1 3

δαβ [ eγ (aγ ) + bγγ ] − γδ(α (e|γ| − 2 a|γ| ) (nβ)δ ) ,

∗

R

:= 2 (2 eα − 3 aα ) (aα ) −

bαβ

:= 2 nαγ nγβ − nγγ nαβ .

1 2

bαα ,

(4.18)

(4.19)

(4.20)

In this set Eq. (4.14) constitutes the (00)-part of the EFE, when the latter are considered in the form given in Eq. (A.31) of Appendix A. Equation (4.14) is commonly called the Raychaudhuri equation. Alternatively, it can be obtained from the 1 + 3 covariant dynamical equation (2.25) of Chapter 2 by employing the 1 + 3 ONF translation rules for spatial derivative terms as given in subsection B.2.1 of Appendix B. The symmetric tracefree (αβ)-part of the EFE (A.31) corresponds to the evolution equation for the fluid rate of shear, Eq. (4.15), while Eq. (4.16), which is obtained from the combination (00) + (11) + (22) + (33) of the EFE (A.31), constitutes a generalised Friedmann equation. Finally, Eq. (4.17) is the (0α)-part of the EFE (A.31). Alternatively, it is obtained from the 1 + 3 covariant Eq. (2.28), by use of the 1 + 3 ONF translation rules given in both subsections B.2.1 and B.3.1 of Appendix B, and, as before, it is interpreted as a spatial divergence equation for the fluid rate of shear.


74

The objects ∗Sαβ and ∗R are the tracefree part and the trace of ∗Rαβ , which, in the case of ω α /c = 0, is just the intrinsic 3-Ricci curvature of the spacelike 3-surfaces orthogonal to the preferred timelike reference congruence u/c, here associated with the fluid matter flow ( see Ref. [104] ). However, if ω α /c 6= 0, then ∗Rαβ is not even a tensor; it is just a symbol defined by its relation to the objects aα and nαβ .

4.2.2

Jacobi identities

The 1 + 3 ONF decomposition of Eq. (A.27) in Appendix A, or, alternatively, constructing the combination Ra[bcd] = 0 from Eq. (4.12), leads to the set: e0 (aα )

= − 31 (δ αβ eβ + u˙ α /c2 + aα ) (Θ/c) + +

e0 (nαβ )

1 αβγ 2

1 2

(eβ + u˙ β /c2 − 2 aβ ) (σ αβ /c)

(eβ + u˙ β /c2 − 2 aβ ) (ωγ /c + Ωγ /c) ,

= − 13 (Θ/c) nαβ + (δ γ(α eγ + u˙ (α /c2 ) (ω β) /c + Ωβ) /c) + 2 σ (αγ /c nβ)γ − δ αβ (eγ + u˙ γ /c2 ) (ω γ /c + Ωγ /c) h i − γδ(α (eγ + u˙ γ /c2 ) (σ β)δ /c) + 2 nβ)γ (ωδ /c + Ωδ /c) ,

e0 (ω α /c)

0

= − 23 (Θ/c) ω α /c + σ αβ /c ω β /c − 12 nαβ u˙ β /c2 + αβγ 12 (eβ − aβ ) (u˙ γ /c2 ) − ωβ /c Ωγ /c , =

(eβ − 2 aβ ) (nαβ ) +

2 3

=

(4.22)

(4.23)

(Θ/c) ω α /c + 2 σ αβ /c ω β /c

+ αβγ [ eβ (aγ ) − 2 ωβ /c Ωγ /c ] ,

0

(4.21)

(eα − u˙ α /c2 − 2 aα ) (ω α /c) .

(4.24)

(4.25)

Note that the evolution and spatial divergence equations for the fluid vorticity, Eqs. (4.23) and (4.25), respectively, can also be obtained from Eqs. (2.26) and (2.29) in Chapter 2, by application of the 1 + 3 ONF translation rules given in subsection B.2.1 of Appendix B.

4.2.3

“Electric” and “magnetic” parts of the Weyl curvature tensor

Analogous to some of the relations provided up to this point, a 1 + 3 ONF expression for the “electric” part of the Weyl curvature tensor is obtained by conversion of the 1 + 3 covariant equation (2.27) of Chapter 2, when the translation rules for spatial derivative terms given in subsections B.2.1 and B.3.1 of Appendix B are used. This yields Eαβ

= − e0 (σαβ /c) −

2 3

(Θ/c) σαβ /c + (e(α + u˙ (α /c2 + a(α ) (u˙ β) /c2 ) − σαγ /c σ γβ /c

− ωα /c ωβ /c + 4πG c4 παβ − 13 δαβ (eγ + u˙ γ /c2 + aγ ) (u˙ γ /c2 ) − 2 (σ/c)2 − (ω/c)2 + γδ(α 2 Ω|γ| /c σβ)δ /c − nβ)γ u˙ δ /c2 .

(4.26)

75

4.3. THE SECOND BIANCHI IDENTITIES Combining this expression with the field equation (4.15) above, one easily derives (Eαβ +

4πG c4

παβ )

=

1 3

(Θ/c) σαβ /c − σαγ /c σ γβ /c − ωα /c ωβ /c + 2 ω(α /c Ωβ) /c + 13 δαβ 2 (σ/c)2 + (ω/c)2 − 2 ωγ /c Ωγ /c + ∗Sαβ . (4.27)

The 1 + 3 covariant equation (2.30) of Chapter 2 which gives an expression for the “magnetic” part of the Weyl curvature tensor converts into Hαβ

4.2.4

= − (e(α + 2 u˙ (α /c2 + a(α ) (ωβ) /c) + 12 nγγ σαβ /c − 3 nγ(α σβ)γ /c + 13 δαβ (eγ + 2 u˙ γ /c2 + aγ ) (ω γ /c) + 3 nγδ σ γδ /c + γδ(α (e|γ| − a|γ| ) (σβ)δ /c) + nβ)γ ωδ /c .

(4.28)

Conformal 3-Cotton–York tensor

If the fluid matter flow identified with u/c is irrotational, ω α /c = 0, then the 1 + 3 ONF form of the 3-Cotton–York tensor of the spacelike 3-surfaces orthogonal to u/c is obtained from converting Eqs. (2.39) and (2.40) in Chapter 2, leading to ∗

Cαβ

= γδ(α (e|γ| − a|γ| ) (∗Sβ)δ ) − 3 nγ(α ∗Sβ)γ + = − e0 (Hαβ ) −

4 3

1 2

nγγ ∗Sαβ + δαβ nγδ ∗S γδ

(4.29)

(Θ/c) Hαβ + 3 σ γ(α /c Hβ)γ − 3 nγ(α σ δβ) /c σγδ /c

γ 4πG γ nγγ σαδ /c σ δβ /c + 2 nαβ (σ/c)2 − 6 4πG c4 n (α πβ)γ + c4 n γ παβ γδ − δαβ σγδ /c H γδ − nγδ σ δ /c σ γ /c + nγγ (σ/c)2 − 2 4πG c4 nγδ π 1 + γδ(α (e|γ| − a|γ| ) (σ β) /c σδ /c + 2 4πG c4 πβ)δ ) − 3 e|γ| (Θ/c) σβ)δ /c − 2 u˙ |γ| /c2 Eβ)δ + 4πG . (4.30) c4 σβ)γ /c qδ /c + 2 Ω|γ| /c Hβ)δ

+

4.3

1 2


Finally, in the same straightforward way, the 1+3 ONF conversion procedure, resting on the translation formulae given in subsections B.2.1 and B.3.1 of Appendix B, is applied to the 1 + 3 covariant form of the second Bianchi identities, Eqs. (2.31), (2.32), (2.35) and (2.36) in Chapter 2, and to the 1 + 3 covariant form of their twice-contracted version, Eqs. (2.33) and (2.34). This leads to:

4.3.1

Bianchi identities for the Weyl curvature tensor

e0 (E αβ +

4πG c4

π αβ )

αβ = − 4πG /c − (Θ/c) (E αβ + c4 (µ + p) σ

−

4πG c4

1 4πG 3 c4

π αβ )

(δ γ(α eγ + 2 u˙ (α /c2 + a(α ) (q β) /c)

+ 3 σ (αγ /c (E β)γ −

1 4πG 3 c4

π β)γ ) +

1 2

nγγ H αβ

− 3 n(αγ H β)γ + 13 δ αβ 4πG ˙ γ /c2 + aγ ) (q γ /c) c4 (eγ + 2 u

+ γδ(α

h

− 3 σγδ /c (E γδ − + 3 nγδ H γδ

1 4πG 3 c4

(eγ + 2 u˙ γ /c2 − aγ ) (H β)δ )

π γδ )


76

+ (ωγ /c + 2 Ωγ /c) (E β)δ + i β) + 4πG n q /c , γ δ c4 e0 (H αβ )

4πG c4

π β)δ ) (4.31)

(α β) = − (Θ/c) H αβ + 3 σ (αγ /c H β)γ + 3 4πG c4 ω /c q /c αβ nγγ (E αβ − 4πG ) + 3 n(αγ (E β)γ − c4 π γ − δ αβ σγδ /c H γδ + 4πG c4 ωγ /c q /c γδ + nγδ (E γδ − 4πG c4 π ) h β) − γδ(α (eγ − aγ ) (E β)δ − 4πG c4 π δ )

−

1 2

+ 2 u˙ γ /c2 E β)δ −

4πG c4

π β)γ )

σ β)γ /c qδ /c i − (ωγ /c + 2 Ωγ /c) H β)δ , 0

=

(eβ − 3 aβ ) (E αβ + +

2 4πG 3 c4

− αβγ

0

=

4πG c4 α

π αβ ) −

(Θ/c) q /c −

4πG c4

4πG c4

2 4πG αβ 3 c4 δ

σ αβ /c q β /c

eβ (µ) − 3 ωβ /c H αβ

σβδ /c H δγ − 3 4πG c4 ωβ /c qγ /c + nβδ (E δγ +

4πG c4

π δγ )

,

4πG c4

π δγ ) (4.34)

Bianchi identities for the source terms e0 (µ) e0 (q α /c)

= − (µ + p) (Θ/c) − (eα + 2 u˙ α /c2 − 2 aα ) (q α /c) − σαβ /c π αβ ,

(4.35)

= − 43 (Θ/c) q α /c − δ αβ eβ (p) − (µ + p) u˙ α /c2 − (eβ + u˙ β /c2 − 3 aβ ) (π αβ ) − σ αβ /c q β /c − αβγ (ωβ /c − Ωβ /c) qγ /c − nβδ π δγ .

4.4

(4.33)

α (eβ − 3 aβ ) (H αβ ) + 2 4πG c4 (µ + p) ω /c αβ α β ) − 4πG + 3 ωβ /c (E αβ − 31 4πG c4 π c4 n β q /c δ + αβγ 4πG c4 (eβ − aβ ) (qγ /c) + σβδ /c (E γ + − nβδ H δγ .

4.3.2

(4.32)

(4.36)

Introducing local coordinates

Since in relativistic cosmology, as in many other branches of physics, one is largely interested in invariant qualitative and quantitative (measurable) features of a particular model, emphasis in this work is laid on 1 + 3 covariant and their complementary 1 + 3 ONF methods. However, given one successfully manages to solve the 1 + 3 dynamical equations for the spacetime dependences of all its dynamical variables, this programme also includes (re-)constructing the local geometry in the neighbourhood of any (non-singular) event on ( M, g, u/c ) as it is encoded in the metric tensor field g and the infinitesimal line element derived thereof. The metric tensor field plays the fundamental role of determining the lapses of physical clocks as carried by (hypothetical) observers with, e.g., worldlines tangent to u/c,

77

4.4. INTRODUCING LOCAL COORDINATES

and it also determines the changes in physical length scales as a consequence of local variations in the spacetime curvature. Furthermore, it is needed to predict the geodesic paths in the resultant spacetime geometry of both light rays (null geodesics) and also small massive objects (timelike geodesics) that do not significantly contribute to the matter content of a particular model (“test particles”).3 For the purpose of (re-)constructing the metric tensor field g it is necessary to introduce a set of local coordinates in the neighbourhood of every event on ( M, g, u/c ). If the introduction of local coordinate systems is intended, then it is advisable to do so in a geometrical way, which reflects specific features of the problem at hand. In the present context it is assumed that there exists a preferred timelike reference congruence u/c. The natural way of introducing local coordinates adapted to this particular structure is the 1 + 3 threading approach, recently discussed by Jantzen et al (1992) [81] and Boersma and Dray (1995) [14]. In this approach one expresses the 1+3 ONF spanning basis fields as follows ( e0 = u/c ): e0

= M −1 ∂ct

(4.37)

eα

= eα i (Mi ∂ct + ∂i ) .

(4.38)

This leads to an expression for the inverse metric tensor field of ( M, g, u/c ) given by g−1

= − e0 ⊗ e0 + δ αβ eα ⊗ eβ = − M −2 ∂ct ⊗ ∂ct + δ αβ eα i eβ j (Mi ∂ct + ∂i ) ⊗ (Mj ∂ct + ∂j ) ,

(4.39)

where M = M (ct, xi ) is the threading lapse function and Mi dxi = Mi (ct, xi ) dxi is the threading shift 1-form ( see Refs. [81] and [14] ). The triad vector components eα i = eα i (ct, xi ) describe the threading metric γ ij = δ αβ eα i eβ j ( this is just the spatial projection tensor expressed in local coordinates; for a discussion on its physical significance, see Refs. [81], [14] and [91] ). Thus, in the present 1 + 3 threading approach, ct is a coordinate along all worldlines within the preferred timelike reference congruence u/c, while the arbitrary parameters xi label the different worldlines themselves ( i.e., xi = const on an individual one ). This is the “dual” formulation to the more familiar 3 + 1 slicing decomposition of ( M, g ) introduced by Arnowitt, Deser and Misner (ADM) in 1962 [2]. In that formulation one imposes a causal condition on 3-surfaces, which are assumed to be spacelike, while no condition is imposed on the flow of time lines threading the 3-surfaces. In the threading approach one imposes a causal condition on the preferred reference congruence u/c, which is assumed to be timelike, while no causal condition is imposed on the 3-surfaces. Inserting the 1 + 3 ONF spanning basis fields in the form of Eqs. (4.37) and (4.38) into the commutators (4.8) and (4.9) yields relations for the commutation functions in terms of the variables M , Mi and eα i . One arrives at eα i (Mi ),ct + M −1 (M,i + Mi M,ct ) = u˙ α /c2 (4.40) (eα i ),ct

M e[α i eβ] j (Mi,j + Mi,ct Mj )

3 On

2 e[α i (eβ] j ),i + (eβ] j ),ct Mi eγ j

= − M eβ i

1 3

(Θ/c) δ βα + σ βα /c

− βαγ (ω γ /c + Ωγ /c)

= − αβγ ω γ /c

=

2 a[α δ γβ] + αβδ nδγ ,

(4.41)

(4.42)

(4.43)

the question of viability of the notion of “test particles” see, e.g., recent work by Tavakol and Zalaletdinov (1996) [153].


78

where f,ct := ∂ct f , f,i := ∂i f , and where eα i is defined through the relation eα i eα j = δ j i .

(4.44)

Note that one can insert the expression for (eα i ),ct given in Eq. (4.41) into Eq. (4.43), thus obtaining an expression involving only spatial derivatives. Note also that Mi 6= 0, if one has a rotating preferred timelike reference congruence u/c, ω α /c 6= 0, which is the case when the local rest 3-spaces do not mesh together to form spacelike 3-surfaces everywhere orthogonal to u/c. Of course, one still has a considerable freedom in choosing the threading shift Mi . This freedom just corresponds to the freedom of choosing a foliation such that the preferred timelike reference congruence u/c is nowhere tangent to it. If the preferred timelike reference congruence u/c is non-rotating, ω α /c = 0, it naturally gives rise to a foliation of ( M, g, u/c ) by spacelike 3-surfaces (“slices”) everywhere orthogonal to it. In this case one can choose to focus on the 3-surfaces rather than u/c. This situation suggests to take the ADM 3 + 1 slicing point of view in which one introduces local spatial coordinates on the 3-surfaces, while introducing flow of time lines that do not, in general, follow the original preferred timelike reference congruence u/c. The different 3-surfaces themselves are labelled by an arbitrary parameter ct ( i.e., ct = const on an individual one ). For an introduction into the ADM 3 + 1 slicing approach see, e.g., Refs. [81] and [14], as well as the standard texts by Misner, Thorne and Wheeler (1973) [114] and York (1979) [165]. In the following, the spatial frame { eα } is expressed with respect to a non-coordinate basis rather than a coordinate one, since this is useful in the context of, e.g., Bianchi cosmology ( see, e.g., MacCallum (1973) [104] or Wainwright and Ellis (1996) [160] ). In the slicing approach the 1 + 3 ONF spanning basis fields can be expressed as follows ( e0 = u/c ):4 e0

¯i e ¯i ) = N −1 (∂ct − N

(4.45)

eα

¯i . = e¯α i e

(4.46)

This leads to an expression for the inverse metric tensor field of ( M, g, u/c ) given by g−1

= − e0 ⊗ e0 + δ αβ eα ⊗ eβ ¯i e ¯j e ¯i ) ⊗ (∂ct − N ¯j ) + δ αβ e¯α i e¯β j e ¯i ⊗ e ¯j , = − N −2 (∂ct − N

(4.47)

¯i = N ¯ i (ct, xi ) is the shift vector. Note that the shift where N = N (ct, xi ) is the lapse function and N vector is proportional to the degree of tilt between the directions given by ∂ct and e0 = u/c. The inverse e¯α i (ct, xk ) to e¯α j (ct, xk ) ( with e¯α i e¯α j = δ j i ) yields the metric tensor gij = δαβ e¯α i e¯β j on ¯i = e¯i j (xk ) ∂j . This basis yields the the spacelike 3-surfaces expressed in the non-coordinate basis e commutation relations ¯i ] [ ∂ct , e ¯i , e ¯j ] [e

= 0 =

¯k γ¯ kij (xl ) e

(4.48) .

(4.49)

4 Note that, deviating from the conventions set up in Appendix A, here i, j, k are also used as spatial indices for the noncoordinate basis.

4.5. COMMENTS ON THE 1 + 3 ONF DYNAMICAL EQUATIONS

79

Inserting the 1 + 3 ONF spanning basis fields in the form of Eqs. (4.45) and (4.46) into the commutators (4.8) and (4.9) and setting ω α /c = 0 yields relations for the remaining commutation functions in ¯ i and e¯α i and the frame derivatives e ¯i . One obtains terms of the variables N , N ¯i (N ) = u˙ α /c2 N −1 e¯α i e

(¯ eα i ),ct

e¯γ k

¯i (¯ 2 e¯[α i e eβ] k ) + e¯[α i e¯β] j γ¯ kij

(4.50)

¯j e ¯ i) + N ¯ j e¯α k γ¯ ijk ¯j (¯ ¯j (N = N eα i ) − e¯α j e − N e¯β i 31 (Θ/c) δ βα + σ βα /c − βαγ Ωγ /c (4.51) = 2 a[α δ γβ] + αβδ nδγ .

(4.52)

The slicing point of view will be employed in an example of a 1 + 3 ONF formulation of LRS class II perfect fluid spacetime geometries in subsection 5.2.1 of Chapter 5, where the spatial frame { eα } will be given with respect to a local coordinate basis.

4.5

Comments on the 1 + 3 ONF dynamical equations

The usual initial value problem in GR is associated with a ADM 3+1 slicing decomposition of ( M, g ) and has been developed and investigated in broad detail ( see, e.g., York (1979) [165] ). The 1 + 3 approach with its associated threading formulation, on the other hand, is less well understood. The related difficulties have been discussed, e.g., by Jantzen et al [82]. Therefore, here properties of the ADM 3+1 slicing approach as applied to the present extended 1+3 ONF formulation for the case of a non-rotating ( ω α /c = 0 ) preferred timelike reference congruence u/c are discussed. In the general ADM 3+1 initial value formulation one usually uses the metric tensor defined on the spacelike 3-surfaces and their extrinsic curvature tensor as variables, together with the fluid matter variables which describe the sources. The lapse function represents the freedom in choosing a spatial foliation (choice of time), while the shift vector represents the freedom in choosing a threading (choice of spatial gauge). In the present formulation the triad components e¯α i replace the spatial metric ( gij = δαβ e¯α i e¯β j ⇒ δαβ e¯α [i e¯β j] = 0 ), while the rate of expansion scalar (Θ/c) and the rate of shear tensor σαβ /c correspond to the trace and tracefree part of the extrinsic curvature tensor, respectively. The frame derivative e0 is just the normal derivative to the spacelike 3-surfaces. The role of derivatives in the 1 + 3 (as any other) ONF formulation becomes trivial, since all frame components of any tensor field are spacetime scalars. Note ¯i e ¯i ) , eα = e¯α i e ¯i ). that it is only e0 that contains a temporal partial derivative ( e0 = N −1 (∂ct − N Thus, one obtains an evolution equation in standard form from a e0 frame derivative equation simply by moving its shift part over to the righthand side. Consequently, equations involving e0 are referred to as evolution equations, while equations not involving e0 are said to be spatial constraints. This terminology is analogous to the usage of Ellis (1971) [45] in the 1 + 3 covariant formulation, even though, as was mentioned earlier, the definition and interpretation of an initial value problem in the rotating 1 + 3 case is not clear. It is seen that by extending the number of variables from the triad, rate of expansion, rate of shear and fluid matter variables to include all commutation functions and also the Weyl curvature variables


80

one obtains additional evolution equations as well as further constraints. However, there are no evolution equations directly obtained for the acceleration of u/c, u˙ α /c2 , and the Fermi-rotation, Ωα /c. This apparent incompleteness of the picture arises from the fact that in general the preferred timelike reference congruence u/c could be chosen so as to obtain any convenient value for the acceleration u˙ α /c. This freedom is also seen when choosing local coordinates, in particular, one can choose any lapse function that may turn out to be useful ( cf. Eq. (4.50) ). However, if, as was assumed to hold throughout, there exists a fluid matter source whose average 4-velocity can be identified with u/c, this might provide an evolution equation for the (fluid) acceleration, u˙ α /c2 . For example, in the perfect fluid case, where 0 = q α /c = παβ , the spatial pressure gradient determines the fluid acceleration. If one has a barotropic equation of state, p = p(µ), one can obtain an evolution equation for u˙ α /c2 by using the commutators in conjunction with the fluid matter source equations of motion (4.35) and (4.36). This leads to e0 (u˙ α /c2 )

∂p ∂p = eα ( ∂µ (Θ/c) ) + ( ∂µ −

1 3

) (Θ/c) u˙ α /c2

− [ σ βα /c − βαγ (ω γ /c + Ωγ /c) ] u˙ β /c2 ,

(4.53)

which, alternatively, can be obtained by applying the 1 + 3 ONF conversion procedure to the 1 + 3 covariant equation (2.79) of Chapter 2, using the translation rules given in subsection B.2.1 of Appendix B. As the 1 + 3 ONF formalism does not prescribe the way of propagation of a spatial frame { eα } along an individual worldline with tangent u/c, the lack of an evolution equation for the Fermi-rotation Ωα /c can be anticipated, since Ωα /c represents the freedom in the choice of { eα }. In this way, it plays a role analogous to the shift vector, which represents the freedom in identifying the spatial local coordinates of two points on adjacent spacelike 3-surfaces within a complete foliation of ( M, g ). As mentioned before, one possible choice for the Fermi-rotation is Ωα /c = 0, i.e., choosing a Fermitransported spatial frame { eα }. Nevertheless, in view of physical and mathematical considerations the (non-rotating) Fermi-frame may not necessarily be the best choice of spatial frame in describing the dynamical properties of a particular spacetime geometry at hand. Other choices may bring about more helpful simplifications in the dynamical equations. In that respect, e.g., spatial frames in which the fluid rate of shear tensor or the “electric” part of the Weyl curvature tensor remain diagonal throughout the entire evolution provide possible alternatives. These spatial frames are sometimes called, respectively, shear eigenframes and tidal eigenframes. Of considerable interest is also the “counter-rotating” spatial frame, defined by Ωα /c = − ω α /c. The absence of separate evolution equations for some of the fluid matter variables in both the imperfect and the perfect case and the arising need to link the full 1 + 3 formalism with a relativistic thermodynamical treatment of the matter degrees of freedom was already commented on in Section 2.4 of Chapter 2, to which the reader is referred. If the 1 + 3 ONF { ea } is explicitly expressed in terms of the lapse function, shift vector and the triad variables (forming the tetrad variables) and one calculates the EFE for a given fluid matter source using the metric tensor field g as derived from inverting Eq. (4.47), one obtains a system of quasilinear second-order partial differential equations. The commutation functions are defined through the commutators, and the Jacobi identities are identically satisfied. However, one can instead view the commutation functions as a set of new independent variables. In this case one obtains a first-order

4.5. COMMENTS ON THE 1 + 3 ONF DYNAMICAL EQUATIONS

81

system of equations in all frame derivatives, constituted by the commutators, the EFE and the Jacobi identities, the latter of which are now “promoted” to non-trivial field equations. The Ricci identities provide a definition for the Riemann curvature tensor. Using this definition one finds that the second Bianchi identities are identically satisfied. However, in principle the Weyl curvature is a directly measurable object and therefore of immediate physical interest. This makes it desirable to consider the Weyl curvature components as variables as well. Thus, one can continue and view the Weyl curvature components as independent variables. In this case the second Bianchi identities are “promoted” to non-trivial field equations. Note that the system of equations for the tetrad variables, commutation functions and fluid matter and Weyl curvature variables is highly redundant. From a physical point of view, in the present formulation it would be desirable to start at the bottom of the derivative ladder5 with the fluid matter and Weyl curvature variables and the second Bianchi identities (including their twice-contracted form) as the relevant field equations. Then the question arises as to whether one can go “upwards” to lower derivatives and obtain a complete description of a spacetime geometry solely by adding the irreducible quantities associated with the covariant derivative of u/c ( cf. Eq. (2.11) in Chapter 2 ) as extra dynamical variables and the Ricci identities as extra field equations. In general, the answer is, unfortunately, no, and one will always need some tetrad variables, commutation functions, commutator equations and Jacobi identities as well. The redundancy, in the general case, is described by the Papapetrou identities ( see Papapetrou (1971) [120, 121] and, for a pedagogical review, Edgar (1980) [36] ). Thus, one can pick out interesting nonredundant subsets of dynamical variables and equations in the general case. However, for practical reasons one is usually interested in special cases, e.g., in algebraically special Petrov types of the Weyl curvature tensor. Unfortunately, these special cases need further consideration and one might have to consider higher derivatives of the dynamical equations in order to check for their consistency. The “silent” cosmological models later discussed in Section 5.1 of Chapter 5 are examples of this. In the context of the derivative picture it should be pointed out that one can take the opposite route. Starting from given values of the dynamical variables at the Riemann curvature (or commutation function) level, one can go “downwards” towards higher (covariant) derivatives of both the dynamical variables and their field equations and obtain a complete description of a spacetime geometry after a finite number of steps. This is (an inverted form of) the equivalence problem approach to invariantly classifying spacetime geometries along the lines of work by Cartan (1946) [27] and Karlhede (1980) [84]. In this approach one successively obtains the information needed to determine the components of the 1 + 3 ONF spanning basis vector fields ea and their dual 1-form fields ω a , i.e., after a finite number of covariant derivatives one has obtained all the information needed to construct the full metric tensor field g ( see Bradley and Karlhede (1990) [16] ). 5 In

this image: derivatives of the underlying metric g.


82

4.6 Irreducible tracefree decomposition In practical applications of the 1 + 3 ONF formalism it proves to be helpful to introduce a new set of variables adapted to the tracefree condition and the magnitude A2 :=

1 2

Aαβ Aβα ≥ 0 of any 1 + 3

invariantly defined rank 2 symmetric tracefree spacelike tensor field A orthogonal to u/c. In particular, with respect to all equations here in Chapter 4, Aαβ represents the variables σαβ /c, Eαβ , Hαβ , and παβ . A new set of irreducible frame components for any A is defined by ( cf. Wainwright and Ellis (1996) [160] ) A+ := − 23 A11 = A1 :=

√

3 2

√

(A22 + A33 )

3 A23

A− := A2 :=

3 2

√

(A22 − A33 )

3 A31

A3 :=

√

(4.54) 3 A12 .

These frame components have the property that the magnitude of A assumes the explicit form A2 =

1 3

[ (A+ )2 + (A− )2 + (A1 )2 + (A2 )2 + (A3 )2 ] .

(4.55)

It should be emphasised that this decomposition arbitrarily adapts to the spatial e1 -axis. However, this is a matter of pure convention, and by a cyclic permutation of indices 1 → 2 → 3 → 1 one can easily adapt to any of the remaining two other spatial axes as well. In principle, one can extend this procedure to also include the spatial commutation functions nαβ , i.e., splitting them into a trace and a tracefree part, where the latter is further subdivided according to the scheme above. However, as nαβ usually has no immediate counterpart in the 1+3 covariant picture, this possibility will not be pursued here any further.

4.7

Conclusion

Chapter 4 motivated and reviewed the 1 + 3 ONF approach to the description of the dynamical properties of spacetime geometries in relativistic cosmology. To bring it in line with the complementary 1 + 3 covariant framework discussed in Chapter 2, it was extended to include the full second Bianchi identities, which serve as field equations for the Weyl curvature variables. In its structure the extended 1 + 3 ONF approach is set up in close analogy to the well-known Newman–Penrose null frame formalism. Related to the presence of a preferred timelike reference congruence u/c, in this chapter ways of introducing local coordinates on ( M, g, u/c ) were outlined in the context of both the 1 + 3 threading and ( for ω α /c = 0 ) the ADM 3 + 1 slicing point of view. Finally, a useful set of irreducible frame components for 1 + 3 invariantly defined rank 2 symmetric tracefree spacelike tensors A orthogonal to u/c was introduced. The 1 + 3 ONF dynamical equations in extended form suggest various applications. For example, in order to gain further insight into the inter-dependences between reductions in internal symmetries of simple cosmological models and a related increase in its degrees of freedom, it would be useful to have the Cartan–Karlhede equivalence problem approach to the description of invariant features of spacetime geometries adapted to the very intuitive 1 + 3 framework. It is conceivable that such a reformulation could additionally reveal further interesting dynamical properties of different spacetime geometries and provide hints to a clearer picture of the definition and propagation of gravitational waves in the cosmological setting. A 1 + 3 ONF adaptation of the Cartan–Karlhede framework may obtain support from a constructive combination with recent considerations by Siklos (1996) [139] on the maximum

4.7. CONCLUSION

83

number of freely specifiable components in the n-th (covariant) derivative of a given set of dynamical field variables, after their evolution equations and constraints and potentially existing gauge freedom have been taken into account. All of these remarks are also relevant to the widely increasing attempts in numerical relativity that address evolutionary features of fluid spacetime geometries. As overall the 1+3 framework employs more physically interesting dynamical variables than the standard ADM 3+1 slicing treatment has to offer, it would be interesting to obtain numerical simulations that are based on the alternative 1 + 3 concept. Wainwright and collaborators have shown that for an expanding (contracting) preferred timelike reference congruence u/c ( where (Θ/c) 6= 0 ) it proves very useful to transform the 1 + 3 ONF dynamical equations into dimensionless form by expansion-normalising all dependent variables and the derivative operators ( see Wainwright and Ellis (1996) [160] ). In this way it is possible to compactify the state space of various cosmological models such that their past and future asymptotic behaviour may be elegantly investigated, which in some cases is of self-similar character. Wainwright et al so far restricted their considerations to spacetime geometries with either a G3 or an Abelian G2 isometry group acting simply-transitively on spacelike orbits. As they play no further role in this thesis, the extended 1 + 3 ONF dynamical equations (4.8) - (4.9), (4.14) - (4.30), and (4.31) - (4.36) are listed in Appendix D in expansion-normalised dimensionless form, thus generalising the 1 + 3 ONF dynamical equations given by the aforementioned authors. In Chapter 5 the extended 1+3 ONF dynamical equations are used to investigate the integrability of a spatial constraint that arises when the so-called “silent” condition is imposed on a spacetime geometry with an irrotational dust fluid matter source, i.e., the condition that demands that the “magnetic” part of its Weyl curvature be zero. To provide a further example of application of the extended 1 + 3 ONF formalism, the LRS spacetime geometries are reformulated in this language and briefly compared to their 1 + 3 covariant counterpart.

84


Chapter 5

Special Perfect Fluid Spacetime Geometries. II. 5.1

So-called “silent” models of the Universe

The idea behind the “silent” cosmological models, introduced and discussed by Matarrese et al (1994/95) ( see Refs. [111], [20], and [21] ), is the following. Starting from any consistent initial configuration for the idealised barotropic perfect fluid spacetime geometries, there exist two physically different phenomena which can convey information between adjacent worldlines within the preferred timelike reference congruence u/c, which, as before, is identified with the average 4-velocity of the fluid matter source. These are either sound waves or gravitational waves. Mathematically these dynamical interactions are represented by the spatial derivative source terms on the righthand sides of the 1 + 3 covariant evolution equations of the models. A careful look at the righthand sides of the perfect fluid reduced time derivative equations in Chapter 2, Eqs. (2.25) - (2.27) between the Ricci identities and Eqs. (2.31) - (2.34) between the second Bianchi identities, reveals that these terms are comprised of either the spatial derivatives of the fluid acceleration, or the spatial rotation terms of both the “electric” and “magnetic” parts of the Weyl curvature. Given the barotropic equation of state assumption, the fluid acceleration, and consequently the spatial 3-gradient of the fluid pressure, are linked to the spatial 3-gradient of the fluid’s total energy density, as seen from Eq. (2.43). Pressure 3gradients and their spatial variations generate propagating sound waves within the fluid matter source, while non-zero values of the spatial rotation of one of the Weyl curvature variables generically induce temporal changes in the other (and vice versa), usually interpreted as propagating gravitational waves. Both from a mathematical and a physical point of view it is of interest and tempting to investigate the case in which these spatial derivative terms vanish such that the resultant 1 + 3 covariant evolution equations reduce to a coupled set of first-order ordinary differential equations. That is, provided the spatial constraints are satisfied in initial rest 3-spaces and remain satisfied, the subsequent evolution along individual worldlines within u/c is decoupled from each other. The fully orthogonally projected covariant time derivatives decouple from the spatial ones. This technically describes what is called the “silent” assumption for cosmological models: the absence of any form of waves (hence, any form of communication) propagating between the worldlines of neighbouring fluid elements within u/c. To 85

86

CHAPTER 5. SPECIAL PERFECT FLUID SPACETIME GEOMETRIES. II.

realise it, Matarrese et al assumed the fluid matter source to be irrotational dust, which generates a spacetime geometry of purely “electric” Weyl curvature, p = 0 ⇒ (u/c ˙ 2) = 0

(ω/c) = 0

H =0.

(5.1)

Well-known examples of exact solutions of the EFE fall into this category. Spatially inhomogeneous representatives are Ellis’ dust subclass of LRS class II spacetime geometries, which contain the Lemaˆıtre–Tolman–Bondi model ( see Ref. [44] and cf. subsection 3.1.8 of Chapter 3 ), as well as the family of dust spacetime geometries given by Szekeres (1975) [151]. In both examples the Weyl curvature tensor is of algebraic Petrov type D ( see, e.g., Ref. [88] ). The Szekeres spacetime geometries were discussed in a nice geometrical reformulation by Goode and Wainwright (1982) [63]. A Petrov type I example of a “silent” model is provided by the dust case of OSH Bianchi Type–I ( cf. subsection 3.2.1 of Chapter 3 ). It is instructive to give the dynamical equations defining irrotational dust “silent” spacetime geometries first in 1 + 3 covariant form; this is done in subsection (5.1.1). In subsection (5.1.2) the “silent” configurations are then formulated in 1 + 3 ONF terminology. By the assumption of a vanishing “magnetic” part of the Weyl curvature a new spatial constraint is generated. The consistency of this constraint with the remaining dynamical equations is investigated in subsection (5.1.3) by making use of algebraic computing facilities. Finally, the results obtained are discussed in subsection (5.1.4).

5.1.1

1 + 3 covariant formulation

Starting from the barotropic perfect fluid reduced form of the 1+3 covariant dynamical equations (2.25) - (2.36) given in Chapter 2 and imposing the “silent” conditions for irrotational and pressure-free fluid matter sources, Eq. (5.1), one obtains the set ( with Λ = 0; see Ref. [111] ) Time derivative equations 2

(Θ/c)˙/c = − 13 (Θ/c) − 2 (σ/c)2 − hµρ

hνσ

ρσ

(σ /c)˙/c = −

hµρ hνσ (E ρσ )˙/c = −

µν

4πG c4 νρ

µ

(5.2)

(Θ/c) σ /c − /c − E + 23 (σ/c)2 hµν µν µν 4πG + 3 σ (µρ /c E ν)ρ − σ ρσ /c E σρ c4 µ σ /c − (Θ/c) E 2 3

σ µρ /c σ

µν

µ/c ˙ = − (Θ/c) µ .

(5.3) hµν (5.4) (5.5)

Constraint equations = hµρ hνσ (∇ν σ ρσ /c) −

0

hµρ

hνσ

(∇ν E

ρσ

)−

2 3

Zµ

2 4πG 3 c4

(5.6) µ

0

=

0

= − µνρσ σντ /c E τρ uσ /c

(5.8)

0

= Kµν

(5.9)

0

= Iµν .

(5.10)

X

(5.7)

Gauß equations and 3-Cotton–York tensor 3

Sµν 3

R

3

Cµν

= Eµν −

1 3


2 3

(σ/c)2 hµν

(Θ/c) + 2 (σ/c)2 = − h1/3 hρ(µ hσν) ρτ κλ ∇τ (σ κξ /c σ ξσ /c) − =

4 4πG c4 µ −

2 3

(5.11)

2

(5.12) 1 3

Z

τ

σ κσ /c

λ

u /c .

(5.13)

5.1. SO-CALLED “SILENT” MODELS OF THE UNIVERSE

87

The constraints in this setting have the following implications. The spatial divergence of the fluid rate of shear is related to the spatial 3-gradient of the fluid rate of expansion, Eq. (5.6), and, analogously, the spatial divergence of the “electric” part of the Weyl curvature to the spatial 3-gradient of the fluid’s total energy density, Eq. (5.7). Both the fluid rate of shear tensor and the “electric” part of the Weyl curvature tensor share a common eigenframe, a property expressed by Eq. (5.8). This result was originally obtained by Barnes and Rowlingson (1989) [5]. Additionally, the constraints (5.9) and (5.10) express the condition that in “silent” configurations as defined by Eq. (5.1) these tensors need to have zero spatial rotation. It was pointed out before that it follows from the analysis of the spatial constraints for barotropic perfect fluid spacetime geometries given in Section 2.7 of Chapter 2, that the constraints (5.6) - (5.9) are preserved along the integral curves of the preferred timelike reference congruence u/c, if all of the constraints (5.6) - (5.10) are satisfied at an initial instant. Equations (5.6) - (5.9) are just reduced forms of the constraints underlying general barotropic perfect fluid spacetime geometries ( cf. Eqs. (2.66) and (2.68) - (2.70) in Chapter 2 ). The character of Eq. (5.10), however, is slightly different. It is a new spatial constraint arising as a consequence of imposing the “silent” conditions (5.1). Namely, the vanishing of the “magnetic” part of the Weyl curvature results in the reduction of the general evolution equation (2.32) in Chapter 2 to become Eq. (5.10). Together with Eq. (2.43) in Chapter 2 and Eq. (3.287) in Chapter 3, this case provides a third example of the conversion process where the supposition that in a particular spacetime geometry a certain dynamical variable shall remain zero throughout, a further integrability condition is obtained from its related general evolution equation. Consistency of Eq. (5.10) with the remaining set of equations is given, if 0 = hµρ hνσ (I ρσ )˙/c

(5.14)

holds throughout. With Eq. (2.63) in Chapter 2 this is equivalent to 0

= h(µρ hν)σ ρτ κλ σ ξτ /c (∇ξ Eκσ ) uλ /c − 3 h(µρ hν)σ ρτ κλ ∇τ [ σ ξ(κ /c Eσ)ξ ] uλ /c −[

4πG c4

σ (µρ /c Xσ + E (µρ Zσ ] ν)ρστ uτ /c .

(5.15)

Lesame et al (1995) [95] recently reported that this condition was identically satisfied. However, below a different conclusion is obtained. After reformulating the set of dynamical equations (5.2) - (5.10) in terms of 1 + 3 ONF variables (which helps to facilitate the subsequent computations), a detailed analysis of the consistency of Eq. (5.10) with the remaining set is given. A spatial constraint similar to Eq. (5.10) arises in the context of the work by Barnes and Rowlingson (1989) [5], who, among the restrictions (5.1), allowed the fluid pressure to be non-zero instead (in which case, of course, a configuration no longer can be called “silent”). However, in their work the integrability of that constraint was never established.

88

5.1.2


1 + 3 ONF formulation

It was proved by Barnes and Rowlingson (1989) [5] that under the given assumptions, Eq. (5.1), a canonical 1 + 3 ONF can be set up in which the fluid rate of shear tensor and the “electric” part of the Weyl curvature tensor are simultaneously diagonal; this follows from Eq. (5.8). Using the tracefreeadapted irreducible frame components introduced in Section 4.6 of Chapter 4 this implies ⇒

0 = A1 = A2 = A3

A2 =

1 3

[ (A+ )2 + (A− )2 ]

(5.16)

for the frame components of both tensors. Then it follows from the on-diagonal components of both the H-constraint (4.28) and the evolution equation for the “magnetic” part, Eq. (4.32), that 0 = n11 = n22 = n33 .

(5.17)

Combining Eqs. (5.16) and (5.17) and bearing in mind that (ω/c) = 0, one obtains with Eq. (4.10) that the 1 + 3 ONF { ea } is spanned by four individually HSO basis fields, which implies that local coordinates can be found on ( M, g, u/c ) with respect to which the metric tensor field g is diagonal. Again, this result was proved by Barnes and Rowlingson, and it holds for Weyl curvature tensors of either algebraic Petrov type I or type D. These authors also showed that the spatial frame { eα } is Fermi-transported along u/c, Ωα /c = 0 ,

(5.18)

a condition obtained from the off-diagonal components of the evolution equations of both the fluid rate of shear, Eq. (4.15) , and the “electric” part, Eq. (4.31). Given these specialisations and using the tracefree-adapted irreducible frame components of Section 4.6, the following set describing “silent” cosmological models according to conditions (5.1) can be derived from the general 1 + 3 ONF dynamical equations of Chapter 4: The commutators: = − 13 ( (Θ/c) − 2 σ+ /c ) e1 √ [ e0 , e2 ] = − 31 ( (Θ/c) + σ+ /c + 3 σ− /c ) e2 √ [ e0 , e3 ] = − 31 ( (Θ/c) + σ+ /c − 3 σ− /c ) e3

[ e0 , e1 ]

(5.19) (5.20) (5.21)

[ e1 , e2 ]

= − (a2 − n31 ) e1 + (a1 + n23 ) e2

(5.22)

[ e2 , e3 ]

= − (a3 − n12 ) e2 + (a2 + n31 ) e3

(5.23)

[ e3 , e1 ]

= − (a1 − n23 ) e3 + (a3 + n12 ) e1 .

(5.24)

Coupled subsystem of ordinary differential evolution equations: e0 (Θ/c)

2

= − 13 (Θ/c) − 2 (σ/c)2 − 1 3

4πG c4

µ 1 3

e0 (σ+ /c)

= − ( 2 (Θ/c) − σ+ /c ) σ+ /c −

e0 (σ− /c)

= − 23 ( (Θ/c) + σ+ /c ) σ− /c − E− 4πG c4

(5.25) 2

(σ− /c) − E+

(5.26) (5.27)

e0 (E+ )

= −

µ σ+ /c − ( (Θ/c) + σ+ /c ) E+ + σ− /c E−

(5.28)

e0 (E− )

= − 4πG c4 µ σ− /c − ( (Θ/c) − σ+ /c ) E− + σ− /c E+

(5.29)

= − (Θ/c) µ .

(5.30)

e0 (µ)

89

5.1. SO-CALLED “SILENT” MODELS OF THE UNIVERSE Note that this set implies σ− /c = 0 ⇔ E− = 0.

Remaining decoupled system of ordinary differential evolution equations: = − 13 ( (Θ/c) + σ+ /c ) a1 − √13 σ− /c n23 √ e0 (a2 ) = 61 ( −2 (Θ/c) + σ+ /c + 3 σ− /c ) a2 − √ e0 (a3 ) = 61 ( −2 (Θ/c) + σ+ /c − 3 σ− /c ) a3 +

e0 (a1 )

(5.31) 1 2

(σ+ /c −

1 2

(σ+ /c +

= − 13 ( (Θ/c) + σ+ /c ) n23 − √13 σ− /c a1 √ e0 (n31 ) = 16 ( −2 (Θ/c) + σ+ /c + 3 σ− /c ) n31 − √ e0 (n12 ) = 61 ( −2 (Θ/c) + σ+ /c − 3 σ− /c ) n12 +

√1 3 √1 3

σ− /c) n31

(5.32)

σ− /c) n12

(5.33)

e0 (n23 )

(5.34) 1 2

(σ+ /c −

1 2

(σ+ /c +

√1 3 1 √ 3

σ− /c) a2

(5.35)

σ− /c) a3 .

(5.36)

Tracefree part and trace of 3-Ricci curvature of spacelike 3-surfaces orthogonal to u/c ( 0 = ∗S1 = ∗

S2 = ∗S3 ): ∗

= − 21 [ 2 e1 (a1 ) − e2 (a2 ) − e3 (a3 ) − 4 (n23 )2 + 2 (n31 )2 + 2 (n12 )2

S+

− 3 (e2 − 2 a2 ) (n31 ) + 3 (e3 − 2 a3 ) (n12 ) ] = E+ − √

∗

S−

=

3 2

1 3

( (Θ/c) + σ+ /c ) σ+ /c +

1 3

(σ− /c)2

(5.37)

[ e2 (a2 ) − e3 (a3 ) − 2 (n31 )2 + 2 (n12 )2 + 2 (e1 − 2 a1 ) (n23 ) − (e2 − 2 a2 ) (n31 ) − (e3 − 2 a3 ) (n12 ) ]

= E− −

1 3

( (Θ/c) − 2 σ+ /c ) σ− /c

(5.38)

0

=

(e2 − 2 a2 ) (n12 ) − (e3 − 2 a3 ) (n31 ) + e2 (a3 ) + e3 (a2 ) + 4 n31 n12

(5.39)

0

=

(e3 − 2 a3 ) (n23 ) − (e1 − 2 a1 ) (n12 ) + e3 (a1 ) + e1 (a3 ) + 4 n12 n23

(5.40)

0

=

(e1 − 2 a1 ) (n31 ) − (e2 − 2 a2 ) (n23 ) + e1 (a2 ) + e2 (a1 ) + 4 n23 n31

(5.41)

R

=

2 [ (2 e1 − 3 a1 ) (a1 ) + (2 e2 − 3 a2 ) (a2 ) + (2 e3 − 3 a3 ) (a3 )

∗

− (n23 )2 − (n31 )2 − (n12 )2 ] =

4 4πG c4 µ −

2 3

2

(Θ/c) + 2 (σ/c)2 .

(5.42)

3-Cotton–York tensor of spacelike 3-surfaces orthogonal to u/c ( 0 = ∗C+ = ∗C− ): ∗

=

∗

= − √13 (σ+ /c − √13 σ− /c) (e2 − a2 ) (σ− /c) √ 1 + 3 n31 (σ+ /c + 3√ σ /c) σ− /c 3 −

(5.44)

= − √13 (σ+ /c + √13 σ− /c) (e3 − a3 ) (σ− /c) √ 1 − 3 n12 (σ+ /c − 3√ σ /c) σ− /c . 3 −

(5.45)

C1

C2

∗

C3

√1 3

σ− /c (e1 − a1 ) (σ+ /c) + n23

(σ+ /c)2 −

2 3

(σ− /c)2

(5.43)

90


The constraint equations:

0 =

√ (e1 − 3 a1 ) (σ+ /c) − 3 n23 σ− /c + e1 (Θ/c) √ (e2 − 3 a2 ) (σ+ /c + 3 σ− /c) + 3 n31 (σ+ /c − √13 σ− /c) − 2 e2 (Θ/c) √ (e3 − 3 a3 ) (σ+ /c − 3 σ− /c) − 3 n12 (σ+ /c + √13 σ− /c) − 2 e3 (Θ/c) √ (e1 − 3 a1 ) (E+ ) − 3 n23 E− + 4πG c4 e1 (µ) √ (e2 − 3 a2 ) (E+ + 3 E− ) + 3 n31 (E+ − √13 E− ) − 2 4πG c4 e2 (µ) √ (e3 − 3 a3 ) (E+ − 3 E− ) − 3 n12 (E+ + √13 E− ) − 2 4πG c4 e3 (µ)

0 =

(e2 − 2 a2 ) (n12 ) + (e3 − 2 a3 ) (n31 ) + e2 (a3 ) − e3 (a2 )

(5.52)

0 =

(e3 − 2 a3 ) (n23 ) + (e1 − 2 a1 ) (n12 ) + e3 (a1 ) − e1 (a3 )

(5.53)

0 =

(e1 − 2 a1 ) (n31 ) + (e2 − 2 a2 ) (n23 ) + e1 (a2 ) − e2 (a1 ) √ (e1 − a1 ) (σ− /c) − 3 n23 σ+ /c √ (e2 − a2 ) (σ+ /c − √13 σ− /c) + n31 (σ+ /c + 3 σ− /c) √ (e3 − a3 ) (σ+ /c + √13 σ− /c) − n12 (σ+ /c − 3 σ− /c) √ (e1 − a1 ) (E− ) − 3 n23 E+ √ (e2 − a2 ) (E+ − √13 E− ) + n31 (E+ + 3 E− ) √ (e3 − a3 ) (E+ + √13 E− ) − n12 (E+ − 3 E− ) .

(5.54)

0

=

0

=

0

=

0

=

0

=

0 = 0

=

0

=

0

=

0

=

0

=

(5.46) (5.47) (5.48) (5.49) (5.50) (5.51)

(5.55) (5.56) (5.57) (5.58) (5.59) (5.60)

Here the following correspondences exist between the 1 + 3 ONF form and the 1 + 3 covariant form of the spatial constraints: Eqs. (5.46) - (5.48) correspond to Eq. (5.6), Eqs. (5.49) - (5.51) to Eq. (5.7), Eqs. (5.55) - (5.57) to Eq. (5.9), and Eqs. (5.58) - (5.60) to Eq. (5.10), respectively. Equations (5.52) (5.54) follow from the Jacobi identities (4.24) in Chapter 4. By use of the commutators (5.19) - (5.21) and re-substitution from known relations, one can show that the Jacobi constraints (5.52) - (5.54) as well as the Gauß constraints (5.37) - (5.42) are preserved along u/c.

5.1.3

Constraint analysis

In 1 + 3 ONF variables the specific “silent” request for an irrotational dust fluid matter source that (additionally to the fluid rate of shear) the “electric” part of the Weyl curvature needs to be of zero spatial rotation, Eq. (5.10), takes the form of Eqs. (5.58) - (5.60). In order for the “silent” assumption, as specified by Eq. (5.1), to lead to a consistent set of dynamical equations, the zero spatial rotation condition(s) must be preserved along the integral curves of the preferred timelike reference congruence u/c. The constraints (5.58) - (5.60) are propagated along u/c by application of the commutators (5.19) - (5.21), and it is straightforward to show that they will be preserved, given that the conditions 0

= E− e1 (Θ/c) +

4πG c4

− 2 (a1 σ+ /c + 0

=

(E+ −

√1 3

σ− /c e1 (µ) − 2 (a1 σ− /c + √1 3

√

3 n23 σ+ /c) E+

n23 σ− /c) E−

E− ) e2 (Θ/c) +

4πG c4

(σ+ /c −

(5.61) √1 3

σ− /c) e2 (µ)

+ 2 (a2 − n31 ) (σ+ /c + √13 σ− /c) E+ √ + √23 a2 (σ+ /c − 3 σ− /c) E− − √23 n31 (σ+ /c + 0 =

(E+ +

√1 3

E− ) e3 (Θ/c) +

4πG c4

(σ+ /c +

√1 3

√5 3

σ− /c) e3 (µ)

σ− /c) E−

(5.62)

91

5.1. SO-CALLED “SILENT” MODELS OF THE UNIVERSE + 2 (a3 + n12 ) (σ+ /c − √13 σ− /c) E+ √ − √23 a3 (σ+ /c + 3 σ− /c) E− − √23 n12 (σ+ /c −

√5 3

σ− /c) E−

(5.63)

are satisfied. These equations are equivalent to the 1 + 3 covariant expression (5.15) derived above. In the following it is assumed that the “electric” part of the Weyl curvature is non-zero. Petrov type D If the spacetime geometry is spatially inhomogeneous and its Weyl curvature tensor is of algebraic Petrov type D ( E− = 0 ⇒ σ− /c = 0 ), then, as Barnes and Rowlingson (1989) [5] proved, it is identical to the dust models of Szekeres (1975) [151], or special subcases thereof. This can be seen as follows. For E− = 0 ⇒ σ− /c = 0, one obtains from Eq. (5.55) that n23 = 0. Then Eq. (5.61) is identically satisfied, while Eqs. (5.62) and (5.63) yield 0

= E+ e2 (Θ/c) +

4πG c4

σ+ /c e2 (µ) + 2 (a2 − n31 ) σ+ /c E+

(5.64)

0

= E+ e3 (Θ/c) +

4πG c4

σ+ /c e3 (µ) + 2 (a3 + n12 ) σ+ /c E+ .

(5.65)

However, in that case one can derive from the spatial constraints that 0

= e2 (Θ/c) + (a2 − n31 ) σ+ /c

0 = e3 (Θ/c) + (a3 + n12 ) σ+ /c =

4πG c4

e2 (µ) + (a2 − n31 ) E+

0 =

4πG c4

e3 (µ) + (a3 + n12 ) E+ ,

0

such that Eqs. (5.64) and (5.65) are identically satisfied as well. A special subcase contained within the Szekeres family are Ellis’ LRS class II dust models ( see Ref. [44], and subsection 3.1.8 of Chapter 3 ). Here, the further conditions 0 = e2 (f ) = e3 (f ), 0 = a3 = n12 and a2 = n31 apply ( cf. subsection 5.2.1 below ), and the equations are again consistent. Petrov type I If, on the other hand, the spacetime geometry is spatially inhomogeneous and its Weyl curvature tensor is of algebraic Petrov type I ( E− 6= 0 ⇒ σ− /c 6= 0 ), then, contrary to what was claimed by Lesame et al (1995) [95], Eqs. (5.61) - (5.63) do not vanish identically, but constitute a new set of spatial constraints. Of course, they are trivially satisfied, if a Petrov type I spacetime geometry is of OSH Bianchi Type–I ( where eα (f ) = 0, 0 = aα = nαβ ). Equations (5.61) - (5.63) can be interpreted as expressions for the spatial 3-gradients of the fluid rate of expansion, eα (Θ/c). Propagating them along u/c and re-substituting from known relations, one obtains algebraic expressions for the spatial 3-gradients of the fluid’s total energy density, eα (µ), in the form e1 (µ)

= f1 [ a1 , n23 , σ+ /c, σ− /c, E+ , E− , µ ]

(5.66)

e2 (µ)

= g1 [ a2 , n31 , σ+ /c, σ− /c, E+ , E− , µ ]

(5.67)

e3 (µ)

= h1 [ a3 , n12 , σ+ /c, σ− /c, E+ , E− , µ ] .

(5.68)

92


Here, f1 , g1 and h1 are multivariate rational expressions of the variables indicated. Each individual term in the numerators therein is linear in either aα or nαβ , and all terms in the expressions contain a factor of either a power of σ− /c or a power of E− . Propagating Eqs. (5.66) - (5.68) along u/c and re-substituting from known relations, one obtains algebraic expressions for the spatial commutation functions aα in the form a1

= n23 f2 [ (Θ/c) , σ+ /c, σ− /c, E+ , E− , µ ]

(5.69)

a2

= n31 g2 [ (Θ/c) , σ+ /c, σ− /c, E+ , E− , µ ]

(5.70)

a3

= n12 h2 [ (Θ/c) , σ+ /c, σ− /c, E+ , E− , µ ] ,

(5.71)

where again f2 , g2 and h2 are multivariate rational expressions of the variables indicated, with each individual term containing a factor of either a power of σ− /c or a power of E− . Finally, propagating Eqs. (5.69) - (5.71) along u/c and re-substituting from known relations, one obtains purely algebraic constraints of the form 0 = n23 f3 [ (Θ/c) , σ+ /c, σ− /c, E+ , E− , µ ]

(5.72)

0

= n31 g3 [ (Θ/c) , σ+ /c, σ− /c, E+ , E− , µ ]

(5.73)

0

= n12 h3 [ (Θ/c) , σ+ /c, σ− /c, E+ , E− , µ ] ,

(5.74)

where f3 , g3 and h3 are high-order multivariate polynomial expressions of the variables indicated, with each individual term containing a factor of either a power of σ− /c or a power of E− . At this stage, in 1 + 3 covariant terms one has taken the fourth fully orthogonally projected covariant time derivative of the condition that the “electric” part of the Weyl curvature needs to be of vanishing spatial rotation, Eq. (5.10). It is clear from Eqs. (5.72) - (5.74) that this condition can be achieved in four different ways ( modulo a cyclic permutation of the axes of the spatial frame { eα } ), depending on the number of non-zero nαβ variables: (i) 0 = n23 = n31 = n12 ; it is straightforward to show that this case simply corresponds to the OSH dust models of Bianchi Type–I. (ii) 0 = n31 = n12 . (iii) 0 = n12 . (iv) all of n23 , n31 and n12 are non-zero. Due to the particular structure of f3 , g3 and h3 , case (iv) could be solved, e.g., by σ− /c = 0 ⇔ E− = 0, which just reproduces the previous Petrov type D situation. However, the interesting case is to see whether non-trivial solutions in the variables (Θ/c), σ+ /c, σ− /c, E+ , E− , and µ to the highly complex algebraic conditions 0 = f3 = g3 = h3 could be found. Additionally, this would involve satisfying Eqs. (5.66) – (5.71), and then showing that the time derivatives of this solution are consistent. Note that the set of time derivatives needed to prove the consistent result generically was not completed, rather the investigation ceased pursuing the consistency conditions beyond Eqs. (5.72) – (5.74) because of the number of terms involved. Given that all of the previous conditions were satisfied, this would establish the existence of spatially inhomogeneous “silent” models with a Weyl curvature tensor of algebraic Petrov type I. Each of the cases (ii) - (iv) requires further investigation.

5.1. SO-CALLED “SILENT” MODELS OF THE UNIVERSE

5.1.4

93

Conclusion

This section is concluded with the following Conjecture: On the basis of the analysis given in the previous subsection it seems unlikely that there exist spatially inhomogeneous “silent” models according to Eq. (5.1), the Weyl curvature tensor of which is of algebraic Petrov type I. In this case the “silent” assumption for irrotational dust fluid matter sources would only reproduce already known classes of spatially inhomogeneous spacetime geometries. In brief, imposing the vanishing “magnetic” part of the Weyl curvature assumption on a spatially inhomogeneous irrotational dust spacetime geometry could be too restrictive a condition as to allow for an algebraically general Weyl curvature tensor. On the other hand, recall that dropping this assumption leads to a consistent set of 1 + 3 covariant dynamical equations describing a (non-“silent”) class of spacetime geometries, as was recently shown by Maartens (1996) [101] and also demonstrated in Section 2.7 of Chapter 2. It has been suggested by Berger et al (1977) [10], that if ( M, g, u/c ) contains an irrotational fluid matter source, (ω/c) = 0, there might then exist a link between the presence of gravitational radiation therein and a non-trivial conformal curvature of the spacelike 3-surfaces orthogonal to u/c, as expressed by the 3-Cotton–York tensor. Their hypothesis arose from an investigation of the aforementioned spatially inhomogeneous, Petrov type D dust models of Szekeres (1975) [151]. In their work they demonstrated that for those spacetime geometries the spacelike 3-surfaces have a vanishing 3-Cotton–York tensor. In the 1 + 3 picture, gravitational radiation phenomena necessarily require a non-zero “magnetic” part of the Weyl curvature, as measured by observers comoving with u/c. If spatially inhomogeneous “silent” models of Petrov type I do indeed exist, then there is no a priori reason given, why their 3-Cotton–York tensor should be zero ( see Eqs. (5.43) - (5.45) ). In that case a counterexample to the Berger et al proposal were provided. However, if the conjecture given above were to hold true, then this proposal would be further strengthened. It is easy to see that the result of Berger et al for the Petrov type D Szekeres dust models is contained in Eqs. (5.43) - (5.45). A final remark should be made. The “silent” criterion for barotropic perfect fluids, as introduced by Matarrese et al (1994/95) [111, 20], and as applied in this section, demanded that in mathematical terms the evolution equations within the set of 1 + 3 covariant dynamical equations of Chapter 2 reduce to a coupled set of ordinary differential equations. However, it should be pointed out that a coupled set of ordinary differential equations describing evolutionary behaviour of relativistic cosmological models can also be obtained from the 1 + 3 ONF dynamical equations of Chapter 4. The best-known example of this kind are the OSH perfect fluid models as discussed, e.g., by Ellis and MacCallum (1969) [49] ( see also Wainwright and Ellis (1996) [160], and Section 3.2 of Chapter 3 ). Here the reduction of the evolution equations to a set of ordinary differential equations is achieved by choosing a 1+3 ONF { ea } such that it is invariant under the motions induced by the G3 isometry group of spacelike translations. Hence, according to the ordinary differential equations criterion, the OSH perfect fluid models can be called 1 + 3 ONF “silent”, but, in general, not 1 + 3 covariantly “silent”. Most of the OSH perfect fluid models have non-zero “electric” and “magnetic” parts of the Weyl curvature, E 6= 0 6= H, and, more importantly, non-zero spatial rotation terms thereof, I 6= 0 6= J, as can be seen from the results given in Section 3.2 of Chapter 3. What the spatial rotation terms in 1+3 ONF language specifically correspond

94


to for the different Bianchi types can be established in conjunction with the group classification given in Ref. [49] from the translation formula Eq. (B.19) presented in subsection B.3.1 of Appendix B.

5.2 Locally rotationally symmetric spacetime geometries The LRS spacetime geometries with barotropic perfect fluid matter source were discussed in broad detail in Section 3.1 of Chapter 3 from the perspective of a 1+3 covariant formulation. To further illustrate the extended 1 + 3 ONF approach, fluid configurations with LRS spacetime symmetries imposed are taken up again, as they are relatively simple to deal with. The original systematic analysis of the LRS fluid spacetime geometries by Ellis (1967) [44] and Stewart and Ellis (1968) [147] already employed the 1 + 3 ONF method, without, however, making use of the Weyl curvature variables. Of course, as mentioned earlier in Chapter 4, they are not essential for a complete treatment, but are of interest from a physical point of view. In this section, first, the extended 1 + 3 ONF dynamical equations of Chapter 4 for an imperfect fluid matter source and Λ = 0 are given with LRS conditions imposed. Secondly, these equations are then specialised to the non-rotating spatially inhomogeneous models with perfect fluid matter source, i.e., the models within LRS class II. The ADM 3 + 1 slicing point of view as discussed in Section 4.4 of Chapter 4 is employed to introduce a set of local coordinates for this class of models. Overall, the relations that occur in this section differ from their 1 + 3 covariant counterparts only by either numerical factors or notation. Under the given assumption of a LRS structure on the spacetime manifold, ( M, g, u/c, e ), it is natural to identify the normalised tangents of the two mutually orthogonal preferred reference congruences with two of the 1 + 3 ONF spanning basis fields. As before, the timelike direction is taken to be e0 = u/c, while now also e1 = e is chosen. This has the effect that all covariantly defined spacelike vector fields V orthogonal to u/c are colinear with e1 . Also all covariantly defined rank 2 spacelike symmetric tracefree tensor fields A orthogonal to u/c have coinciding eigenframes with two equal eigenvalues. In terms of the tracefree-adapted irreducible frame components introduced in Section 4.6 of Chapter 4 this property is represented by the relations ⇒

0 = A− = A1 = A2 = A3

A2 =

1 3

(A+ )2

⇒

A = ± √13 A+ .

(5.75)

Ellis (1967) [44] and Stewart and Ellis (1968) [147] proved that by further exploiting the frame freedom, the 1 + 3 ONF { ea } can be brought into a form such that only the following commutation functions are non-zero: (Θ/c)

u˙ 1 /c2 := u/c ˙ 2 ω1 /c = − Ω1 /c := ω/c

σ11 /c = − 2 σ22 /c = − 2 σ33 /c := − 32 σ+ /c

(5.76)

a1 := a

(5.77)

a2 = n31

n11 .

The non-zero fluid matter and Weyl curvature variables are given by { µ, p, q1 /c, π+ , E+ , H+ } .

(5.78)

This particular choice of a 1 + 3 ONF also has the consequence that frame derivatives in the e2 - and e3 -directions vanish, if they act on the frame components of any 1+3 ( 1+1+2 ) covariant geometrical object A, i.e., 0 = e2 (A) = e3 (A). Now, the commutators for LRS fluid spacetime geometries can be cast into the form:

95

5.2. LOCALLY ROTATIONALLY SYMMETRIC SPACETIME GEOMETRIES The commutators: [ e0 , e1 ]

= u/c ˙ 2 e0 −

[ e0 , e2 ]

1 3

= − ( (Θ/c) + σ+ /c ) e2

(5.80)

[ e0 , e3 ]

= − 13 ( (Θ/c) + σ+ /c ) e3

(5.81)

[ e1 , e2 ]

= a e2

(5.82)

[ e2 , e3 ]

= 2 ω/c e0 + n11 e1 + 2 n31 e3

(5.83)

[ e3 , e1 ]

= − a e3 .

(5.84)

1 3

( (Θ/c) − 2 σ+ /c ) e1

(5.79)

Note the similarities between the commutators (5.79) and (5.83) on the one hand, and Eqs. (3.48) and (3.43), respectively, on the other. Here, e3 (n31 ) = 0, and n11 is proportional to the magnitude of the spatial rotation of the preferred spacelike reference congruence e, while a relates to the magnitude of its spatial divergence. The notation for some dynamical variables used in this section slightly deviates from the one employed in Section 3.1. Those differences are summed up in Table 5.1 below. Also notational differences to the work by Stewart and Ellis are included. From the commutators (5.79) (5.84) one can immediately read off that the basis fields e2 and e3 are HSO ( cf. Eq. (4.10) in Chapter 4 ). In the zero-vorticity LRS case, ω/c = 0, it follows that the spatial frame { eα } is Fermi-transported along u/c ( cf. Eq. (5.77) ). If both 0 = ω/c = n11 , all four frame spanning basis fields ea are HSO ( cf. Eq.(4.10) ).

Stewart and Ellis (1968) [147]

Section 3.1

This section

α β a k s B2 K

√ [ (Θ/c) + 2√ 3 (σ/c) ] / 3 [ (Θ/c) − 3 (σ/c) ] / 3 − a/2 −k — K

( (Θ/c) − 2 σ+ /c ) / 3 ( (Θ/c) + σ+ /c ) / 3 a − n11 2 n31 2 (e2 − 2 n31 ) (n31 ), K

Table 5.1: Notational differences. Note that K is a 1 + 3 covariantly defined variable only if 0 = ω/c = n11 . The LRS-reduced extended 1 + 3 ONF dynamical equations of Chapter 4 for an imperfect fluid matter source take the form: The evolution equations: e0 (Θ/c)

2

= − 31 (Θ/c) + (e1 + u/c ˙ 2 − 2 a) (u/c ˙ 2) − + 2 (ω/c)2 −

e0 (σ+ /c)

4πG c4

e0 (a)

(σ+ /c)2

(µ + 3p)

(5.85)

= − 13 ( 2 (Θ/c) − σ+ /c ) σ+ /c − (e1 + u/c ˙ 2 + a) (u/c ˙ 2) + (ω/c)2 − (E+ −

e0 (ω/c)

2 3

4πG c4

π+ )

= − 23 ( (Θ/c) + σ+ /c ) ω/c − 1 3

1 2

(5.86)

n11 u/c ˙ 2 2

= − ( (Θ/c) + σ+ /c ) (a + u/c ˙ )+

1 2

n11 ω/c −

(5.87) 4πG c4

q1

(5.88)

96

CHAPTER 5. SPECIAL PERFECT FLUID SPACETIME GEOMETRIES. II. e0 (n11 ) e0 (n31 ) 4πG c4

e0 (E+ +

π+ )

= − 13 ( (Θ/c) + 4 σ+ /c ) n11 = − ( (Θ/c) + σ+ /c ) n31

3 2

1 4πG 3 c4

π+ ) −

4πG c4

n11 (E+ −

4πG c4

3 2

n11 H+

(5.91)

ω/c q1 /c

π+ )

(5.92) 2

= − (µ + p) (Θ/c) − (e1 + 2 u/c ˙ − 2 a) (q1 /c) 2 3

− e0 (q1 /c)

π+ )

(e1 + 2 u/c ˙ + a) (q1 /c)

= − ( (Θ/c) + σ+ /c ) H+ − 3 +

1 4πG 3 c4

2

− σ+ /c (E+ −

e0 (µ)

(5.90)

= − 4πG c4 (µ + p) σ+ /c − (Θ/c) (E+ + + 4πG c4

e0 (H+ )

(5.89)

1 3

σ+ /c π+

(5.93)

= − 34 (Θ/c) q1 /c − e1 (p) − (µ + p) u/c ˙ 2 3

+

(e1 + u/c ˙ 2 − 3 a) (π+ ) +

2 3

2

σ+ /c q1 /c .

(5.94)

The constraint equations: 0 0 0

(e1 − u/c ˙ 2 − 2 a) (ω/c)

=

(5.95)

(e1 − 3 a) (σ+ /c) + e1 (Θ/c) +

=

(e1 − a) (a) +

=

+

1 3

1 9

(E+ +

2

4πG c4

2 3

=

(e1 − 2 a) (n11 ) +

0

=

(e1 − a) (n31 )

0

= H+ − 3 (u/c ˙ 2 + a) ω/c +

q1 /c

(5.96) 1 4

(n11 )2

µ

(5.97)

( (Θ/c) − 2 σ+ /c ) ω/c

(5.98) (5.99)

(e1 − 3 a) (E+ +

=

n11 ω/c − 3

4πG c4

( (Θ/c) + 2 σ+ /c ) σ+ /c +

2 4πG 3 c4

π+ ) −

0

0

1 9

(Θ/c) −

3 2

4πG c4

3 2

n11 σ+ /c 4πG c4

π+ ) +

e1 (µ) −

(5.100) 4πG c4

( (Θ/c) + σ+ /c ) q1 /c

− 3 ω/c H+ 0

(5.101)

(e1 − 3 a) (H+ ) − 3

=

+

3 4πG 2 c4

4πG c4

(µ + p) ω/c + 3 ω/c (E+ −

1 4πG 3 c4

π+ )

n11 q1 /c .

(5.102)

Tracefree part and trace of ∗Sαβ ( 3-Ricci curvature of spacelike 3-surfaces orthogonal to u/c, if ω/c = 0 ): ∗

S+

= − e1 (a) − (n11 )2 + 2 (e2 − 2 n31 ) (n31 ) =

∗

R

(E+ +

4πG c4

π+ ) −

1 3

=

2 (2 e1 − 3 a) (a) −

=

4 4πG c4 µ −

2 3

2

( (Θ/c) + σ+ /c ) σ+ /c − 3 (ω/c)2 1 2

(5.103)

2

(n11 ) + 4 (e2 − 2 n31 ) (n31 )

(Θ/c) +

2 3

(σ+ /c)2 − 6 (ω/c)2 .

(5.104)

3-Cotton–York tensor of spacelike 3-surfaces orthogonal to u/c ( if ω/c = 0 ): ∗

C+

= − 23 n11 ∗S+ = − e0 (H+ ) −

4 3

( (Θ/c) + σ+ /c ) H+ − 3 4πG c4 n11 π+ .

(5.105)

97


Up to a constant factor the variables (Θ/c), σ+ /c, ω/c, a, n11 , E+ and H+ correspond directly to the 1 + 3 ( 1 + 1 + 2 ) invariantly defined 3-scalars given in the discussion of the LRS perfect fluid spacetime geometries in Section 3.1. Hence, as mentioned already, their frame derivatives in the e2 and e3 -directions vanish. Thus, acting on this set of variables with the commutator (5.83), and subsequently substituting from the evolution equations (5.85) - (5.94) and constraints (5.95) - (5.102), one successively derives consistency conditions which need to be satisfied by any solution to the LRS dynamical equations ( cf. Ref. [147] ). In this way, applying Eq. (5.83) to ω/c one obtains ( cf. Eq. (3.46) ) 0 =

2 3

( (Θ/c) + σ+ /c ) ω/c − a n11 ,

(5.106)

which is reproduced when the commutator (5.83) acts upon n11 . Application of Eq. (5.83) to ( (Θ/c) + σ+ /c ), using Eq. (5.106), yields 0

= ω/c [ (E+ −

4πG c4

π+ ) +

1 3

( (Θ/c) − σ+ /c ) (Θ/c) −

+ 3 a u/c ˙ 2+

3 4

(n11 )2 +

4πG c4

2 3

(σ+ /c)2 − 3 (ω/c)2

(µ + 3 p) ] −

3 4πG 2 c4

n11 q1 /c ,

(5.107)

while applied to a, using Eq. (5.106), one derives 0

= n11 [ (E+ +

4πG c4

π+ ) +

1 3

( (Θ/c) − σ+ /c ) (Θ/c) −

+ 3 a u/c ˙ 2+

3 4

2 3

(σ+ /c)2 − 3 (ω/c)2

4πG (n11 )2 − 2 4πG c4 µ ] + 6 c4 ω/c q1 /c .

(5.108)

Combining Eqs. (5.107) and (5.108) then yields the condition ( see Ref. [147] ) 0

=

[ 2 (ω/c)2 +

1 2

(n11 )2 ] q1 /c − [ (µ + p) −

2 3

π+ ] ω/c n11 .

(5.109)

This shows that if hypersurface orthogonality properties of the two preferred reference congruences are chosen as a classification criterion for consistent LRS spacetime geometries, either ω/c = 0 or n11 = 0 impose strong restrictions on the fluid matter source, since in this case the energy current density is required to vanish, q1 /c = 0. This is possible, e.g., for sourcefree magnetic Maxwell fields, which have a vanishing Poynting vector. No such restriction arises for 0 = ω/c = n11 . Finally, applying Eq. (5.83) to E+ , one arrives at 0

= ω/c [ (µ + p) σ+ /c −

2 3

( (Θ/c) + 2 σ+ /c ) π+ − (e1 + 2 u/c ˙ 2 + a) (q1 /c) ]

− n11 [ e1 (µ) − ( (Θ/c) + σ+ /c ) q1 /c ] ,

(5.110)

while applying it to H+ reproduces Eq. (5.109). Under the assumption that the matter source is a perfect fluid ( 0 = q1 /c = π+ ) with tangent u/c and (µ + p) > 0, the consistency condition (5.109) reduces to the familiar relation 0 = ω/c n11 ,

(5.111)

the first condition in Eq. (3.57) of subsection 3.1.5, on which the classification of the LRS perfect fluid spacetime geometries into three disjoint groups is based. For more details the reader is referred back to Section 3.1 of Chapter 3. In the next subsection, a brief look is taken at the perfect fluid models of LRS class II from the point of view of the 1 + 3 ONF description.

98

5.2.1


LRS class II perfect fluid models

The models in this class satisfy 0 = ω/c = n11 — all four frame fields ea are HSO and the metric tensor field g can be diagonalised — such that it follows from Eq. (5.100) that H+ = 0 .

(5.112)

Dust configurations contained in this set thus belong to the “silent” class of Section 5.1. It was stated in subsection 3.1.8 that the underlying spacetime symmetry group of LRS class II is a multiply-transitive G3 with 2-D spacelike orbits of either constant positive, zero, or negative Gaußian curvature, here defined by K := 2 (e2 − 2 n31 ) (n31 ) . (5.113) Using K as a dynamical variable in favour of E+ , the sets of equations (5.79) - (5.84), (5.85) - (5.94), and (5.95) - (5.102) specialise to: The commutators: [ e0 , e1 ]

= u/c ˙ 2 e0 −

[ e0 , e2 ]

1 3

= − ( (Θ/c) + σ+ /c ) e2

(5.115)

[ e0 , e3 ]

= − 13 ( (Θ/c) + σ+ /c ) e3

(5.116)

[ e1 , e2 ]

= a e2

(5.117)

[ e2 , e3 ]

= 2 n31 e3

(5.118)

[ e3 , e1 ]

= − a e3 .

(5.119)

1 3

( (Θ/c) − 2 σ+ /c ) e1

(5.114)

The evolution equations ( cf. Eqs. (3.106) - (3.110) ): e0 (Θ/c)

2

= − 31 (Θ/c) + (e1 + u/c ˙ 2 − 2 a) (u/c ˙ 2) − −

e0 (σ+ /c)

4πG c4

e0 (u/c ˙ ) e0 (a) e0 (K) e0 (µ)

(5.120)

= − (Θ/c) − ( (Θ/c) − 3 2 3 2 a − 2 K ∂p ∂µ (Θ/c) ) +

+ 2

= e1 (

(σ+ /c)2

(µ + 3p) 2

1 6

2 3

1 6

2

2

σ+ /c ) σ+ /c − (e1 + u/c ˙ + a) (u/c ˙ )

4πG c4 µ ∂p 1 ˙ 2 ∂µ − 3 ) (Θ/c) u/c 2

+ (

(5.121) +

2 3

σ+ /c u/c ˙

2

(5.122)

= − 13 ( (Θ/c) + σ+ /c ) (a + u/c ˙ )

(5.123)

2 3

= − ( (Θ/c) + σ+ /c ) K

(5.124)

= − (µ + p) (Θ/c) .

(5.125)

The constraint equations ( cf. Eqs. (3.111) - (3.114) ): 0

=

(e1 − 3 a) (σ+ /c) + e1 (Θ/c) 3 2

0

=

(e1 −

0

=

(e1 − 2 a) (K)

0

a) (a) +

1 6

2

(Θ/c) −

1 6

(5.126) 2

(σ+ /c) +

1 2

K−

4πG c4

µ

(5.127) (5.128)

2

= e1 (p) + (µ + p) u/c ˙ .

(5.129)

The evolution equation (5.122) for u/c ˙ 2 derives from the LRS-reduced form of Eq. (4.53) in Chapter 4. Note that the evolution equation (5.124) for the Gaußian 2-curvature K has involutive character, as

99


expected. The magnitude of the “electric” part of Weyl curvature, E+ , and K are related by ( cf. Eq. (3.105) ) E+ =

3 2

K+

1 6

( (Θ/c) + σ+ /c )2 −

3 2

a2 −

4πG c4

µ.

(5.130)

Now taking the ADM 3 + 1 slicing point of view, as reviewed in Section 4.4, and choosing a set of local coordinates (ct, x, y, z), the 1 + 3 ONF spanning basis fields can be expressed as ( e0 = u/c ) e0 = N −1 (∂ct − Nx ∂x ) e1 = X −1 ∂x

e2 = Y −1 ∂y e3 = (Y Z)−1 ∂z ,

(5.131)

which is just the 1 + 3 ONF underlying the form of the line element (3.127) in subsection 3.1.8, but with the freedom in the choice of the shift vector remaining. The following functional dependences of the metric functions on the local coordinates exist: N = N (ct, x), Nx = Nx (ct, x), X = X(ct, x), Y = Y (ct, x), and Z = Z(y). Inserting the 1 + 3 ONF spanning basis fields in the form of Eq. (5.131) into the commutators (5.114) - (5.119), one obtains ( cf. Eqs. (3.128) - (3.131) ) u/c ˙ 2

= N −1 e1 (N )

(5.132)

=

1 3

( (Θ/c) − 2 σ+ /c ) X + X 2 N −1 e1 (Nx )

(5.133)

e0 (Y )

=

1 3

( (Θ/c) + σ+ /c ) Y

(5.134)

0

=

(e1 + a) (Y )

(5.135)

0

=

(e2 + 2 n31 ) (Z) .

(5.136)

e0 (X)

Then, demanding that K be constant for ct = const = x, Eq. (5.113), combined with Eq. (5.136), yields the condition 0 = Z,yy + C1 Z

C1 = const ,

(5.137)

from which it follows that K = C1 / Y 2 ( cf. Eq. (3.132) ). Obviously, if C1 > 0, the group orbits are spherically symmetric 2-surfaces, if C1 = 0, they are flat 2-planes, and if C1 < 0, the 2-surfaces have hyperbolic geometry. In light of the question as to how much invariant geometrical information on the metric tensor field g of ( M, g, u/c ) is contained in the general (perfect fluid) 1 + 3 ONF dynamical equations, the LRS class II perfect fluid models as described by Eqs. (5.120) - (5.129) provide an instructive example. From the commutators one finds that the metric function Y occurs implicitly through the dynamical variable K, if K 6= 0, i.e., in the spherically and hyperbolically symmetric cases. However, there is no analogous information given about the metric function X, which implicitly appears in the set (5.120) (5.129) through the frame derivative e1 . Thus, in order to obtain a complete set of dynamical equations one also has to include the commutator which corresponds to Eq. (5.133). It is also worth noting that the e0 and e1 frame derivatives of Y are encoded through the combination ( (Θ/c) + σ+ /c ) ) and the variable a, respectively. This is in stark contrast to X, for which only its e0 frame derivative is determined by the combination ( (Θ/c) − 2 σ+ /c ), assuming that the shift vector was specified. Note, furthermore, that in Eqs. (5.120) - (5.125) the shift vector occurs implicitly through e0 . The fact that one does not have a variable corresponding to a e1 frame derivative of X is completely analogous to the non-existence of a e0 frame derivative of the lapse function N . The non-existence of a e1 frame derivative of X is associated with the freedom of making a coordinate re-parametrisation x = x(¯ x).

100

5.2.2


Conclusion

In this section, fluid spacetime geometries with LRS symmetry were formulated in extended 1 + 3 ONF terminology for illustrative purposes. In the case of a perfect fluid, most of the 1 + 3 ONF dynamical equations differ from their 1 + 3 covariant counterparts in Section 3.1 of Chapter 3 only by constant numerical factors. This arises because in the LRS case the frame components of 1 + 3 ( 1 + 1 + 2 ) invariantly defined geometrical quantities become directly proportional to 3-scalars ( see, e.g., Eq. (5.75) ). Subsequently, perfect fluid models of LRS class II were briefly discussed again, and local coordinates were introduced by adopting the ADM 3 + 1 slicing point of view.

Chapter 6

Generalisation to f (R)-Type Lagrangean Theories of Gravitation A central assumption generally made in cosmological modelling, which also underlies the investigations of the previous chapters, is that ever since one unit of the Planck time, 10−43 s, had elapsed after the event of the initial singularity, gravitational phenomena in classical physical terms can be accurately described by Einstein’s GR. As outlined in Appendix A, in a metric focused treatment of this theory the dynamical equations — the EFE — can be derived via a variational principle from a so-called Hilbert action of the form given in Eq. (A.29). The particular feature of this Hilbert action is the linearity of the related Lagrangean density for the gravitational field in the Ricci curvature scalar R. As is wellknown, this has the (convenient) consequence that although R contains second-order derivatives of the metric tensor field g on ( M, g ), the resultant field equations themselves, the EFE according to Eq. (A.28), only contain second-order derivatives of g, rather than derivatives of fourth or higher order. However, motivated by different considerations, for a long time various authors have discussed modifications and generalisations of GR, both in classical and so-called semi-classical (half-quantised) terms. These include, e.g., actions of the gravitational field of higher order and also considerations of more than four spacetime dimensions. An early work in this connection is the seminal article by Buchdahl (1970) [22] on classical 4-D higher-order Lagrangean theories of gravitation, in which he contemplates the possibility of avoiding the generic initial singularity of relativistic cosmological models ( see, e.g., Hawking and Ellis (1973) [70] ) by introducing additional non-linear spacetime invariants of the Riemann curvature tensor into the Hilbert action (A.29). In view of the fact that in pursuit of this target it is not obvious what the most effective non-linear generalisation of the gravitational Lagrangean density might be, Buchdahl confined himself to considerations of a theory in which the generalised Hilbert action takes the form ( Λ = 0 ) S [ gµν (x), . . . ]

=

1 4

4πG −1 c4 +

Z

M

Z

M

p

LM (x)

−g(x) f [ R(x) ] d4 x , c

d4 x c (6.1)

which leaves the form of the matter source field Lagrangean density untouched. Here, f (R) denotes an arbitrary differentiable function in the Ricci curvature scalar. As before, the associated field equations 101

CHAPTER 6. GENERALISATION TO f (R)-THEORIES

102

follow from taking a variational derivative of Eq. (6.1) with respect to the components of the metric tensor field, gµν , leading to ( f 0 (R) := df (R)/dR ) Rµν

= f 0 −1

1 2

f gµν + f 00 ( ∇µ ∇ν R − gµν ∇ρ ∇ρ R ) + f 000 (∇µ R ∇ν R − gµν ∇ρ R ∇ρ R) + 2 4πG c4 Tµν

,

(6.2)

the trace of which is given by R = f 0 −1

2f − 3f 00 ∇µ ∇µ R − 3f 000 ∇µ R ∇µ R + 2 4πG c4 T

.

(6.3)

This procedure has the immediate consequences that the stress-energy-momentum tensor of the matter √ source fields is still defined according to Eq. (A.30) in Appendix A by T µν := (2/ −g) δLM /δgµν ( with T denoting its trace ), and, equally importantly, that the twice-contracted second Bianchi identities, Eq. (A.33), continue to apply ( cf. Ref. [22] ), ∇µ T µν = 0 .

(6.4)

It is clear from Eqs. (6.2) and (6.3) that it is the Ricci parts of the Riemann curvature tensor ( cf. Eqs. (A.15) and (A.22) ) that are directly modified. Indirectly, however, the Weyl curvature tensor is affected through its source terms, as can be seen from Eq. (A.24). Terms non-linear in the Riemann curvature tensor and its traces in the action for the gravitational field are now known to arise in the low-energy limit of modern quantum field theories such as superstrings ( see, e.g., Candelas et al (1985) [24] ), and in attempts at renormalising GR by application of standard perturbation expansions ( see, e.g., Barth and Christensen (1983) [8], and Antoniadis and Tomboulis (1986) [3] ). In all cases these purely geometrical terms are interpreted as classical phenomenological manifestations of fluctuations in the quantum state of the vacuum gravitational field. Generically, not only do these terms contain powers of the Ricci curvature scalar R, but also contractions between the components of the Weyl and Ricci curvature tensors, Cµνρσ and Rµν , respectively. To quadratic order one can show that the generalised Lagrangean density of the gravitational field takes the form ( cf. Ref. [71] ) LG =

1 4

4πG −1 c4

[ R − α Cµνρσ C µνρσ + 1 R2 ] ,

(6.5)

where both α and 1 are coupling constants of physical dimension [ length ]2 . As indicated, advocates of classical higher-order Lagrangean modifications to GR argue that non-linear curvature terms could be of relevance in extremely strong gravitational fields such as occur during the very early stages in the evolution of the Universe, where the spacetime curvature length scales are just short of the Planck length scale, 10−35 m. Given physical conditions of this kind, in higher-order Lagrangean scenarios ordinary matter source fields such as relativistic photons and fermionic particles are thought to be of negligible dynamical importance as compared to the non-linear curvature terms, thus leading to an effective vacuum theory. Starobinskiˇı (1980) [141], in an investigation of an isotropic cosmological model without singularity ( for a more recent model with similar features see Brandenberger et al (1993) [17] ), pointed out that a R2 -term in the Lagrangean density of the gravitational field, as it is capable of violating the strong energy condition, can lead to a natural inflationary phase of superluminal expansion, whenever this term is dominant. When the expansion phase has sufficiently progressed such that the spacetime curvature length scales have been vastly increased, the higher-order corrections are thought

103 to lose their influence and standard GR becomes appropriate for the description of the subsequent dynamics. The dynamical effects of R2 -terms on spacetime geometries with a FLRW-type metric Ansatz were also discussed by, e.g., Bicknell (1974) [12] and Mijić et al (1986) [112]. Schmidt (1987) [133] proved that the field equations of a (vacuum) higher-order Lagrangean theory of gravitation which neglects non-linear contractions of the Weyl and Ricci curvature tensors — i.e., a so-called f (R)-theory whose action is given by Eq. (6.1) and where f 0 (R) f 00 (R) 6= 0 is assumed — are conformally equivalent to those of a theory that minimally couples a massive, self-interacting scalar field to the Hilbert action of ordinary GR. The metric tensor fields of the physical and nonphysical spacetime geometries are related by a conformal transformation, the conformal factor of which contains the dynamical information on the terms non-linear in R, and, additionally, so does the “scalar field” self-interaction potential ( see also Maeda (1989) [110] ). This formal trick proves to be quite useful, as the characteristics of minimally coupled scalar field configurations have been widely studied and discussed ever since the advent of the so-called inflationary cosmological models ( see Chapter 7 below ). As an aside it should be mentioned that for 4-D spacetime geometries curvature terms of quadratic order, as they occur in the Lagrangean density of Eq. (6.5), were often assumed to be the generic prototypes of higher-order Lagrangean theories. However, the conformal method revealed that this is not so, but that in 4-D the R2 -terms are rather special indeed ( see Barrow and Cotsakis (1991) [7] ). The “scalar field” potential corresponding to quadratic models tends asymptotically, i.e., for large values of the Ricci curvature scalar R, to a finite non-zero constant limiting value ( thus mimicking a non-zero positive cosmological constant in the non-physical spacetime geometry ), whereas for cubic and higher-order models it tends to zero, which renders realisation of the aforementioned phase of inflationary expansion problematic in the latter cases. To round off the discussion on aspects of a 1 + 3 decomposition of ( M, g ), this chapter now turns to presenting the generalisation of the 1+3 covariant dynamical equations (2.25) - (2.36) of Chapter 2 to Lagrangean theories of gravitation which are described by an action as given in Eq. (6.1). These are the models considered in the paper by Buchdahl (1970) [22]. The specific form of the matter field sources is not further addressed, and although this immediately raises the question as to the invariant definition of a preferred timelike reference congruence u/c, it is assumed that one such u/c can always be found in the given situation, even if it is not unique. In fact, as the emphasis in non-linear Lagrangean theories of gravitation is laid on evolutionary epochs in which the higher-order curvature terms are dominant ( i.e., suppressing any ordinary form of matter ), situations of effective vacuum spacetime geometries with a non-unique u/c are likely to prevail. Maartens and Taylor (1994) [102] recently derived the 1 + 3 covariant dynamical equations for quadratic Lagrangean theories with an imperfect fluid matter source. In that respect the new equations given below extend their work to the general f (R)-situation coupled to an arbitrary stress-energymomentum tensor. In general, f (R) can be viewed as a Taylor polynomial expansion of an arbitrary Pn analytic function of R, but finite polynomials of the form f (R) = k=1 k−1 Rk prove to be more tractable and provide many interesting test cases. The following restrictions arise for a finite f (R)-

polynomial. Stelle (1978) [143] showed that a f (R)-type model cannot evolve into Minkowski spacetime, if 1 < 0 ( see also Ref. [7] ), and that also particles propagating at superluminal velocities could be generated. Hence, 1 > 0. Furthermore, to be consistent with the conformal methods it is required


104

that n−1 > 0 as well as Rn−1 > 0. Finally, also using conformal methods, Maeda (1989) [110] demonstrated that it is sufficient to confine oneself to regimes where f 0 (R) > 0. The conformal transformation breaks down whenever R assumes the critical value such that f 0 (R) = 0, but there exist indications that at all these points singularities will develop in the physical spacetime geometry, if it has non-zero Weyl curvature ( see Nariai (1973) [116] and also Schmidt (1987) [133] ). In the derivation of the 1 + 3 covariant dynamical equations for f (R)-type theories the following expressions are needed. First, taking the covariant derivative of the Ricci curvature scalar (6.3) one obtains ∇µ R =

3f 00 ∇µ (∇ν ∇ν R) + 6f 000 (∇µ ∇ν R) ∇ν R − 2 4πG c4 ∇µ T . 0 00 000 σ 0000 f − f R − 3f ∇σ ∇ R − 3f ∇σ R ∇σ R

(6.6)

Secondly, contracting the Ricci curvature tensor, as determined by Eq. (6.2), with both the timelike vector u/c and its associated orthogonal projection tensor h yields Rµν uµ /c uν /c = f 0 −1 [ − 12 f + f 00 hµν ∇µ ∇ν R + f 000 hµν ∇µ R ∇ν R µ ν + 2 4πG c4 Tµν u /c u /c ]

(6.7)

˙ hνµ ∇ν R Rνρ hνµ uρ /c = f 0 −1 [ f 00 hνµ (∇ν ∇ρ R) uρ /c + f 000 R/c ν ρ + 2 4πG c4 Tνρ h µ u /c ]

Rρσ hρµ hσν

= f 0 −1 [

1 2

(6.8)

f hµν + f 00 hρµ hσν ( ∇ρ ∇σ R − hρσ ∇τ ∇τ R ) + f 000 hρµ hσν ( ∇ρ R ∇σ R − hρσ ∇τ R ∇τ R ) ρ σ + 2 4πG c4 h µ h ν Tρσ ] .

(6.9)

Given these expressions one now obtains the following form for the f (R)-generalised 1 + 3 covariant Ricci identities and second Bianchi identities, respectively.

6.1 6.1.1

The Ricci identities Time derivative equations 2

2

2

(Θ/c)˙/c = − 13 (Θ/c) + hµν ∇µ u˙ ν /c2 + (u/c ˙ 2 )2 − 2 (σ/c) + 2 (ω/c) − f 0 −1 [ − 12 f + f 00 hµν ∇µ ∇ν R

µ ν + f 000 hµν ∇µ R ∇ν R + 2 4πG c4 Tµν u /c u /c ] (6.10)

hµν (ω ν /c)˙/c = − 23 (Θ/c) ω µ /c + σ µν /c ω ν /c −

1 µνρσ 2

∇ν u˙ ρ /c2 uσ /c

(6.11)

hµρ hνσ (σ ρσ /c)˙/c = − 32 (Θ/c) σ µν /c + h(µρ hν)σ ∇ρ u˙ σ /c2 + u˙ µ /c2 u˙ ν /c2 − σ µρ /c σ νρ /c − ω µ /c ω ν /c − E µν

2

2

−

1 3

[ hρσ ∇ρ u˙ σ /c2 + (u/c ˙ 2 )2 − 2 (σ/c) − (ω/c) ] hµν

+

1 2

f 0 −1 ( hµρ hνσ −

1 3

hµν hρσ )

ρσ × [ f 00 ∇ρ ∇σ R + f 000 ∇ρ R ∇σ R + 2 4πG ]. c4 T

(6.12)

105

6.2. THE SECOND BIANCHI IDENTITIES

6.1.2

Constraint equations

0

= hµρ hνσ (∇ν σ ρσ /c) −

2 3

Z µ − µνρσ

∇ν ωρ /c + 2 u˙ ν /c2 ωρ /c uσ /c

˙ hµν ∇ν R − f 0 −1 [ f 00 hµν (∇ν ∇ρ R) uρ /c + f 000 R/c ν µ ρ + 2 4πG c4 T ρ h ν u /c ]

(6.13)

0

= hµν ∇µ ω ν /c − u˙ µ /c2 ω µ /c

0

= Hµν + 2 u˙ (µ /c2 ων) /c + hρ(µ hσν) (∇ρ ωσ /c) − Kµν −

1 3

(6.14)

[ 2 u˙ ρ /c2 ω ρ /c + hρσ ∇ρ ω σ /c ] hµν .

(6.15)

Note that neither the vorticity evolution equation, Eq. (6.11), nor the spatial divergence equations for the vorticity, Eq. (6.14), nor the H-constraint, Eq. (6.15), have changed in form as compared to the equations in Chapter 2. This feature can easily be traced down to the fact that the f (R)-modifications only affect Rµρνσ uρ /c uσ /c, which is symmetric in µ and ν.

6.2


6.2.1

Time derivative equations

hµρ hνσ (E ρσ )˙/c = − (Θ/c) E µν + 3 σ (µρ /c E ν)ρ + J µν − σ ρσ /c E σρ hµν − h(µκ ν)ρστ 2 u˙ ρ /c2 H κσ + ωρ /c E κσ uτ /c +

4πG c4

f 0 −1 [ h(µρ hν)σ (∇ρ T στ ) uτ /c − h(µρ hν)σ (T ρσ )˙/c +

+

1 2

˙ ( Rρσ − f 0 −1 f 00 h(µρ hν)σ [ R/c

1 3

1 3

T˙ /c hµν ]

R g ρσ )

− ∇ρ R Rστ uτ /c + ∇ρ (∇τ ∇σ R) uτ /c − (∇ρ ∇σ R)˙/c ]

(6.16)

hµρ hνσ (H ρσ )˙/c = − (Θ/c) H µν + 3 σ (µρ /c H ν)σ − I µν − σ ρσ /c H σρ hµν + h(µκ ν)ρστ 2 u˙ ρ /c2 E κσ − ωρ /c H κσ uτ /c −

4πG c4

−

1 2

f 0 −1 h(µκ ν)ρστ (∇ρ T κσ ) uτ /c

f 0 −1 f 00 h(µκ ν)ρστ × [ (∇ρ R) Rκσ − ∇ρ ∇σ (∇κ R) ] uτ /c .

(6.17)


106

6.2.2


= hµρ hνσ (∇ν E ρσ ) − 3 ων /c H µν + µνρσ σντ /c H τρ uσ /c 0 −1 µ h ρ [ hνσ (∇[ρ T σ]ν ) − − 2 4πG c4 f

−

1 2

1 3

(∇ρ T ) ]

˙ Rρν uν /c − ( Rνσ uν /c uσ /c + 1 R ) (∇ρ R) f 0 −1 f 00 hµρ [ R/c 3 − ∇ρ (∇σ ∇σ R) − uν /c (∇ρ ∇ν R)˙/c + hνσ ∇ρ (∇ν ∇σ R) ]

(6.18)

0 = hµρ hνσ (∇ν H ρσ ) + 3 ων /c E µν − µνρσ σντ /c E τρ uσ /c +

4πG c4

+

1 2

f 0 −1 µνρσ (∇ν T τρ ) uσ /c uτ /c

f 0 −1 f 00 µνρσ [ ∇ν ∇ρ (∇τ R) − (∇ν R) Rτρ ] uσ /c uτ /c .

(6.19)

The twice-contracted second Bianchi identities are given by Eq. (6.4) and can be further 1 + 3 decomposed once the particular form of the stress-energy-momentum tensor of the matter source fields was specified, alongside with an invariant definition of the preferred timelike reference congruence u/c.

6.3

Vanishing vorticity

If u/c is irrotational, (ω/c) = 0, then in f (R)-type Lagrangean theories of gravitation the spacelike 3-surfaces orthogonal to it have intrinsic 3-Ricci curvature according to 3

Sµν

= Eµν −

1 3


+ f 0 −1 ( hρµ hσν −

1 3

2 3

2

(σ/c) hµν

hµν hρσ )

× [ f 00 ∇ρ ∇σ R + f 000 ∇ρ R ∇σ R + 2 4πG c4 Tρσ ] 3

R

= −

2 3

(6.20)

2

(Θ/c) + 2 (σ/c)2

+ f 0 −1 [ f + f 00 ( 2 hµν ∇µ ∇ν R − 3 ∇µ ∇µ R ) ˙ 2 − ∇µ R ∇µ R ) + f 000 ( 2 (R/c) µ ν + 2 4πG c4 ( 2 Tµν u /c u /c + T ) ] .

6.4

(6.21)

Discussion

Maartens and Taylor (1994) [102] employed the 1 + 3 covariant dynamical equations for R2 -type theories to derive the following two results. First, they computed the modified shear evolution equation for OSH models with isotropic 3-Ricci curvature ( cf. subsection 3.2.1 of Chapter 3 ), coupling them to fluid matter sources with both zero bulk viscosity and energy current density, and pointed out qualitative changes with respect to the GR case. Secondly, they demonstrated that the 1 + 3 covariant characterisation of FLRW perfect fluid spacetime geometries in GR, given by 0 = (σ/c) = (ω/c) = (u/c ˙ 2 ), for R2 -type theories requires the extra condition that the equation of state is of barotropic form, p = p(µ). The second result leads to the possibility of extending the Ehlers–Geren–Sachs (EGS) theorem ( see Ehlers, Geren and Sachs (1968) [39] ) of relativistic cosmology to R2 -type theories. The EGS theorem

6.4. DISCUSSION

107

states that if a family of observers comoving with an irrotational and geodesic, but expanding, timelike congruence with tangent u/c all measure a distribution of photons in thermal equilibrium to be exactly spatially isotropic, then the spacetime geometry is exactly FLRW. Of course, this theorem assumes that interactions between the photons and fermionic matter particles have come to a stand still, by which time in the state of expansion of the Universe the non-linear curvature correction terms in the action probably have become negligible. Subsequent to the work of Maartens and Taylor, Rippl et al (1996) [130]1 showed that the second R2 -result of the former two authors can be generalised to the f (R)-case, even accommodating the EGS theorem in its “almost” version in the sense of Stoeger et al (1995) [148]. Concluding this chapter, it has to be said that results obtained on the basis of classical higher-order Lagrangean theories of gravitation overall have not gained widespread acclaim so far, although in view of the conformal transformation method there does exist a special subclass of the f (R)-type theories which provide the same physical effects as those induced in GR spacetime geometries which are coupled to a (“slow rolling”) scalar field matter source — the much celebrated inflationary models. The question of which between the two frameworks contains fewer idealising assumptions may be a matter of debate. Standard inflationary scenarios and the effect of simple spatial anisotropy perturbations thereupon will be discussed in the following chapter.

1 To cover the content of the work by Rippl et al (1996) [130] completely, this paper has been bound in at the end of this thesis.

108


Chapter 7

Inflationary Evolution in Bianchi Type–I and Type–V Spacetime Geometries After taking a detour into an exposition of modifying Einstein’s GR while maintaining the principle of least action — modifications that were motivated by specific cosmological considerations such as avoidance of initial singularities — in this chapter GR (and the EFE) itself will be the basis of cosmological modelling again. As mentioned in Chapter 1, for over three decades the standard model in relativistic cosmology has involved an expanding spatially homogeneous and spatially isotropic FLRW fluid spacetime geometry (bearing in mind the subtleties relating to the values of its observational parameters). Besides their mathematical simplicity the main reason for adopting such models is that they can easily account for most observational features of cosmological relevance. However, apart from the structure formation problem there are two other important properties of the observable part of the Universe which the FLRW spacetime geometries fail to explain by themselves. For a long time these explanatory difficulties have been known under the keywords “flatness problem” and “horizon problem” ( see, e.g., Lightman and Brawer (1990) [97] ). The first could be paraphrased as the question: “Why is the magnitude of the local spatial 3-curvature in the Universe so near the value zero, when on the basis of the values of the fundamental constants of Nature the value 1070 m−2 related to the Planck length scale would occur to be more natural?”1 The second is related to the fact that in an ordinary FLRW spacetime geometry the light rays of the CMBR coming in from opposite directions on the celestial sphere could never have been in causal contact with each other, simply because not sufficient time has passed since the Big Bang in order for mutual causal interactions to become possible. Hence, the question: “Why can the CMBR be so very nearly at the same temperature, no matter what direction it is measured in?”. A proposal to resolve these difficulties, which received phenomenal recognition, was put forward by Guth in 1981 [66]. Inspired by the successes of modern quantum field theories which accurately describe the interactions of elementary particles at extraordinarily high energy scales, he suggested that a new bosonic elementary particle of spin 0 could have existed at a time between 10−43 s and 10−30 s after the Big Bang, the effect of which was to initiate a brief phase of accelerated, near-exponential expansion of the Universe at around 10−35 s. Mathematically, during this phase the Universe could be modelled by a de Sitter spacetime geometry, which has a matter source consisting of 1 Of

course, under such conditions life as we know it would be rather uncomfortable for wo/man-kind.

109

110

CHAPTER 7. INFLATION IN BIANCHI TYPE–I AND TYPE–V MODELS

a positive cosmological constant, Λ > 0, only ( see de Sitter (1917) [140] ). Convenient consequences of this enormous expansion phase — called “inflation” — would be (i) to vastly increase the size of the so-called particle horizon at a time long before physical processes between photons (later turning into the CMBR) and fermionic matter particles ceased to take place (“decoupling”), thus rendering causal contacts on exponentially larger scales possible ( see, e.g., the very nice, pedagogically oriented discussion given by Ellis and Rothman (1993) [54] ), and (ii) to drastically increase the radius of the local spatial 3-curvature such that to any observer comoving with the fluid matter flow the Universe would appear to be nearly spatially flat. Additionally it is argued that the inflationary dynamics would push the value of the dimensionless cosmological density parameter, Ω, for a fluid spacetime geometry defined by ( Ω ≥ 0 ) −2 4πG c4

Ω := 6 (Θ/c)

µ,

(7.1)

almost indistinguishably close to Ω ≈ 1, a value that would effectively remain constant throughout the subsequent evolution. The mechanism to get the inflationary phase started has been refined many times since the original work of Guth, and a unique form of realisation has not been agreed upon so far ( see, e.g., Olive (1990) [119] ). Its most commonly employed variant is the so-called “chaotic inflation” scenario due to Linde (1983) [98]. Here “chaotic” refers to the assumption that the spacetime manifold ( M, g ) emerges from the initial singularity in a highly irregular, spatially inhomogeneous state, i.e., the Universe starts from arbitrary initial conditions. Beginning with the classical phase of evolution at the Planck time, the existence of a non-quantised massive scalar field φ on ( M, g ) is postulated, which is minimally coupled to the gravitational field, and which self-interacts through a (non-linear) potential V (φ) ( generally, V (φ) > 0 ). Provided that the 4-gradient (∇µ φ) is timelike, the scalar field contributes to the relativistic Hilbert action, Eq. (A.29) in Appendix A, through a Lagrangean density of the form Lφ = − 12

√

−g [ g µν (∇µ φ) (∇ν φ) + 2 V (φ) ] ,

(7.2)

which leads to a related stress-energy-momentum tensor given by Tµν = (∇µ φ) (∇ν φ) −

1 2

[ g ρσ (∇ρ φ) (∇σ φ) + 2 V (φ) ] gµν .

(7.3)

The physical dimensions of φ and its self-interaction potential V (φ) are [ mass ]1/2 × [ length ]1/2 × [ time ]−1 and [ mass ] × [ length ]−1 × [ time ]−2 , respectively.2 Note that for a “massless” scalar field, i.e., V (φ) = 0, expression (7.3) has non-zero trace. The equation of motion for φ follows from the twice-contracted second Bianchi identity, Eq. (A.33), yielding g µν (∇µ ∇ν φ) −

∂V (φ) =0, ∂φ

(7.4)

which is the Klein–Gordon equation. The “chaotic inflation” scenario then assumes that on sufficiently small scales ( M, g ) has locally a FLRW-like structure such that the scalar field can be treated to be spatially homogeneous in any small neighbourhood. Inflationary expansion is said to be induced in only those (in general disconnected) FLRW-like regions, in which at about 10−35 s the kinetic part in the Lagrangean density (7.2) of the scalar field, as well as any other matter sources present, have become dynamically negligible with respect to the self-interaction potential, − (∇µ φ) (∇µ φ) V (φ). As a consequence of the Klein–Gordon equation (7.4), the gradient ∂V (φ)/∂φ would also be small so 2 In

geometrised units, where c = 1 = G/c2 , φ is dimensionless, while V (φ) has physical dimension [ length ]−2 .

111 that the self-interaction potential acquires the anti-gravitating features of an effective (positive) cosmological constant, driving the accelerated growth of all physical length scales. In precise terms, inflation would start wherever the dimensionless relative slope and the dimensionless relative curvature of the self-interaction potential satisfy the conditions ( see Steinhardt and Turner (1984) [142] ) 4πG c4

V

−1

∂V (φ) (φ) ∂φ

2

1

4πG c4

2 −1 V (φ) ∂ V (φ) ∂φ2

1.

(7.5)

Note that in an exact de Sitter spacetime geometry one has R = 4 Λ > 0 for the Ricci curvature scalar, −30 while during a “chaotic” inflationary phase R ≈ 8 4πG s the c4 V (φ) > 0. At a time of roughly 10

inflationary expansion is stopped via decay of φ ( and so V (φ) ), and ordinary (phenomenologically modelled) matter takes over the dynamically important role. In the “chaotic inflation” scenario the observable part of the Universe emerges from just a single FLRW-like region, i.e., even though in its entirety the Universe would be highly irregular, with possibly fractal-like structure, these features would only occur on scales far beyond any (current) detectability. Despite their promising possibilities, the inflationary models of the very early phases in the life of the Universe contain a number of unresolved problems. For example, in its “chaotic” variant it is essential that there already do exist small FLRW-like regions on ( M, g ). However, there is no a priori reason why this should indeed be so. If ( M, g ) had structure on all possible scales, then the FLRW assumption would be unlikely to hold. In fact, the inflationary models would be far more convincing if they could also provide a smoothing effect under truly spatially inhomogeneous conditions. This aspect has long been under scrutiny, but the genericity of the occurrence of inflation under spatially inhomogeneous conditions is so far not entirely clear. Another major problem is related to the rather vague dynamical mechanism that induces the anti-gravitational effect, including the exit from the state of inflation. The problems here arise partially from the current experimental impossibility of probing physics at the extreme energy scales as they prevailed at 10−35 s after the Big Bang. Hence, neither the existence of the (hypothetical) scalar field, nor the functional form of its self-interaction potential can be tested for, leaving a rather unsatisfactorily wide range of different inflationary models at one’s disposal, rather than establishing a unique framework. In the “chaotic” scenario, in general self-interaction potentials of φ2 , φ4 , or exp( λ φ ) dependence are assumed ( see, e.g., Linde (1983) [98] ), or one starts from a particular Ansatz for the functional form of the expansion length scale parameter and inverts the EFE to solve for the related V (φ) ( see, e.g., Lucchin and Matarrese (1985) [100], or Barrow (1990) [6] ). Furthermore, as mentioned in Chapter 1, it is difficult to reconcile the inflationary models with the WCH ( see Penrose (1989) [127] ), assuming that the latter was valid. Finally, current observational evidence on visible fermionic and dynamically detected dark matter hints at a value of the cosmological density parameter at the present time, τ0 , in the range 0.1 ≤ Ω0 ≤ 0.3, rather than Ω0 ≈ 1 ( see Peebles (1993) [124] and Coles and Ellis (1994) [28] ). Madsen and Ellis (1988) [108] and later Madsen et al (1992) [109] demonstrated by investigating the full phase space of simple inflationary FLRW perfect fluid models that, firstly, in general Ω is a dynamical quantity (rather than a constant), and, secondly, that the inflationary forecast of Ω0 ≈ 1 need not be obligatory, but that in principle any value Ω0 > 0 could be realised from different given sets of initial conditions. In support of these investigations, by integrating the dynamical equations underlying these models, Ellis (1988) [47], and in a refined form Hübner und Ehlers (1991) [78], showed that in particular the “horizon problem” could still be solved, if relevant parameters were appropriately adjusted. Ellis (1988) [47] argues that eventually the question

112


of the value of Ω0 is a matter of a dynamically invariant measure of probability over the set of all possible initial data, yet to be devised.

7.1 Dynamical consideration of inflationary 2-fluid model in presence of anisotropy FLRW settings are based on the rather idealised assumptions of spatial isotropy and spatial homogeneity. In reality, however, one would expect the real Universe to deviate, at least partially, from such idealisations. The question then arises as to how robust the dynamical behaviour of Ω (as deduced from FLRW spacetime geometries) is with respect to physically motivated perturbations such as spatial anisotropy. This question is particularly important in view of results from dynamical systems theory which imply that the structural stability of dynamical systems cannot be assumed a priori ( see Tavakol and Ellis (1988) [152] ), in the sense that small perturbations may cause qualitative changes in their behaviour. In fact cosmological dynamical systems are likely to be fragile ( see Coley and Tavakol (1992) [29] ). Consequently, it is necessary to check the stability of each model concretely with respect to physically relevant perturbations. This section extends the phase plane analyses on the dynamical evolution of the cosmological density parameter Ω, given by Madsen and Ellis (1988) [108] and Madsen et al (1992) [109] for inflationary FLRW perfect fluid spacetime geometries, to the (slightly) less idealised anisotropic OSH perfect fluid models with isotropic 3-Ricci curvature, discussed in subsection 3.2.1 of Chapter 3. In line with these works, the inflationary phase is assumed to be driven by an effective (positive) cosmological constant, but the particle physics providing the inflationary mechanism is not further specified. A preferred timelike reference congruence u/c is invariantly defined as the normalised tangent to the fluid matter flow lines. Being of Bianchi Type–I and Type–V, the OSH perfect fluid models with isotropic 3-Ricci curvature are known to contain the flat ( 3R = 0 ) and open ( 3R < 0 ) FLRW models, respectively ( see Ref. [49] ). In this sense the former can be regarded as (exact) anisotropic perturbations of the latter. The following consideration addresses the stability of the FLRW results under these perturbations.

7.1.1

Dynamical equations

In line with the FLRW cases, it is helpful to introduce an average expansion length scale parameter S in the local rest 3-spaces orthogonal to u/c, defined in terms of the fluid rate of expansion by ˙ S/c := S

1 3

(Θ/c) ,

(7.6)

which is representative of physical length scales on ( M, g, u/c ) at a given instant in time. Rather than working with (Θ/c) as a dynamical variable, here, for reasons of comparison with the analysis of, e.g., Madsen et al (1992) [109], the standard Hubble rate of expansion parameter H will be employed, which is defined by ˙ S/c = 13 (Θ/c) . (7.7) S Under the simplifying assumption that the perfect fluid matter source obeys a linear barotropic equation H/c :=

of state, p(µ) = (γ − 1) µ ,

(7.8)

113

7.1. ANISOTROPIC INFLATIONARY 2-FLUID MODEL

with 0 ≤ γ < 2, the 1 + 3 covariant dynamical equations (3.283) - (3.285) of subsection 3.2.1 assume the form (H/c)˙/c = − (H/c)2 −

2 3

(σ/c)2 −

1 4πG 3 c4

(3 γ − 2) µ

(σ/c)˙/c = − 3 (H/c) (σ/c)

(7.9)

(7.10)

µ/c ˙ = − 3 γ (H/c) µ ,

(7.11)

thus providing a 3-D autonomous dynamical system in the variables ( (H/c), (σ/c), µ ). This system is constrained by the generalised Friedmann equation (H/c)2 =

2 4πG 3 c4

µ−

1 3 6 R

+

1 3

(σ/c)2 .

(7.12)

In terms of the length scale parameter S one can integrate the rate of shear scalar evolution equation (7.10) to obtain ( see Ref. [49] ) 2

(σ/c) =

C S3

2

,

(7.13)

where C = const has physical dimension [ length ]2 . Also, the 3-Ricci curvature scalar 3R can be shown to be expressible as 3

R=6

k , S2

(7.14)

which is identical to the FLRW situation. The length scale parameter S is generally adjusted so that in the FLRW cases the dimensionless spatial curvature parameter k takes discrete values in the set { − 1, 0, 1 }, while for models of Bianchi Type–I and Type–V it is k = 0 and k = − 1, respectively.3 Using the cosmological density parameter Ω defined by Eq. (7.1), the generalised Friedmann equation (7.12) can be re-written as a dimensionless conserved “energy equation” in the form −

1 2

k=

1 2

(H/c)2 S 2 (1 − Ω) −

1 6

C S2

2

,

(7.15)

which reduces to the FLRW cases when C = 0. In those cases the immediate implications k = − 1, 0, + 1 ⇔ Ω < 1, = 1, > 1 follow, as is well-known. In the anisotropic case when C 6= 0, these correspondences are modified. For geometrical reasons k = + 1 is excluded, and if k = 0, it no longer follows that Ω = 1, except in the asymptotic limit S → ∞, when the fluid rate of shear becomes negligible. Only the implication for k = − 1 remains qualitatively unchanged. The first term on the righthand side of Eq. (7.15) can be interpreted as a kinetic part ( “escape velocity”; see, e.g., Ref. [31] ), while the terms proportional to Ω and C constitute gravitational potentials relating to the self-gravitational and intrinsic shearing effects of the fluid matter source, respectively. 3 The

3.

spatial curvature parameter k in this section should not be confused with the LRS variable k in Section 3.1 of Chapter

114


Using Eq. (7.12) - (7.14), the set (7.9) - (7.11) can be transformed into a 2-D (plane) autonomous dynamical system in the variables ( S, Ω ), which is given by ˙ S/c = S −2 [ 3 k S 4 − C 2 ]1/2 [ 3 (Ω − 1) ]−1/2 ˙ Ω/c =

Ω S −3

h

√1 3

(7.16)

( 3 γ(S) − 2 ) [ (3 k S 4 − C 2 ) (Ω − 1) ]1/2 +

4 3

C 2 [ 3 (Ω − 1) ]1/2 [ 3 k S 4 − C 2 ]−1/2

i

.

(7.17)

In contrast to Eqs. (7.9) - (7.11), here the righthand sides involve (rather inconvenient) non-polynomial components. For a complete analysis of this system the equation of state function γ(S) and the rate of shear parameter C require further specification. Combining Eqs. (7.16) and (7.17) one obtains the equivalent 1-D non-autonomous dynamical system dΩ Ω 4 C2 = − (1 − Ω) 3 γ(S) − 2 + . (7.18) dS S 3 k S4 − C2 This ordinary differential equation provides the basis for studying the effects of simple rate of shear perturbations on the evolution of the cosmological density parameter Ω in k = 0 and k = − 1 inflationary FLRW perfect fluid spacetime geometries as discussed by Madsen and Ellis (1988) [108] and Madsen et al (1992) [109]. The stationary solutions Ω = 0 and Ω = 1, which are obtained in the FLRW cases, continue to be valid. On the other hand, Ω is stationary for γ(S) = 2 (the “stiff matter” case, which was excluded) when k = 0, and for γ(S) = 2/3 + (4 C 2 /3) / (3 S 4 + C 2 ) when k = − 1. Nevertheless, the latter is S dependent, and so there exists no value of γ for which Ω = const throughout. In the case where the matter content considered is a single adiabatic perfect fluid such that γ = const, the ordinary differential equation (7.18) can be integrated to yield 1 Ω (S) = (7.19) h i h i3γ−6 , 4 3 k S −C 2 S 0 1 + 1−Ω 4 2 Ω0 3 k S0 −C S0 where Ω0 and S0 denote the values of Ω and S at the present time τ0 , respectively. Equation (7.19) generalises the result given by Madsen et al (1992) [109] for C = 0. To determine whether a particular expansion phase in the evolution of a cosmological model is inflationary, the standard procedure is to define the dimensionless cosmological deceleration parameter q by 1 1 q := 3 ˙/c − 1 = ˙/c − 1 , (7.20) (Θ/c) (H/c) which in terms of the length scale parameter S can be converted into the form q=−

¨ 2S S/c . ˙ 2 (S/c)

(7.21)

An expansion phase is said to be inflationary (i.e., accelerated), if q < 0. For the models with spacetime geometry of either Bianchi Type–I or Type–V under discussion here, q can be shown, using Eqs. (7.9) - (7.11) and (7.13) - (7.15), to assume the form 1 − Ω(S) q(S) = 12 [ 3 γ(S) − 2 ] Ω(S) − 2 C 2 , (7.22) 3 k S4 − C2 where Ω(S) has to be determined from Eq. (7.18). The zeros of q(S) corresponding to the transitions from decelerated to inflationary expansion phases and vice versa, i.e., transitions q > 0 ←→ q < 0, depend, in particular, on the form of the equation of state function γ(S) and the value of the rate of shear parameter C.


7.1.2

115

2 phase/ 2 fluid setting

The concrete inflationary models analysed in this and in the following subsection by qualitative methods provide an extension of a class of 2 phase/ 2 fluid inflationary FLRW models discussed by Madsen et al (1992) [109]. Here the effect of adding a simple form of spatial anisotropy is studied. The particular features of the (non-flat) 2 phase/ 2 fluid FLRW version were put forward by Hübner and Ehlers (1991) [78] in a refined calculation of the conditions under which inflationary models can solve the “horizon problem”, while leading to a current value of the cosmological density parameter different from unity, Ω0 6= 1. These authors followed a line of investigation given earlier by Ellis (1988) [47] in which they assumed that a spatially homogeneous and spatially isotropic dynamical model of the Universe beginning at the Big Bang can be split up into two qualitatively distinct phases. In phase I, starting at the Planck time, τP , the dynamics is determined by a spatial curvature term, a perfect fluid with radiation equation of state ( which invariantly defines a preferred timelike reference congruence u/c ) and a positive cosmological constant. The latter ingredient soon plays the dominant role and initiates inflationary expansion at a time τi such that qI (τ > τi ) < 0. The end of phase I at τf is marked by a discontinuous change in the effective equation of state, when the cosmological constant is set to zero, and the matter content is modelled by two non-interacting (decoupled) perfect fluids comoving with the same u/c, representing radiation and pressure-free fermionic matter (dust), respectively. Also a spatial curvature term is taken into account during phase II, which covers the period from τf to the present time τ0 . The discontinuous change at τf stops the inflationary phase instantaneously.4 On the spacelike 3-surface orthogonal to u/c at τf , the total energy densities of the matter content in both phases are matched, µI (τf ) = µII (τf ) ⇒ ΩI (τf ) = ΩII (τf ), so that the dynamical equations of the model (equivalently, the EFE) remain well-defined ( see Refs. [47] and [78] ). For a general mixture of individually adiabatic, non-interacting perfect fluids that all comove with the same u/c, an effective equation of state function γ can be defined by γ :=

Σi γi µi . Σi µi

(7.23)

Here γi = const denotes the barotropic equation of state parameter of the different fluid components. Each of the single fluids separately satisfies Eq. (7.11), which with the help of the Hubble rate of expansion parameter given in Eq. (7.7) can be integrated to give µi = µi (S). For example, for a multi-component fluid consisting of non-interacting radiation and dust and a (positive) cosmological constant, Eq. (7.23) takes the form γ(S) =

4/3 + (B/A) S , −1 Λ S 4 1 + (B/A) S + (2 4πG c4 A)

(7.24)

−1 which contains the two free dimensional parameters (B/A) and (2 4πG Λ. At an arbitrary value c4 A)

of the length scale parameter S ( corresponding to an arbitrary moment in cosmic proper time τ ), both ratios encode the relative magnitudes between the total energy density of the dust matter source and the cosmological constant on the one hand, and the total energy density of the radiation matter source on the other. The particular 2 phase/ 2 fluid inflationary simulations for models with spacetime geometry of either Bianchi Type–I or Type–V can now be realised. The focus of the investigation is on the dynamical 4 According to Guth (1981) [66] the “horizon problem” would be solved, if during the inflationary phase physical length scales had been increased by a logarithmic factor N := ln(Sf /Si ) > 65.

116


evolution of the cosmological density parameter Ω with respect to the length scale parameter S, as determined by Eq. (7.18), and on the associated values of the cosmological deceleration parameter q, as obtained from Eq. (7.22). Phase I Phase I spans the interval from the Planck time τP to the end of the inflationary expansion phase at τf . The length scale parameter is assumed to be scaled such that S(cτP ) = 0. The effective equation of state function given in Eq. (7.24) specialises to γI (S) =

4/3 , −1 Λ S 4 1 + (2 4πG c4 A)

(7.25)

−1 and the ratio (2 4πG Λ can be specified, e.g., at S(cτf ) = Sf . Here, starting at S = 0 and letting c4 A) S → ∞, γI (S) decreases from 4/3 (pure radiation) to asymptotically approach the value 0 (pure cosmological constant). Including a cosmological constant in Eq. (7.12) ( cf. Eq. (2.38) in Chapter 2 ) and using µ = A/S 4 , one can show that the state of near-exponential expansion starts ( at Si ) as soon 4 2 3 2 as Λ > 2 4πG c4 A/S − 3 k/S + (C/S ) . To successfully solve the “horizon problem”, the value of 4πG the free parameter (2 c4 A)−1 Λ has to be chosen such that N := ln(Sf /Si ) > 65 can be obtained.

Phase II Phase II spans the interval from the end of the inflationary expansion phase at τf up to the present time τ0 . The effective equation of state function given in Eq. (7.24) specialises to γII (S) = 1 +

1/3 , 1 + (B/A) S

(7.26)

and the ratio (B/A) can be specified, e.g., at S(cτ0 ) = S0 . Since Sf > 0, γII (S) assumes values in the range 4/3 > γII (S) ≥ 1 as S → ∞, i.e., values between pure radiation and pure dust. The discontinuous jump in the effective equation of state at τf manifests itself in the inequalities dΩI dΩII γI (Sf ) 6= γII (Sf ) ⇒ 6= ⇒ qI (Sf ) 6= qII (Sf ) . (7.27) dS S=Sf dS S=Sf However, by necessity ΩI (Sf ) = ΩII (Sf ). The value of the cosmological density parameter is continuous at τf .

7.1.3

Discussion

Separated into the Bianchi Type–I and Type–V cases, Eqs. (7.18) and (7.22) were solved numerically by specifying a range of data for ( S0 , Ω0 ), as well as for the values of the parameters C, A, B and Λ, and then integrating back towards S = 0. The changeover in the effective equation of state was chosen at an arbitrary but fixed value S = Sf , where the matching condition ΩI (Sf ) = ΩII (Sf ) was imposed. The qualitative features of this investigation are illustrated by the non-compactified phase plane plots given in Figs. 7.1 - 7.4. The particular values of the parameters involved are given in the captions. For technical reasons only one trajectory representative of the whole dynamical flow is plotted in both the ( S, Ω ) phase planes and the associated q(S) graphs. In the Type–V case this trajectory is contrasted with the related k = − 1 FLRW result ( C = 0 ). The k = 0 FLRW subcase of the Type–I model is


117

trivial, as in that case Ω = 1 throughout. As is well-known ( see, e.g., Ref. [160] ), it is a common feature of both the Type–I and the Type–V models that in the limit S → 0 the fluid rate of shear dominates over the total energy density, the spatial curvature ( cf. Eq. (7.15) ) and also the cosmological constant, thus leading to a quasi-vacuum state described by a Kasner (1921) [85] spacetime geometry of Bianchi Type–I. Consequently, here in phase I, independent of γI as given by Eq. (7.25), one obtains lim Ω(S) → 0 ,

S→0

as can be seen in Figs. 7.1 and 7.3. This is to be contrasted with the FLRW asymptotic value limS→0 Ω(S) → 1 for γ > 2/3, where all trajectories emerge from a k = 0 FLRW model with pure radiation equation of state. Then, from Eq. (7.22) one finds that lim q(S) → 2 ,

S→0

which is also reflected in Figs. 7.2 and 7.4. In the FLRW case one has limS→0 q(S) → (3γ − 2)/2 instead. Following the phase plane equation (7.18) and the associated deceleration parameter (7.22) towards large values of S in phase II, where γII is given by Eq. (7.26), the influence of the fluid rate of shear falls off and the Type–I and Type–V models approach their related FLRW subcases. That is, in the limit S → ∞ one obtains for Type–I lim Ω(S) → 1 ,

S→∞

and therefore lim q(S) →

S→∞

1 2

;

these models asymptotically approach an Einstein–de Sitter dust state ( see Einstein and de Sitter (1932) [42] ), and for Type–V lim Ω(S) → 0 ,

S→∞

and therefore lim q(S) → 0 ,

S→∞

leading to a Milne vacuum state, which is equivalent to the flat Minkowski spacetime geometry. Note that for Bianchi Type–I models the onset of an inflationary phase is (implicitly) strongly −1 dependent on the values of the free parameters (2 4πG Λ and C. Unlike the FLRW case, where c4 A) γ < 2/3 ⇔ q < 0, here γI < 2/3 in phase I does not necessarily lead to qI < 0. Assuming 0 ≤ γI ≤ 2/3 and setting k = 0, one finds from Eq. (7.22) that 2 − 2 ΩI (S) ≥ qI ≥ 2 − 3 ΩI (S) .

(7.28)

118


The trajectory exhibited in Fig. 7.1 is a non-inflationary one, as can easily be seen from Fig. 7.2. Note also that Figs. 7.1 - 7.4 clearly depict the discontinuities in dΩ(S)/dS and q(S) at S = Sf , as listed in Eq. (7.27).


119

Figure 7.1: 2 phase/ 2 fluid - ( S, Ω ) phase plane for the Bianchi Type–I case. The parameter values are: S0 = 1, Ω0 = 0.9, Sf = 0.1, C = 10−3 , A = 2.38 × 10−5 , B = 5.02 × 10−5 , Λ = 10. A and B were fixed at S = Sf . Units are such that c = 1 = G/c2 .

Figure 7.2: 2 phase/ 2 fluid - q(S) plot for the Bianchi Type–I case. The parameter values are: S0 = 1, Ω0 = 0.9, Sf = 0.1, C = 10−3 , A = 2.38 × 10−5 , B = 5.02 × 10−5 , Λ = 10. A and B were fixed at S = Sf . Units are such that c = 1 = G/c2 .

120


Figure 7.3: 2 phase/ 2 fluid - ( S, Ω ) phase plane for the Bianchi Type–V case. The parameter values are: S0 = 1, Ω0 = 0.1, Sf = 0.1, C = 10−3 , A = 2.63 × 10−6 , B = 5.59 × 10−6 , Λ = 10. A and B were fixed at S = Sf . Units are such that c = 1 = G/c2 .

Figure 7.4: 2 phase/ 2 fluid - q(S) plot for the Bianchi Type–V case. The parameter values are: S0 = 1, Ω0 = 0.1, Sf = 0.1, C = 10−3 , A = 2.63 × 10−6 , B = 5.59 × 10−6 , Λ = 10. A and B were fixed at S = Sf . Units are such that c = 1 = G/c2 .

7.2. CONSTRAINTS ON ANISOTROPIC SCALAR FIELD MODELS

121

A weakness of the 2 phase/ 2 fluid inflationary models is the implicit assumption that independent of the value Ωi at the onset of the inflationary phase the same logarithmic factor N := ln(Sf /Si ) in the growth of physical length scales will result ( see Ref. [109] ). Apart from this, in order to determine admissible values of the free parameters for which N > 65 is attained such that the “horizon problem” can be solved, separate considerations are needed. This information could be obtained by (numerically) integrating the full set of dynamical equations, analogous to the investigations by Ellis (1988) [47] and by Hübner and Ehlers (1991) [78] in the FLRW case. Concluding this section, one can summarise that overall the main result of Madsen et al (1992) [109] for inflationary FLRW prefect fluid models, namely that Ω is a dynamical quantity which rarely takes values Ω ≈ 1, remains stable with respect to (exact) anisotropic perturbations. However, the details of the earlier work are modified in the limit S → 0, when the fluid rate of shear becomes dominant. In particular, with anisotropy imposed on the spatially flat, k = 0 FLRW model, no longer does Ω = 1 apply throughout, and the values of the free parameters have to be carefully adjusted in order to achieve q < 0 at an early stage of its evolution. This situation provides a simple but instructive example of fragility of a cosmological dynamical system with respect to slight changes within the set of its non-zero parameters in the sense of Tavakol and Ellis (1988) [152] ( see also Ref. [29] ). Another aspect illustrated by the ( S, Ω ) phase plane analysis of this section, which is interesting to note, is that, irrespective of whether they are inflationary or not, both the Bianchi Type–I and the Type–V model act against the spirit of the WCH in that in these cases the Weyl curvature (as induced by the fluid rate of shear) grows large in the limit S → 0 ( τ → τP ), but vanishes asymptotically as S → ∞ ( τ → ∞ ). Finally, of course a dynamically invariant measure is also needed for the models just discussed, in order to determine whether anisotropic inflationary models that solve, e.g., the “horizon problem” but lead to Ω0 6= 1 are probable or not.

7.2

Constraints on inflationary scalar field models in presence of anisotropy

During the hypothetical inflationary expansion phase — the phase of accelerated growth of physical length scales in the early life of the Universe — a non-quantised self-interacting scalar field matter source minimally coupled to the gravitational field is said to dominate all dynamical processes. Hence, it is legitimate to investigate solutions of the EFE where the matter source is constituted by such a scalar field only. The standard inflationary models in general couple this viewpoint with the further assumption that the scalar field is spatially homogeneous and spatially isotropic such that a spacetime geometry with FLRW symmetry will result from solving the dynamical equations ( see, e.g., Olive (1990) [119] ). As mentioned at the beginning of this chapter, a shortcoming of the inflationary scenarios is the nonexistence of a unique framework that gives a precise form to the physical mechanisms involved. Given the large number of different inflationary models that have been put forward ( see Ref. [119] ), it would be useful if restrictions or constraints of any kind could be obtained, which lead to a reduction in the number of viable alternative models of inflation. This is particularly important in view of the absence of sufficient observational and experimental data. Hence, theoretical considerations provide an important way in which constraints may be obtained. Assuming the nature of the matter source and its equation of state are specified (e.g., a dust fluid) and that its equation of motion has also been integrated, then what remains to be determined in the

122


FLRW case is the functional form of the metric length scale parameter S as a function of the cosmic proper time variable cτ . For inflationary expansion to exist, S(cτ ) needs to satisfy the condition q < 0, with q given in Eq. (7.21). However, if, in particular, a scalar field matter source is assumed, then this requires solving the FLRW-reduced form of the Klein–Gordon equation (7.4), which, depending on the functional form of the self-interaction potential V (φ), may not always be possible in a closed form. Nevertheless, as V (φ) has to act as an effective (positive) cosmological constant in order to drive an inflationary expansion, this problem is often bypassed by invoking the so-called “slow-rolling” approximation, mathematically expressed by conditions (7.5) above. As for the uncertainty of the precise functional form of V (φ), an alternative approach to obtaining inflationary models is to assume a specific functional form of S(cτ ) satisfying q < 0 and to invert the FLRW-reduced Einstein–Scalar– Field equations to solve for φ and V (φ) instead. That is, the matter sources are obtained from the dynamical equations, starting with a particular Ansatz for the FLRW line element. This approach was discussed, e.g, by Ellis and Madsen (1991) [52]. Here, analogous to the discussion in the previous section, in order to investigate the influence of simple rate of shear perturbations on the inflationary FLRW spacetime geometries, the latter case is extended to the case of OSH spacetime geometries of Bianchi Type–I and Type–V. For these models the Einstein–Scalar–Field equations can be written in the form ( see Ellis, Skea and Tavakol (1991) [53] ) " 2 # k C 2 4πG ˙ = − (H/c)˙/c − 2 + c4 (φ/c) S S3 2 4πG c4 V (φ)

=

(H/c)˙/c + 3 (H/c)2 + 2

k S2

(7.29)

.

(7.30)

By virtue of the twice-contracted second Bianchi identities, a solution to these equations will automatically satisfy the related Klein–Gordon equation. Note that the rate of shear ( cf. Eq. (7.13) ) in this case has a direct influence only on the form of the kinetic part of the scalar field. For pure scalar field spacetime geometries an invariant definition of a preferred timelike reference congruence u/c rests on the assumption that the 4-gradient (∇µ φ) is timelike; u/c is then given as the normalised congruence orthogonal to the spacelike 3-surfaces φ = const, uµ /c := (∇µ φ) / (− ∇ν φ ∇ν φ)1/2 ,

(7.31)

i.e., by definition u/c is irrotational. The irreducible dynamical variables associated with the covariant derivative of u/c can then be defined analogously to Eqs. (2.7) - (2.10) in Chapter 2. In fact, the scalar field matter source can be conveniently treated as a perfect fluid comoving with u/c, where its total ˙ 2) energy density and isotropic pressure are defined by ( note that − (∇µ φ) (∇µ φ) = (φ/c) µφ =

1 2

˙ 2 + V (φ) (φ/c)

pφ =

1 2

˙ 2 − V (φ) . (φ/c)

(7.32)

This approach underlies the derivation of the particular form of Eqs. (7.29) and (7.30) above. A systematic discussion of these aspects was given by Madsen (1988) [107]. Equations (7.29) and (7.30) establish a dynamical correspondence between all possible functional forms of the length scale parameter S(cτ ) and the inflation driving self-interaction potential V (φ), respectively, which contain k = − 1, 0 and C as arbitrary parameters. Consequently, given a functional form for the length scale parameter, together with a specification of the parameters k and C, in principle one can find the corresponding self-interaction potential. Equation (7.30) gives V = V (cτ ), while


123

cτ = cτ (φ) can be obtained from Eq. (7.29). However, in practice integrating and inverting the latter often may not be possible analytically. The existence of this correspondence might suggest that one is free to choose any S(cτ ) and values for k and C and find the corresponding V (φ). However, since in the classical regime the scalar field has to be real-valued, it is clear that with fixed k and C, constraints on the specific functional form of S(cτ ) and restrictions on its range of applicability for modelling an inflationary phase may arise from the demand that the kinetic energy density of the scalar field be ˙ 2 > 0. In the context here this so-called “reality condition” assumes the positive definite, i.e., (φ/c) form ( cf. Ref. [53] ) −

"

k (H/c)˙/c − 2 + S

C S3

2 #

≥0.

(7.33)

A well-known example of the restrictiveness of this reality condition under FLRW assumptions ( C = 0 ) is provided by the familiar exponential case, S(cτ ) = A exp(ω cτ ). Here, Eq. (7.33) reduces to the condition k ≥ 0, i.e., only the spatially flat ( k = 0 ) and the spatially closed ( k = + 1 ) FLRW spacetime geometries provide viable backgrounds for the exponential inflationary scenario ( cf., e.g., Ref. [52] ). Given the multiplicity of the functional forms of S(cτ ), which have been employed to produce inflationary effects, and the fact that V (φ) usually is not known in detail, it becomes evident that constraints arising from Eq. (7.33) could be of great interest, as well as having physical consequences for the form of a consistent inflationary framework. Considerations of this kind can be linked to the further demand that the self-interaction potential be positive during the inflationary phase, V (φ) > 0, and the need for a sufficient amount of increase in physical length scales to be attained to solve the “horizon problem”, N = ln(Sf /Si ) > 65 ( cf. Section 7.1 ). In the following a systematic look at the consequences of the reality condition (7.33) with C = 0 for various functional forms of S(cτ ) found in the literature within the FLRW framework is taken ( cf. Ellis and Madsen (1991) [52] ). For a given S(cτ ) the reality condition is studied as a function of the system parameters. The important point is that for certain values of the parameters the reality condition may exclude the solution under consideration from the inflationary epoch it aims to describe. It turns out that, as opposed to the case with non-zero rate of shear, in general the reality condition has little to no effect on most of the FLRW inflationary scenarios discussed in the literature, mainly due to the fact that spatial curvature is neglected throughout. As will be seen below, assuming k = 0 corresponds to the least restrictive reality conditions. For completeness the analytic forms of the related self-interaction potential V (φ) are given where they are known. With the aim of introducing spatial anisotropy as a perturbative effect thereafter, this part of the discussion is confined to those cases which allow the spatial curvature parameter k to assume the values 0 and − 1, respectively. The FLRW results are then contrasted with those pertaining for OSH spacetime geometries of Bianchi Type–I and Type–V ( C 6= 0 ), and qualitative changes are pointed out where they occur. The different Ansätze for the functional form of the length scale parameter S(cτ ) occurring in the following either lead to temporal variations of the Hubble rate of expansion parameter where (H/c)˙/c < 0, or, in the exponential case, (H/c)˙/c = 0. Restricting to k = 0, − 1 only, it is clear from Eq. (7.33) that “super-inflation” models with (H/c)˙/c > 0 are excluded even in the FLRW case ( where C = 0 ).

124


7.2.1

FLRW ( C = 0 )

Power-law expansion In power-law inflation ( see Lucchin and Matarrese (1985) [100] ), the length scale parameter takes the form S(cτ ) = A (cτ )n A, n const > 0 n > 1 , (7.34) with the corresponding expression for V (φ) in the spatially flat case k = 0 given by q −1 2/(n−1) 4πG −1 ) A n (3n − 1) exp ± 4 n (φ − φ ) , V (φ) = 12 ( 4πG c c4 c4

(7.35)

and where φc is an integration constant. Note that for k 6= 0, Eq. (7.29) cannot be inverted analytically, which is the reason why no explicit form of V (φ) is known in those cases. The inverse relation for the cosmic proper time variable cτ along u/c, obtained from Eq. (7.34), is 1/n S cτ (S) = , (7.36) A which allows the reality condition (7.33) to be written in the form n A2/n S (2n−2)/n ≥ − k .

(7.37)

Clearly this can only be satisfied for all ranges of S, n and A, in the cases k = + 1 and k = 0. For k = − 1 there is a lower bound (LB) on S depending upon the values of A and n given by S≥

1 nn/(2n−2)

A1/(n−1)

.

(7.38)

As a result, early inflationary epochs are excluded in the k = − 1 case. de Sitter expansion from a singularity The length scale parameter in this case is of the form ( see Ellis and Madsen (1991) [52] ) S(cτ ) = A sinh(ω cτ )

A, ω const > 0 ,

(7.39)

with the corresponding potential valid for any of the three permissible values of the spatial curvature parameter given by V (φ)

=

3 2

−1 2 ( 4πG ω c4 ) −1 + ( 4πG c4 )

k ω2 + 2 A



sinh2  2 q



ω −1 ω 2 + ( 4πG c4 )

k A2

 (φ − φc ) , (7.40)

and where φc is an integration constant. The inverse of Eq. (7.39) can be written as cτ (S) =

1 S arsinh , ω A

(7.41)

which allows the reality condition (7.33) to be written in the form ω 2 A2 ≥ − k .

(7.42)

Again this is satisfied for all ranges of values of S, ω and A in the cases k = + 1 and k = 0. For the case k = − 1, A and ω are required to be constrained by the condition ω ≥ 1/A.

125

7.2. CONSTRAINTS ON ANISOTROPIC SCALAR FIELD MODELS Intermediate de Sitter expansion from a singularity

As a slight modification of the sinh–expansion given in Eq. (7.39) above, a length scale parameter of functional form S(cτ ) = A sinh( ω (cτ ) )

A, ω, const > 0

(7.43)

is considered. By inverting this relation, for the cosmic proper time variable cτ one finds the expression cτ (S) =

1 S arsinh ω A

1/

.

(7.44)

It follows that the reality condition (7.33) can be written as − ( − 1) ω

1 ω

√

A2 + S 2 + 2 ω 2 A2 S (2−)/ arsinh A

S

S 1 arsinh ω A

(2−2)/

≥ −k .

(7.45)

In general, is allowed to take any positive value whatsoever, but it is particularly instructive to take a look at the behaviour of the reality condition when takes values in the neighbourhood of 1. Suppose one takes -values given by = 1 + α and = 1 − α, respectively, where 0 < α 1 is fixed. Now as S → 0, then qualitatively one recovers from Eq. (7.45) the reality condition of the ordinary sinh–expansion, Eq. (7.42), due to the smallness of α and hence the discussion in the paragraph above would go through for the different values of k. However, for large S the behaviour depends on the case chosen: for = 1 + α, the condition (7.45) would necessitate that an upper bound (UB) be placed on S for all values of the spatial curvature parameter k. On the other hand, when = 1 − α, condition (7.45) would be satisfied for k = 0 and k = + 1, while for fixed values of the positive parameters it imposes a lower bound (LB) on S when k = − 1. This change in behaviour arises from the fact that the first term in Eq. (7.45), which is the dominant one for large S, changes sign as passes from < 1 to > 1. Concluding, this example is strongly parameter dependent (PD) and it demonstrates how a small change in the model ( here in the functional form of the length scale parameter, as compared to Eq. (7.39) ) can qualitatively change the consequences of the reality condition and hence the physical status of the model. Intermediate exponential expansion The length scale parameter in this case is given by ( see Barrow (1990) [6] ) S(cτ ) = A exp( ω (cτ ) )

A, ω, const > 0 ,

0

p C/ω A

(LB), whereas for k = − 1 there exists a lower bound on S and an extra constraint given by S > ( C 2 /(ω 2 A2 − 1) )1/4 (LB) and ω 2 A2 − 1 > 0, respectively. Intermediate sinh–expansion The reality condition (7.33) becomes − ( − 1) ω

1 ω

√

A2 + S 2 + 2 ω 2 A2 S (2−)/ arsinh A

S

1 S arsinh ω A

(2−2)/

−

C2 ≥ −k . S4

(7.56)

The rate of shear term in Eq. (7.56) only becomes important as S → 0. Of course, compared to the results for the FLRW subcase Eq. (7.45), this implies quite a substantial change, because the reality condition would now require a lower bound on S for both k = 0 and k = − 1. On the other hand, as S increases, the rate of shear term vanishes asymptotically and the results inferred from Eq. (7.45) would hold here equally as well, implying that changes in the value of in the neighbourhood of = 1 would produce a significant effect (PD).

Intermediate exponential expansion The reality condition (7.33) in this case becomes (1 − ) ω

1 ω

S2 C2 − ≥ −k . S (2−)/ S4 ln A

(7.57)

Again, the overall situation is very complicated and depends on the specific values of the free parameters , ω, A and C (PD). For example, depending on A and C the rate of shear term could invoke a lower bound on S when k = 0. However, for → 1 and for sufficiently large ω, similar results to those of the related FLRW subcase hold, and so at least in this particular case the FLRW results are robust under non-zero shear perturbations, C 6= 0. Classical de Sitter exponential expansion Due to the presence of the rate of shear term in the reality condition (7.33), in addition to the the k = − 1 case, exponential expansion is now also prohibited in the spatially flat situation where k = 0.

128


7.2.3

Discussion

The mathematical solutions invoked to account for inflationary expansion are not unique. Hence, it is of interest to obtain constraints on the different existing models, which ideally would help to determine a single consistent inflationary framework. A possible approach to support this overall target is provided by exploiting the reality condition for pure scalar field spacetime geometries. Here, the reality condition (7.33) was first investigated for various functional forms of the length scale parameter S(cτ ) encountered in FLRW oriented inflationary treatments ( C = 0 ). The results show that the degree to which the reality condition imposes restrictions depends on the nature of the solution under consideration. The FLRW relevant results are summarised in Tab. 7.1. The effect of the reality condition is different for both different solutions and the same solution with different values of the spatial curvature parameter k. A similar investigation was then repeated for the spatially anisotropic OSH spacetime geometries ( C 6= 0 ) which relate to the k = − 1 and k = 0 FLRW cases. The outcome of this analysis is summarised in Tab. 7.2. It is clear that the inclusion of the rate of shear term in Eq. (7.33) can have a significant effect on some of the FLRW cases discussed before. The important point is that the spatial anisotropy has the overall effect of enhancing the prohibitive character of the reality condition. This can be seen, e.g., by comparing the k = 0 rows in Tabs. 7.1 and 7.2. Another important fact is that the effect of the reality condition on inflationary solutions with very similar functional forms can produce qualitatively different restrictive effects. This can be seen by comparing the results relating to the exponential and the intermediate exponential solutions in Tabs. 7.1 and 7.2. First of all, it is surprising that in the k = − 1 FLRW case a small deviation in the exponent of cτ from 1 to a smaller value renders the k = − 1 model conditionally allowable (compare columns 5 and 6 in Tab. 7.1). And, secondly, imposition of an anisotropic perturbation of Bianchi Type–I on the k = 0 FLRW model excludes the possibility of a de Sitter exponential expansion phase (compare columns 6 in Tabs. 7.1 and 7.2). This latter observation is in line with the result obtained for Bianchi Type–I models in subsection 7.1.3 above, where it was found that the values of the free parameters had to be carefully chosen in order to induce inflationary evolution. As another example one may consider the sinh and intermediate sinh solutions in the FLRW case of Tab. 7.1 where also a small deviation in the exponent of cτ from 1 has important consequences, e.g., in the k = 0 case (compare columns 3 and 4). Again, the cases discussed here provide simple examples for the occurrence of dynamical instabilities in specific cosmological models, when slight changes to the initial assumptions have been introduced, an aspect emphasised by Tavakol and Ellis (1988) [152]. Concluding this discussion, it would be of interest to link the results obtained from the investigation of the reality condition , e.g., to an examination which addresses the conditions under which the “horizon problem” could be solved. FLRW k = −1 k=0 k = +1

A (cτ )n LB Y Y

A sinh(ω cτ ) ω > 1/A Y Y

A sinh( ω (cτ ) ) PD PD PD

A exp( ω (cτ ) ) PD Y Y

A exp(ω cτ ) N Y Y

Table 7.1: Consequences of the reality condition for the FLRW cases, separated according to the different values of the spatial curvature parameter k (N implies the solution is excluded by the reality condition; Y implies the reality condition imposes no restrictions on the solution; PD implies that the condition can be satisfied for certain ranges of parameters; and LB indicates there is a lower bound on S). Parameter range in fifth column: 0 < < 1.

129


OSH k = −1 k=0

A (cτ )n LB LB

A sinh(ω cτ ) LB, ω > 1/A LB

A sinh( ω (cτ ) ) PD PD

A exp( ω (cτ ) ) PD PD

A exp(ω cτ ) N N

Table 7.2: Consequences of the reality condition for the OSH Bianchi Type–V ( k = − 1 ) and Type–I ( k = 0 ) cases. The notation is identical to Tab. (7.1). Parameter range in fifth column: 0 < < 1. Note that de Sitter exponential expansion is inconsistent with both spacetime geometries.

130


Chapter 8

Conclusions Relativistic cosmological modelling is based on two main conceptual assumptions. The first is given by the belief that the classical laws of physics take the same form in all places of the (observable part of the) Universe and thus can be called laws of Nature in the true sense. Specifically, it is assumed that they are constant in space and, furthermore, that they are also constant in time since one unit of the Planck time, 10−43 s, had elapsed after the event of the Big Bang singularity. The second supposition states that since the Planck epoch the evolution of the Universe in the large is entirely determined by gravitational interactions which can be described in terms of Einstein’s GR, although this theory has only been reliably tested on physical distance scales that are negligibly small compared to those that pertain in cosmology. Within this framework, 1 + 3 “threading” decomposition methods of the spacetime manifold ( M, g ) based on the existence of an invariantly defined (future pointing) preferred timelike reference congruence with normalised tangent u/c have proved to be powerful tools in the investigation of cosmological dynamical processes, as seen from the particular perspective of (a family of) observers comoving with a phenomenologically modelled continuous fluid matter source. In the first part of this thesis, extensions to the 1 + 3 decomposition formalism for a spacetime geometrical setting ( M, g, u/c ) have been developed, both within the so-called 1 + 3 covariant and 1 + 3 ONF frameworks, which were subsequently employed to examine various problems of interest in relativistic cosmology. More precisely, in Chapter 2, in support of the general 1 + 3 covariant dynamical equations for barotropic perfect fluid matter sources, new evolution equations for the spatial derivatives of various covariant spatial 3-gradients, 3-vectors and 3-tensors of dynamical relevance were obtained. These relations provided the basis for an explicit proof that for barotropic perfect fluid spacetime geometries the spatial constraints between the 1 + 3 covariant dynamical equations are preserved along the integral curves of u/c. The integrability of a number of different special subcases, which have been under intensive study in the past as well as in the present, could easily be derived from this general result. In Chapter 3 a new 1 + 3 covariant formulation of two classes of well-known barotropic perfect fluid cosmological models was given. These are the families of the locally rotationally symmetric (LRS) and the orthogonally spatially homogeneous (OSH) spacetime geometries, respectively. By application of the 1 + 3 covariant method the LRS models and many of their subcases arising through either dynamical specialisation or the existence of further spacetime symmetries were put on a neat and intuitive unified footing. This formulation allowed, e.g., the investigation of simple spacetime 131

132

CHAPTER 8. CONCLUSIONS

geometries of purely “magnetic” Weyl curvature, a dynamical case which had not been explicitly given before. Also, in this formulation simple tilted SH LRS models could be treated in a transparent way by means of investigating functional dependencies between the dynamical variables. For the OSH models, on the other hand, the attempt at providing a fully 1 + 3 covariant formulation remained incomplete. The central problem which has to be solved here is to give an explicit set of dynamical equations for a physical state space of finite dimension, in line with the standard treatment of the OSH spacetime geometries in terms of 1 + 3 ONF variables. If this target was achieved, it would be possible that more general tilted SH perfect fluid models could also be discussed within the 1 + 3 covariant picture. In Chapter 4, first the motivations for and basic dynamical relations of the well-established 1 + 3 ONF formalism were reviewed. Then the set of 1 + 3 ONF dynamical equations was extended to include also the second Bianchi identities in fully 1 + 3 decomposed form. In this way the 1 + 3 ONF method was brought onto an equal level of applicability as its 1 + 3 covariant complement, in that, in particular, useful dynamical relations for the Weyl curvature variables were provided. Subsequently, the introduction of a set of local coordinates on ( M, g, u/c ) was discussed within both the 1 + 3 threading picture and the more familiar ADM 3 + 1 slicing picture, assuming an irrotational u/c in the latter case. The material of Chapter 4 was used in Chapter 5 to investigate the integrability of the 1 + 3 dynamical equations which define “silent” models for irrotational pressure-free fluid matter sources. A particular assumption in these models is that the resulting spacetime geometry possesses purely “electric” Weyl curvature, implying that the “electric” part of the Weyl curvature tensor has to be spatial rotation-free in order not to generate a non-zero “magnetic” part (and, hence, gravitational waves). In the detailed analysis presented, evidence was obtained that there may not exist any spatially inhomogeneous “silent” models for irrotational dust matter sources which have a Weyl curvature tensor of algebraic Petrov type I. A rigorous proof of this conjecture, however, could not be obtained, and so the existence of a new class of spatially inhomogeneous “silent” models remains an open question. Chapter 5 also reformulated LRS spacetime geometries with an imperfect fluid matter source in terms of the extended 1 + 3 ONF framework and in the perfect fluid subcase a short comparison was made with the results obtained in the earlier Chapter 3. In Chapter 6, the GR based 1 + 3 covariant dynamical equations of Chapter 2 were extended to those (classical) higher-order Lagrangean theories of gravitation, where the Lagrangean density of the gravitational field is proportional to an arbitrary differentiable function f (R) in the Ricci curvature scalar. Generalisations of GR of this particular (and many other) kind could be of relevance in the description of the high spacetime curvature regime in the very early life of the Universe. By application of a conformal transformation method a special subclass of the f (R)-theories can in fact be shown to obtain the same near-exponential expansion effects as the much more common inflationary models, which couple a GR based gravitational field to a self-interacting scalar field matter source. Finally, in Chapter 7 a thorough investigation of the dynamical evolution of the cosmological density parameter Ω in simple inflationary fluid spacetime geometries of Bianchi Type–I and Type–V was given. Here, in particular, the influence of a non-zero rate of shear within the matter content on known results for FLRW conditions was examined in detail. It was found that the (exact) anisotropic perturbations can induce important effects during the phase of inflationary expansion which can lead to significant qualitative changes in the dynamical behaviour of a particular (FLRW) model. The influence of anisotropic perturbations on inflationary evolution in FLRW spacetime geometries was then

133 probed from a different perspective by investigating the so-called reality condition for pure Einstein– Scalar–Field configurations. This was done by employing a number of different functional forms for the expansion length scale parameter S to derive restrictions pertaining to FLRW conditions. These results were then contrasted with the situation where the rate of shear term in the reality condition was assumed to be non-vanishing. Overall the qualitative observations of the first part of Chapter 7 relating to the important dynamical effects as caused by the assumed spatial anisotropy in the spacetime geometry were confirmed by the results obtained in the latter analysis. Classical relativistic cosmology offers a wide field of unresolved problems which are open to future research. One of its main goals is to provide a detailed (global) picture of the dynamical evolution of spacetime geometries containing physically reasonable matter sources, which start from arbitrary sets of regular initial conditions. Scenarios of this kind would involve a variety of very interesting physical effects such as viscous thermodynamical interactions between the matter sources, the growth of matter accumulations around overdense regions which may eventually lead to the formation of black holes, effects of gravitational lensing on light signals, and the propagation of gravitational waves as a consequence of the first two phenomena. Supposing the ideal situation that this state of sophistication had been attained, accurate predictions of so far unknown or unobserved effects could become available which, on the basis of an increasing amount of observational data, could help to further separate cosmological models that are theoretically conceivable but unrealistic from the ones that are in very good agreement with observations. In this way one would like to develop a refined version of the standard FLRW models of relativistic cosmology to obtain a more diverse (global) description of (the observable part of) the Universe and its whole evolution since the incident of the Big Bang. Most qualitative methods employed to date, which aim to obtain information on the generic properties of a model of this kind, have restricted themselves to considerations of spatially homogeneous situations. These restrictions arise to a large extent from the enormous complexity of the mathematics relating to spatially inhomogeneous models. With the advent of powerful computing facilities as well as observational devices of significantly improved sensitivity at the end of the twentieth century, simulations of the characteristic properties of spatially inhomogeneous cosmological models and the possibility to test them against observations now have a better prospect. The work presented in this thesis may be of use in various areas of future applications. Although the 1 + 3 covariant reformulation of the well-understood LRS and OSH barotropic perfect fluid spacetime geometries does not give rise to any new class of solutions to the EFE (both classes still contain a number of unknown solutions), it sheds some light on the dynamical properties of these models from a different, as well as pedagogically interesting angle. Since for both classes the 1 + 3 dynamical equations are relatively simple, they could be used as test cases of numerical simulations that employ the 1 + 3 framework. In numerical methods it is standard to use the ADM 3 + 1 slicing picture, but as the 1 + 3 threading picture contains a number of dynamical variables of direct physical interest, it is conceivable that the latter framework could provide a fruitful alternative, at least in cases where the fluid matter source is assumed to be non-rotating. The extended 1 + 3 ONF dynamical equations given in Chapter 4 would provide a basis for investigations of that kind. Furthermore, they provide a starting point for a 1 + 3 ONF adaptation of the Cartan–Karlhede equivalence problem approach to the invariant classification of different spacetime geometries. This would be very useful in the context of relativistic cosmology in order to obtain a more transparent picture of how deviations from either a FLRW or

134

CHAPTER 8. CONCLUSIONS

a spatially homogeneous model affect the invariant properties of a spacetime geometry. If such an adaptation was established, one could also study a number of implications related to the WCH and its dynamical consequences. In the latter part of this thesis properties of inflationary evolution in simple spatially homogeneous but anisotropic models were investigated. As the idea of a phase of inflationary expansion during the early life of the Universe contains a number of attractive (explanatory) features, it is important to demonstrate that the inflationary concept also works under far less idealised, truly generic conditions than are usually assumed, in order to strengthen (or disprove) the point of view it conveys. The best way to find out about the true global state of the Universe at the present time would be to explore its different parts directly. However, on the basis of the laws of Special Relativity it seems reasonable to assume that wo/man-kind will be confined to the Milky Way within the foreseeable future. As such she/he has to content her/him-self with the situation that information about the large-scale structure of our local spacetime neighbourhood can only be obtained indirectly. That is, the predictions of a particular mathematical model of the Universe can only be verified through interpretation of observational data, rather than through direct exploration and experiment. Given this situation, it will probably always prove difficult to avoid both philosophical bias and a speculative element in cosmological modelling.

“ . . . This galaxy’s better than not having a place 2 go . . . ” (Prince Rogers Nelson in ‘Lady Cab Driver’, 1982)

Appendix A

Sign and Index Conventions Let spacetime be represented by a set ( M, g(x) ), where M is a 4-D pseudo-Riemannian manifold endowed with a metric tensor field g(x) of Lorentzian signature ( − + + + ) . Two complementary formulations for expressing physical relationships between geometrical objects on M of dynamical significance feature prominently in this work: the covariant approach and the orthonormal frame (ONF) approach. In the former the components of all geometrical objects are understood to be projections with respect to an arbitrary local basis { dxµ } ( and its dual basis { ∂µ } such that h dxµ , ∂ν i = δ µ ν ) at each point of M. In practice, however, this is often chosen to be a local coordinate basis, the distinguishing feature of which will be discussed below. For example, the line element encoding infinitesimal distances in the neighbourhood of a point on M is given by ds2 (x) = gµν (x) dxµ dxν .

(A.1)

In the ONF approach the components of all geometrical objects at each point of M are understood to be projections with respect to a set of four linearly independent 1-form fields { ω a } ( and their dual vector fields { ea } such that h ω a , eb i = δ a b ). The 1-forms spanning the ONF are distinguished by the fact that with respect to them the infinitesimal line element is given by ds2 (x) = ηab ω a (x) ω b (x) ,

(A.2)

i.e., at each point of M the components of the metric tensor reduce to the constant components of a Minkowskian metric in local Cartesian coordinates, ηab = diag [ − 1, 1, 1, 1 ]. In this work, following Misner, Thorne and Wheeler (1973) [114], covariant (and local coordinate basis) spacetime indices of geometrical objects are denoted by letters from the second half of the greek alphabet ( µ, ν, ρ, . . . = 0 − 3 ), with spatial coordinate indices symbolised by letters from the second half of the latin alphabet ( i, j, k, . . . = 1 − 3 ). Following Ellis (1967) [44], Kramer, Stephani, MacCallum and Herlt (1980) [88], and MacCallum (1973) [104], orthonormal frame spacetime indices of geometrical objects are denoted by letters from the first half of the latin alphabet ( a, b, c, . . . = 0 − 3 ), with spatial frame indices chosen from the first half of the greek alphabet ( α, β, γ, . . . = 1 − 3 ). The units chosen for the physical dimensions [ length ], [ time ] and [ mass ] are the SI units 1m, 1s and 1kg, respectively, in which the fundamental constants c, G and 2 4πG c4 take the (approximate) values ( see, e.g., Falk and Ruppel (1983) [55] ) c G 2 4πG c4

:= 2.9979 × 108 m/s := 6.672 × 10−11 m3 /(kg s2 ) = 2.076 × 10−43 s2 /(kg m) . 135

(A.3)

136

APPENDIX A. SIGN AND INDEX CONVENTIONS

The physical dimension of the infinitesimal line element as given in Eqs. (A.1) and (A.2) is [ length ]2 . Consequently, as both local coordinates xµ and the constant Minkowskian frame metric ηab are dimensionless (i.e., have no direct physical significance), gµν and its inverse g µν in a local coordinate basis, and ω a have physical dimensions [ length ]2 , [ length ]−2 and [ length ]1 , respectively. With respect to a local coordinate basis the ONF spanning 1-form fields and their dual vector fields can be expressed as ω a = ω a µ (x) dxµ and ea = ea µ (x) ∂µ , i.e., ω a µ (x) has physical dimension [ length ]1 and ea µ (x) [ length ]−1 . The ONF components of a general geometrical object A(x) with local coordinate basis components Aµ...ν... (x) follow by simple contraction with the local coordinate basis components ω a µ (x) and ea µ (x); Aa... b... (x) = Aµ... ν... (x) ω a µ (x) . . . eb ν (x) . . . .

(A.4)

The Aa... b... , which are spacetime invariants with respect to arbitrary local coordinate transformations, are often also called the “physical components” of A ( see Pirani (1956) [129] ), as they are, in principle, measurable with respect to a local (Cartesian) inertial frame with Minkowskian metric ηab . Hence, they carry the proper physical dimension of A. Local coordinate basis (and covariant) indices are raised and lowered with g µν and gµν , respectively, while ONF indices are raised and lowered with η ab and ηab . The question of the physical dimensions of the local coordinate basis components of A was discussed in some detail by, e.g., Eardley (1974) [34], and parts of his results are tabulated at the end of this chapter. Note that in a local coordinate basis the physical dimension of a geometrical object is sensitive to the position of its indices, since gµν has physical dimension [ length ]2 , while its inverse g µν has physical dimension [ length ]−2 , as mentioned before. A torsionfree connection field Γ(x) is chosen such that the covariant derivative operator ∇, defined by ∇ν V µ ∇ ν Vµ

:= ∂ν V µ + Γµρν V ρ := ∂ν Vµ − Γρµν Vρ ,

(A.5)

satisfies the metricity condition ∇µ gνρ = 0 .

(A.6)

With respect to a local coordinate basis the connection has 40 algebraically independent components, which in terms of derivatives of the metric can be expressed by Γµνρ (x) :=

1 2

g µσ [ ∂ρ gνσ + ∂ν gρσ − ∂σ gνρ ] .

(A.7)

They are dimensionless and have the property that Γµ[νρ] = 0. In an ONF, where the ea are understood to act as differential operators ea (A) on the frame components of any geometrical object A, the covariant derivative is given by ∇b V a ∇b Va

:= eb (V a ) + Γacb V c := eb (Va ) − Γcab Vc .

(A.8)

It follows from ∇c ηab = 0 = − Γdac ηdb − Γdbc ηad

(A.9)

that Γ(ab)c = 0, which implies that 24 components of Γabc are algebraically independent. The ONF components of the connection are often also called the Ricci rotation coefficients and have physical dimension [ length ]−1 .

137 The totally anti-symmetric infinitesimal spacetime volume element (“epsilon-tensor”) (x) has one algebraically independent component. Its local coordinate basis components µνρσ (x) :=

p

−g(x) ηµνρσ ,

(A.10)

where ηµνρσ is the infinitesimal volume element of Minkowski spacetime with η0123 = 1 in a local Cartesian coordinate basis and g(x) denotes the determinant of g(x), have physical dimension [ length ]4 . The infinitesimal spacetime volume element is covariantly constant, ∇µ νρστ = 0. For its contravariant local coordinate basis components it follows that 1

µνρσ (x) = p

−g(x)

η µνρσ

(A.11)

( η 0123 = − 1 ), and ∇µ νρστ = 0. The product between two epsilon tensors is expanded in terms of the relation µνρσ κλξη = − 4 ! δ µ[κ δ νλ δ ρξ δ ση] . (A.12) In an ONF the infinitesimal volume element is dimensionless and its components ηabcd are identical to the aforementioned ηµνρσ of Minkowski spacetime. It is the distinguishing feature of a local coordinate basis that its (dual) basis fields { ∂µ } commute; [ ∂µ , ∂ν ] = 0. On the other hand, in an ONF setting the (dual) basis fields { ea } define 24 algebraically independent commutation functions γ abc (x) of physical dimension [ length ]−1 by [ ea , eb ] := γ cab (x) ec .

(A.13)

They are related to the Ricci rotation coefficients through Γabc =

1 2

ηad γ dcb − ηbd γ dca + ηcd γ dab

⇐⇒

γ abc = − [ Γabc − Γacb ] .

(A.14)

The local curvature properties of the spacetime manifold ( M, g(x) ) are encoded in the Riemann curvature tensor field R(x). In a local coordinate basis its 20 algebraically independent components are defined via the connection components Γµνρ by1 Rµνρσ (x) := ∂ρ Γµνσ − ∂σ Γµνρ + Γµτ ρ Γτνσ − Γµτ σ Γτνρ .

(A.15)

Starting with Eqs. (A.5), this definition arises from the Ricci identities 2 ∇[µ ∇ν] V ρ 2 ∇[µ ∇ν] Vρ

:= Rµν ρ σ V σ := − Rµν σ ρ Vσ .

(A.16)

The Rµνρσ are dimensionless, while the local coordinate basis components Rµνρσ have physical dimension [ length ]2 . Locally the Riemann curvature tensor can be determined via the equation of geodesic deviation ( see Synge and Schild (1949) [149] ) uν /c∇ν [ uρ /c∇ρ ξ µ ] = − Rµνρσ uν /c ξ ρ uσ /c ,

(A.17)

for two nearby timelike geodesics, one of which having tangent u/c, and with a mutual spatial separation ξ ( ξµ uµ /c = 0 ). Rµνρσ has the symmetry properties R[µνρ]σ = 0 , 1 Schmidt

(A.18)

(1996) [134] recently proposed that, consistent with the anti-symmetrisation operation over pairs of tensor indices, a factor 1/2 should be introduced in the definition of the Riemann curvature tensor.

138


called the first Bianchi identities, and Rµνρσ = R[µν][ρσ] = Rρσµν .

(A.19)

By contraction one obtains from Eq. (A.15) the 10 algebraically independent components of the symmetric Ricci curvature tensor Rµν (x) := Rρµρν

(A.20)

R(x) := g µν Rµν ,

(A.21)

and the Ricci curvature scalar

which are the traces of the Riemann curvature tensor. The local coordinate basis components of Rµν are dimensionless while R has physical dimension [ length ]−2 . The completely tracefree part of Eq. (A.15) is given by the Weyl conformal curvature tensor field C(x) defined as C µν ρσ := Rµν ρσ − 2 δ [µ [ρ Rν] σ] +

1 3

R δ µ [ρ δ ν σ] .

(A.22)

C µν ρσ has 10 algebraically independent components, which, in a local coordinate basis, have physical dimension [ length ]−2 , and the local coordinate components Cµνρσ of physical dimension [ length ]2 share the symmetry properties (A.18) and (A.19). The Riemann curvature tensor satisfies the differential second Bianchi identities ∇[µ Rνρ]στ = 0 ,

(A.23)

which Kundt and Trümper (1962) [90] showed to be equivalent to the form ∇σ Cµν ρσ = − ∇[µ [ Rρ ν] −

1 6

R δ ρ ν] ] .

(A.24)

In an ONF the components of the Riemann curvature tensor, which have physical dimension [ length ]−2 , are given by Rabcd := ec (Γabd ) − ed (Γabc ) + Γaec Γebd − Γaed Γebc − Γabe γ ecd .

(A.25)

The symmetry property Ra[bcd] = 0

(A.26)

has to be imposed explicitly on (A.25), as the Ricci rotation coefficients are not symmetric in their last two indices. This condition leads to the Jacobi identities e[a (γ d bc] ) + γ e [ab γ d c]e = 0

(A.27)

for the commutation functions γ abc . In the metric focused treatment of General Relativity the Einstein field equations (EFE) describing gravitational interactions,2 Rµν − 2 Schmidt’s

1 2

R gµν + Λ gµν = 2 4πG c4 Tµν ,

proposal (see previous footnote) that the EFE be written as Rµν − (1/2) R gµν + Λ gµν = that in Eq. (A.29) R → 2 R and Λ → 2 Λ.

(A.28) 4πG c4

Tµν suggests

139 are derivable from a variational principle, in agreement with the principle of least action. The Hilbert action (functional) S(x) of a gravitational field and its matter source fields, which is extremal under variations with respect to gµν , δS/δgµν = 0, takes the form Z p −1 d4 x S [ gµν (x), . . . ] = 14 4πG −g(x) [ g µν (x) Rµν (x) − 2 Λ ] c4 c M +

Z

LM (x)

M

d4 x , c

(A.29)

and has physical dimension [ mass ] × [ length ]2 × [ time ]−1 . Λ denotes a (hypothetical) cosmological constant, which, if non-zero, can be either positive or negative. Its physical dimension is [ length ]−2 . The stress-energy-momentum tensor T(x) on the RHS of the EFE (A.28), which is symmetric and has 10 algebraically independent components, is defined as the variational derivative 2 δLM T µν (x) := √ −g δgµν

(A.30)

of the matter source field Lagrangean (scalar) density LM (x). The physical dimension of LM is [ mass ] × [ length ]3 × [ time ]−2 . As Rµν has dimensionless components with respect to a local coordinate basis, it follows from the EFE (A.28), together with Eqs. (A.3), that the local coordinate components of Tµν have physical dimension [ mass ] × [ length ] × [ time ]−2 . Alternatively, the EFE (A.28) can be written as 1 (A.31) Rµν = 2 4πG c4 [ Tµν − 2 T gµν ] + Λ gµν , with T being the trace of Eq. (A.30). It is a consequence of twice contracting the second Bianchi identities (A.23) that the LHS of the EFE (A.28) has vanishing covariant divergence, i.e. ∇µ [ Rµν −

1 2

R δµ ν ] = 0 .

(A.32)

Hence, it follows from the EFE (A.28) that ∇µ T µν = 0 .

(A.33)

This relation is often also called the “equations of motion” for the matter source fields. Furthermore, by use of the EFE (A.31) the second Bianchi identities (A.24) assume the form ρ ∇σ Cµν ρσ = − 2 4πG c4 ∇[µ [ T ν] −

1 3

T δ ρ ν] ] .

(A.34)

The physical dimensions of the components of any geometrical object A occurring in covariantly formulated equations (arbitrary basis fields) can be interpreted to be analogous to their ONF counterparts. Ending this consideration, a brief summary of the physical dimensions of the local coordinate basis components as well as the ONF components of the most commonly occurring geometrical objects in General Relativity is given below. Physical dimensions. A: local coordinate basis components • [ length ]4 :

√

−g, µνρσ

• [ length ]3 : − µνρσ uσ /c

140


• [ length ]2 : ds2 , gµν , Rµνρσ , Cµνρσ , hµν • [ length ]1 : uµ /c, ∇µ uν /c, σµν /c • [ length ]0 : xµ , δ µ ν , ∇µ , Γµνρ , Rµνρσ , C µνρσ , Rµν ,

4πG c4

Tµν , Eµν , Hµν , 3Sµν

• [ length ]−1 : uµ /c, (Θ/c), 3Cµν • [ length ]−2 : g µν , Rµν ρσ , C µν ρσ , Rµν , R,

4πG c4

T µν , Λ, hµν , u˙ µ /c2 , ω µ /c,

Physical dimensions. B: ONF components • [ length ]2 : ds2 • [ length ]1 : ω a • [ length ]0 : ηab , η ab , δ a b , ηabcd , ua /c, hab , hab , − ηabcd ud /c • [ length ]−1 : ea , ∇a , Γabc , γ abc , ∇a ub /c, u˙ a /c2 , (Θ/c), σab /c, ω a /c • [ length ]−2 : Rabcd , Cabcd , Rab , R, • [ length ]−3 : ∗Cab .

4πG c4

Tab , Λ, Eab , Hab , ∗Sab ,

4πG c4

µ

4πG c4

µ .

Appendix B

Useful Relations The relations collected in this appendix hold for general (imperfect) fluid spacetime geometries, which invariantly define a preferred timelike reference congruence with normalised tangent u/c ( uµ /c uµ /c = − 1 ⇒ uν /c (∇µ uν /c) = 0 ):

B.1

3-scalar derivatives

As in Chapter 2, let f denote a 1 + 3 invariantly defined 3-scalar field. Then, an evolution equation for its associated spatial 3-gradient, Fµ := hνµ ∇ν f , is given by: hνµ [ Fν ]˙/c = hνµ [ ∇ν + u˙ ν /c2 ] [ f˙/c ] − 13 (Θ/c) hµν + σµν /c − µνρσ ω ρ /c uσ /c F ν .

(B.1)

For the commutator of two orthogonally projected covariant derivatives acting on f one obtains ( cf. Ellis, Bruni and Hwang (1990) [51] ): hρ[µ hσν] ∇ρ Fσ = − µνρσ ω ρ /c uσ /c f˙/c

− µνρσ (∇ν Fρ ) uσ /c = 2 ω µ /c f˙/c .

⇔

(B.2)

This relation states that, for vanishing vorticity, (ω/c) = 0, in the spacelike 3-surfaces orthogonal to u/c covariant derivatives acting on scalars commute. On the other hand, if, e.g., f = µ, µ/c ˙ 6= 0, then ν µ Xµ := h µ ∇ν µ = 0 ⇒ ω /c = 0.

B.2

3-vector derivatives

Let an arbitrary 1 + 3 invariantly defined spacelike vector field orthogonal to u/c be denoted by V. In component form it satisfies Vµ V µ ≥ 0 and Vµ uµ /c = 0. The evolution of its totally orthogonally projected covariant derivative along u/c is given by: hρµ hσν [ hκρ hλσ (∇κ Vλ ) ]˙/c = hρµ hσν [ ∇ρ + u˙ ρ /c2 ] [ hτσ (Vτ )˙/c ] − 31 (Θ/c) hµρ + σµρ /c

− µρκλ ω κ /c uλ /c

× [ hρτ hσν (∇τ Vσ ) − 2 hρ[ν hστ ] u˙ σ /c2 V τ ] − H ρµ V σ νρστ uτ /c − +

hµν 4πG c4 141

ρ

qρ /c V .

4πG c4

qν /c Vµ (B.3)

142

APPENDIX B. USEFUL RELATIONS

This is the exact form of the FLRW-linearised version given in the appendix of Ellis et al (1990) [51]. Contracting on the indices µ and ν one obtains: Evolution of spatial divergence terms: [ hµν ∇µ V ν ]˙/c = hµν [ ∇µ + u˙ µ /c2 ] [ hνρ (V ρ )˙/c ] −

1 3

(Θ/c) hµν ∇µ V ν

− σ µν /c (∇µ V ν ) − ωµ /c µνρσ (∇ν Vρ ) uσ /c +

2 3

(Θ/c) u˙ µ /c2 V µ − σµν /c u˙ µ /c2 V ν

µ − µνρσ u˙ µ /c2 ω ν /c V ρ uσ /c + 2 4πG c4 qµ /c V .

(B.4)

Furthermore, one can derive: Evolution of spatial rotation terms: hµν [ − νρστ (∇ρ Vσ ) uτ /c ]˙/c = − µνρσ uσ /c [ ∇ν + u˙ ν /c2 ] [ hτρ (Vτ )˙/c ] + +

µνρσ 1 (∇ν Vρ ) uσ /c 3 (Θ/c) µνρσ τ σ ν /c (∇τ Vρ ) uσ /c

+ hµν ωρ /c (∇ν V ρ ) − ω µ /c hνρ ∇ν V ρ + µνρσ [

1 3

(Θ/c) hντ + σντ /c ] u˙ ρ /c2 uσ /c V τ

+ ω µ /c u˙ ν /c2 V ν + ων /c u˙ ν /c2 V µ + H µν V ν −

4πG µνρσ qν /c Vρ c4

uσ /c .

(B.5)

and: Divergence of spatial rotation terms: hµν ∇µ [ − νρστ (∇ρ Vσ ) uτ /c ]

= 2 ωµ /c (V µ )˙/c +

2 3

(Θ/c) ωµ /c V µ

+ 2 σµν /c ω µ /c V ν .

(B.6)

Note that for an irrotational (expanding) general fluid matter flow, (ω/c) = 0, the Euclidian 3-space ~ ] ≡ 0 still holds in the spacelike 3-surfaces orthogonal to u/c, even if 3R 6= 0. identity ∇ · [ ∇ × V Also of interest is: Spatial derivative commutator: hσ[µ hκν] hλρ ∇σ [ hξκ hηλ (∇ξ Vη ) ]

= [ (Eρ[µ + −

4πG c4 1 3

πρ[µ ) +

2 4πG 3 c4

(Θ/c) σρ[µ /c −

1 3

µ hρ[µ −

1 9

2

(Θ/c) hρ[µ

(Θ/c) στ ρ[µ ω σ /c uτ /c

+ ωρ /c ω[µ /c − (ω/c)2 hρ[µ ] Vν] + [ hρ[µ (Eν]σ +

4πG c4

πν]σ ) −

1 3

(Θ/c) hρ[µ σν]σ /c

− σρ[µ /c σν]σ /c +

1 3

(Θ/c) hρ[µ ν]σκλ ω κ /c uλ /c

+ σρ[µ /c ν]σκλ ω κ /c uλ /c −σσ[µ /c ν]ρκλ ω κ /c uλ /c

143

B.3. 3-TENSOR DERIVATIVES + hρ[µ ων] /c ωσ /c ] V σ − µνκλ ω κ /c uλ /c hσρ (Vσ )˙/c .

(B.7)

If the preferred timelike reference congruence u/c is irrotational, (ω/c) = 0, this becomes the 3-Ricci identity 2 3∇[µ 3∇ν] V ρ = 3Rµν ρ σ V σ , where 3Rµν ρσ = 4 h[µ[ρ 3S ν]σ] + 31 3R hµ[ρ hνσ] , Λ = 0, and 3Sµν and 3R are given in Eqs. (2.37) and (2.38), respectively.

Note that the “magnetic” part of the Weyl curvature tensor, as well as the energy current density vector, enter — via application of the Ricci identities (A.16), their extensions to tensors, and the Riemann curvature tensor, as given in Eq. (2.21) — in those relations that time-propagate a spatial derivative of an indexed geometrical object off an initial 3-rest space orthogonal to u/c ( like, e.g., in Eq. (B.3) ). Alongside, if non-zero, the acceleration vector of the preferred timelike reference congruence u/c occurs, which, for a barotropic perfect fluid matter source, is linked to the spatial 3-gradient of its isotropic pressure scalar. The “electric” part of the Weyl curvature tensor, the anisotropic pressure tensor of the fluid matter source and its total matter energy density scalar, on the other hand, contribute, if two spatial derivatives act on an indexed geometrical object within the initial 3-rest space orthogonal to u/c ( like, e.g., in Eq. (B.7) ). These features are also present in the relations of Section B.3 below.

B.2.1

3-vector derivatives in 1 + 3 ONF form

When converting the 1 + 3 covariant dynamical equations of Chapter 2 into their 1 + 3 ONF analogues, as listed in Chapter 4, the following translation rules for the derivative terms of 1 + 3 invariantly defined 3-vectors are used:

hµν (V ν )˙/c

e0 (V α ) − αβγ Ωβ /c Vγ .

hµν ∇µ V ν

=⇒

(eα − 2 aα ) (V α ) .

− µνρσ (∇ν Vρ ) uσ /c

=⇒

αβγ (eβ − aβ ) (Vγ ) − nαβ V β .

h(µρ hν)σ (∇ρ V σ )

B.3

=⇒

=⇒

(δ γ(α eγ + a(α ) (V β) ) − δ αβ aγ V γ − γδ(α nβ)γ Vδ .

(B.8)

(B.9)

(B.10)

(B.11)

3-tensor derivatives

Let an arbitrary 1 + 3 invariantly defined rank 2 symmetric tracefree spacelike tensor field orthogonal to u/c be denoted by A. In component form it satisfies Aµν = A(µν) , Aµµ = 0, Aµν Aνµ ≥ 0 and

144


Aµν uν /c = 0. The evolution of its totally orthogonally projected covariant derivative along u/c is given by: hσµ hκν hλρ [ hτσ hξκ hηλ (∇τ Aξη ) ]˙/c = hσµ hκν hλρ [ ∇σ + u˙ σ /c2 ] [ hξκ hηλ (Aξη )˙/c ] − 13 (Θ/c) hµσ + σµσ /c − µσξη ω ξ /c uη /c × [ hστ hκν hλρ (∇τ Aκλ )

+ 4 h[στ hκ] (ν hλρ) u˙ κ /c2 Aτλ ] − 2 H σµ Aκ(ν ρ)σκλ uλ /c − 2 4πG c4 Aµ(ν qρ) /c σ + 2 4πG c4 hµ(ν q /c Aρ)σ .

(B.12)

This is the exact form of the FLRW-linearised version given in the appendix of Ellis et al (1990) [51]. Contracting on the indices µ and ρ one obtains: Evolution of spatial divergence terms: hµν [ hνσ hρτ (∇ρ Aστ ) ]˙/c = hµρ hνσ [ ∇ν + u˙ ν /c2 ] [ hρκ hσλ (Aκλ )˙/c ] −

1 3

(Θ/c) hµρ hνσ (∇ν Aρσ ) − hµρ σ νσ /c (∇ν Aρσ )

− hµκ ων /c νρστ (∇ρ Aκσ ) uτ /c + (Θ/c) u˙ ν /c2 Aµν + σ µν /c Aνρ u˙ ρ /c2 − Aµν σ νρ /c u˙ ρ /c2 − u˙ µ /c2 σ νρ /c Aρν + 2 u˙ ν /c2 A(µρ ν)ρστ ωσ /c uτ /c µν − µνρσ Hντ Aτρ uσ /c + 3 4πG . c4 qν /c A

(B.13)

Furthermore, one can derive: Evolution of spatial rotation terms: h(µρ hν)σ [ − hρκ hσλ κτ ξη (∇τ Aλξ ) uη /c ]˙/c = − h(µρ hν)σ ρτ κλ uλ /c × [ ∇τ + u˙ τ /c2 ] [ hξσ hηκ (Aξη )˙/c ] +

1 3

(Θ/c) h(µρ hν)σ ρτ κλ (∇τ Aσκ ) uλ /c

+ h(µρ hν)σ ρτ κλ σ ξτ /c (∇ξ Aσκ ) uλ /c + h(µρ hν)σ (∇ρ Aστ ) ω τ /c − ω (µ /c hν)ρ hτσ (∇τ Aρσ ) + 3 H (µρ Aν)ρ − H ρσ Aσρ hµν +

4πG c4

+

1 3

A(µρ ν)ρστ qσ /c uτ /c

(Θ/c) A(µρ ν)ρστ u˙ σ /c2 uτ /c

− σ (µρ /c ν)ρστ Aσκ uτ /c u˙ κ /c2 + A(µκ ν)ρστ σ κρ /c u˙ σ /c2 uτ /c + u˙ (µ /c2 ν)ρστ σρκ /c Aκσ uτ /c + Aµν ωρ /c u˙ ρ /c2 + 2 ω (µ /c Aν)ρ u˙ ρ /c2

145

B.3. 3-TENSOR DERIVATIVES + u˙ (µ /c2 Aν)ρ ω ρ /c − hµν Aρσ ωρ /c u˙ σ /c2 .

(B.14)

and:

Divergence of spatial rotation terms: h(µρ hν)σ ∇ν [ − hρκ hσλ κτ ξη (∇τ Aλξ ) uη /c ] = −

1 µνρσ 2

uσ /c ∇ν [ hκρ hλτ (∇λ Aτκ ) ]

+ 2 hµρ ων /c (Aνρ )˙/c − µνρσ [ (Eντ + −

4πG c4

1 3

πντ )

(Θ/c) σντ /c

+ σνκ /c σ κτ /c + ων /c ωτ /c ] Aτρ uσ /c + [ (Θ/c) ων /c + 3 σνρ /c ω ρ /c ] Aµν .

(B.15)

Also of interest is:

Spatial derivative commutator: hτ[µ hκν] hρλ hσξ ∇τ [ hηκ hλζ hξω (∇η Aζω ) ] = 2 [ (E (ρ[µ +

4πG c4

π (ρ[µ ) +

2 4πG 3 c4

2

µ h(ρ[µ

−

1 9

(Θ/c) h(ρ[µ −

+

1 3

(Θ/c) ω κ /c uλ /c κ λ [µ (ρ

1 3

(Θ/c) σ (ρ[µ /c

+ ω (ρ /c ω[µ /c − (ω/c)2 h(ρ[µ ] Aσ)ν] + 2 [ h(ρ[µ (Eν]τ + −

1 3

4πG c4

πν]τ )

(Θ/c) h(ρ[µ σν]τ /c

− σ (ρ[µ /c σν]τ /c +

1 3

(Θ/c) h(ρ[µ ν]τ κλ ω κ /c uλ /c

+ σ (ρ[µ /c ν]τ κλ ω κ /c uλ /c + ω κ /c uλ /c κ λ [µ (ρ σν]τ /c + h(ρ[µ ων] /c ωτ /c ] Aσ)τ − µνκλ ω κ /c uλ /c hρξ hση (Aξη )˙/c . (B.16) If the preferred timelike reference congruence u/c is irrotational, (ω/c) = 0, this becomes the 3-Ricci identity 2 3∇[µ 3∇ν] Aρσ = 3Rµν ρ τ Aτ σ + 3Rµν σ τ Aρτ .

146

B.3.1


3-tensor derivatives in 1 + 3 ONF form

When converting the 1 + 3 covariant dynamical equations of Chapter 2 into their 1 + 3 ONF analogues, as listed in Chapter 4, the following translation rules for the derivative terms of 1 + 3 invariantly defined rank 2 symmetric tracefree 3-tensors are used: hµρ hνσ (Aρσ )˙/c

hµρ hνσ (∇ν Aρσ )

e0 (Aαβ ) − 2 γδ(α Ωγ /c Aβ)δ .

(B.17)

(eβ − 3 aβ ) (Aαβ ) − αβγ nβδ Aδγ .

(B.18)

=⇒

=⇒

− h(µρ hν)σ ρτ κλ (∇τ Aσκ ) uλ /c

=⇒

γδ(α (eγ − aγ ) (Aβ)δ ) − 3 n(αγ Aβ)γ +

B.4

1 2

nγγ Aαβ + δ αβ nγδ Aδγ .

(B.19)

Identities

Let V and W denote arbitrary spacelike vector fields orthogonal to u/c, satisfying 0 = Vµ uµ /c = Wµ uµ /c, and let A and B denote arbitrary rank 2 symmetric tracefree spacelike tensor fields orthogonal to u/c, satisfying Aµν = A(µν) , Bµν = B(µν) , 0 = Aµµ = B µµ , 0 = Aµν uν /c = Bµν uν /c. Then the following algebraic and differential identities apply ( see also Maartens (1996) [101] ): 0

= − µνρσ Aντ Vρ uσ /c W τ + µνρσ Aντ Wρ uσ /c V τ + Aµν νρστ Vρ Wσ uτ /c

(B.20)

0

= − µνρσ Bνκ Aρλ uσ /c B κλ − B µν νρστ Bρκ Aκσ uτ /c

0

= − V (µ ν)ρστ (∇ρ Vσ ) uτ /c − h(µκ ν)ρστ Vρ (∇σ V κ ) uτ /c + hκ(µ ν)ρστ Vρ (∇κ Vσ ) uτ /c + hµν Vρ ρσκλ (∇σ Vκ ) uλ /c

0

(B.23)

= − µνρσ Vν hκλ (∇κ Aλρ ) uσ /c − hµκ Vν νρστ (∇ρ Aκσ ) uτ /c + µνρσ (∇ν Aτρ ) uσ /c Vτ

0

(B.22)

= − µνρσ Aτν (∇ρ Vτ ) uσ /c + µνρσ Aτν (∇τ Vρ ) uσ /c + Aµν νρστ (∇ρ Vσ ) uτ /c

0

(B.21)

= − µνρσ Bντ (∇ × A)τρ uσ /c +

(B.24)

1 2

B µρ hνσ (∇ν Aρσ )

+ hµρ B νσ (∇ν Aρσ ) − hµρ B νσ (∇ρ Aσν ) .

(B.25)

147

B.5. SOME CONTRACTIONS

B.5

Some contractions µνρσ uσ /c κλξη uη /c =

3 ! hκ[µ hλν hξρ]

(B.26)

µνρσ uσ /c κλρη uη /c = 2 ! hκ[µ hλν]

(B.27)

µνρσ uσ /c κνρη uη /c =

2 hκµ

(B.28)

µνρσ uσ /c µνρη uη /c =

6

(B.29)

148


Appendix C

1 + 3 ONF Relations in Explicit Form C.1

1 + 3 ONF dynamical variables

In the geometrical 1 + 3 decompositions of the dynamical variables and equations characterising a particular spacetime geometry ( M, g, u/c ), as performed in Chapter 4, the typical 1 + 3 ONF { ea } which underlied these methods was chosen such that its timelike direction was aligned with the preferred timelike reference congruence, e0 = u/c ( ua /c = δ a0 , ua /c = − δ 0a ). As a result of this decomposition procedure, the commutation functions, the Ricci rotation coefficients, and the components of the Riemann curvature tensor assumed a particular form. The components of these geometrical objects are explicitly listed in the following subsections.

C.1.1

Commutation functions

In 1 + 3 ONF decomposed form the commutation functions of Eq. (4.3) are given by ( cf. Eqs. (4.8) and (4.9) ) γ 001

=

u˙ 1 /c2

γ 201

= − σ12 /c − (ω3 /c + Ω3 /c)

γ 002

=

u˙ 2 /c2

γ 202

=

γ 003

=

u˙ 3 /c2

γ 203

= − σ23 /c + (ω1 /c + Ω1 /c)

γ 023

=

2 ω1 /c

γ 223

=

− (a3 − n12 )

γ 031

=

2 ω2 /c

γ 231

=

n22

γ 012

=

2 ω3 /c

γ 212

=

(a1 + n23 )

γ 101

=

γ 301

= − σ31 /c + (ω2 /c + Ω2 /c)

γ 102

= − σ12 /c + (ω3 /c + Ω3 /c)

γ 302

= − σ23 /c − (ω1 /c + Ω1 /c)

γ 103

= − σ31 /c − (ω2 /c + Ω2 /c)

γ 303

=

γ 123

=

n11

γ 323

=

(a2 + n31 )

γ 131

=

(a3 + n12 )

γ 331

=

− (a1 − n23 )

γ 112

=

− (a2 − n31 )

γ 312

=

n33 .

− [ σ11 /c +

1 3

(Θ/c) ]

149

− [ σ22 /c +

1 3

(Θ/c) ]

(C.1)

− [ σ33 /c +

1 3

(Θ/c) ]

APPENDIX C. 1 + 3 ONF RELATIONS IN EXPLICIT FORM

150

C.1.2

Ricci rotation coefficients

Then it follows from Eq. (4.5) that in 1 + 3 ONF decomposed form the Ricci rotation coefficients assume the form

C.1.3

− u˙ 1 /c2

Γ010

=

Γ120

=

Ω3 /c

Γ011

= − [ σ11 /c +

Γ121

=

− (a2 − n31 )

Γ012

=

− (σ12 /c − ω3 /c)

Γ122

=

(a1 + n23 )

Γ013

=

− (σ31 /c + ω2 /c)

Γ123

=

Γ020

=

− u˙ 2 /c2

Γ230

=

Γ021

=

− (σ12 /c + ω3 /c)

Γ231

=

Γ022

= − [ σ22 /c +

Γ232

=

− (a3 − n12 )

Γ023

=

− (σ23 /c − ω1 /c)

Γ233

=

(a2 + n31 )

Γ030

=

− u˙ 3 /c2

Γ310

=

Ω2 /c

Γ031

=

− (σ31 /c − ω2 /c)

Γ311

=

(a3 + n12 )

Γ032

=

− (σ23 /c + ω1 /c)

Γ312

=

Γ033

= − [ σ33 /c +

Γ313

=

1 3

1 3

1 3

(Θ/c) ]

(Θ/c) ]

(Θ/c) ]

1 2

(n33 − n11 − n22 ) Ω1 /c

1 2

1 2

(n11 − n22 − n33 ) (C.2)

(n22 − n33 − n11 ) − (a1 − n23 ) .

Riemann curvature tensor

From Eq. (4.12) one obtains that in 1 + 3 ONF decomposed form the components of the Riemann curvature tensor are given by π11 ) +

1 4πG 3 c4

R0102 =

(E12 −

4πG c4

π12 )

R0103 =

(E31 −

4πG c4

π31 )

R0101 =

(E11 −

4πG c4

(µ + 3p) −

R0131 =

− (H12 +

4πG c4

q3 /c)

R0112 =

− (H31 −

4πG c4

q2 /c)

π22 ) +

1 4πG 3 c4

R0203 =

(E23 −

4πG c4

R0223 =

− (H12 −

R0231 =

Λ

1 3

Λ

− H11

R0123 =

R0202 =

1 3

(E22 −

4πG c4

π23 )

4πG c4

− H22

(µ + 3p) −

q3 /c)

151

C.2. EQUATION OF GEODESIC DEVIATION − (H23 +

R0212 = R0303 =

(E33 −

4πG c4

π33 ) +

q1 /c)

1 4πG 3 c4

− (H31 +

4πG c4

q2 /c)

R0331 =

− (H23 −

4πG c4

q1 /c)

1 3

Λ

− H33 − (E11 +

R2323 =

4πG c4

π11 ) +

2 4πG 3 c4

R2331 =

− (E12 +

4πG c4

π12 )

R2312 =

− (E31 +

4πG c4

π31 )

− (E22 +

R3131 =

4πG c4

π22 ) +

− (E23 +

R3112 = R1212 =

(C.3)

(µ + 3p) −

R0323 =

R0312 =

C.2

4πG c4

− (E33 +

4πG c4

2 4πG 3 c4

4πG c4

π33 ) +

µ+

1 3

Λ

µ+

1 3

Λ

π23 )

2 4πG 3 c4

µ+

1 3

Λ.

Equation of geodesic deviation

The equation of geodesic deviation ( cf. Eq. (A.17) in Appendix A ) gives a direct measure of the focusing or diverging of two initially parallel fiducial timelike geodesics due to gravitational effects, as one follows along one of them towards, say, increasing values of its affine parameter. This equation was discussed in some length, e.g., by Synge and Schild (1949) [149], Pirani (1956) [129], and Szekeres (1965) [150], and serves as an operational instruction for measuring the physical components of the Riemann curvature tensor for a given spacetime geometry, hence, the local curvature of a setting ( M, g ). The separation of the geodesics can be encoded in a spacelike vector field ξ ( such that ξα ξ α > 0 ), which is orthogonal to the tangent u/c of the reference geodesic ( where u˙ α /c2 = 0 ). In principle, the latter represents the tangent to the worldline of a potential observer. Then, on using the decomposition of the Riemann curvature tensor with respect to a 1 + 3 ONF { ea } as given in Eq. (4.12), the acceleration of the spacelike vector field ξ can be expressed by e0 [ e0 (ξ α ) ]

= − Rα 0β0 ξ β − Ωα /c Ωβ /c ξ β + Ωβ /c Ωβ /c ξ α + αβγ [ e0 (Ωβ /c) ξγ + 2 Ωβ /c e0 (ξγ ) ] = − E α β (u/c) − +

1 3

Λ ξα

4πG c4

π α β (u/c) ξ β −

1 4πG 3 c4

(C.4)

[ µ(u/c) + 3 p(u/c) ] ξ α

− Ωα /c Ωβ /c ξ β + Ωβ /c Ωβ /c ξ α + αβγ [ e0 (Ωβ /c) ξγ + 2 Ωβ /c e0 (ξγ ) ] .

(C.5)


152

This 1 + 3 ONF adapted form of the equation of geodesic deviation suggests that in measuring the physical components of the Riemann curvatute tensor it may be most convenient for an observer to Fermi-transport ( Ωα /c = 0 ) her/his spatial frame { eα } along u/c, although other choices such as a tidal eigenframe ( Eαβ diagonal ) are possible as well. Supposing that ξ represents three eigendirections of a reference volume forming body, Eq. (C.5) provides a nice illustration of the different possible distortions the body may experience as it moves geodesically in the presence of non-zero spacetime curvature, specifically as measured in the local rest 3-spaces orthogonal to u/c. In these rest 3-spaces an observer can distinguish between three different effects which contribute linearly to the acceleration of ξ: (i) the volume-preserving, shearing effects characteristic of tidal forces, as exerted by the “electric” part of the Weyl curvature Eαβ and the anisotropic pressure παβ of the fluid matter source ( note that παβ acts in diametrically opposed directions to Eαβ and thus tries to balance the shearing action of the latter ), (ii) the volume-shrinking effects of the total energy density µ and isotropic pressure p ( given that (µ + 3 p) > 0 ), and (iii) the volume-inflating, anti-gravitating effects induced by a non-zero, positive cosmological constant Λ, which is an invariant spacetime constant and thus does not depend on the choice of u/c.1 However, Eq. (C.5) only provides partial information on the Riemann curvature of ( M, g ) ( cf. Eqs. (C.3) above ). The missing components, namely the values of the components of the “magnetic” part of the Weyl curvature Hαβ and the energy current density q α /c are obtained from analogous measurements made with respect to two further 1+3 ONF { ˜ ea } and { ¯ ea }, where ˜ e0 = u ˜ /c and ¯ e0 = u ¯ /c, which move in different spatial directions relative to { ea }, ( see, e.g., Szekeres (1965) [150] and Misner, Thorne and Wheeler (1973) [114] ). Relating the results of this procedure back to the original 1 + 3 ONF { ea } by taking into account the Lorentz transformation rules, one can solve for the values of the remaining Riemann curvature components. The relevant Lorentz transformation relations are explicitly given in the next subsection. For the case of a non-geodesic preferred timelike reference congruence u/c, the following (1 + 3 covariant) expressions for the relative velocity and the relative acceleration between two nearby individual flow lines can be derived ( see Ehlers (1961) [37] ): hµν (ξ ν )˙/c = σ µν /c ξ ν +

hµν [ hνρ (ξ ρ )˙/c ]˙/c = − (E µ ν −

1 3

(Θ/c) ξ µ − µνρσ ωρ /c uσ /c ξν

4πG c4

πµ ν ) ξν −

1 4πG 3 c4

(µ + 3 p) ξ µ +

+ hµρ hσν (∇σ u˙ ρ /c2 ) ξ ν + u˙ µ /c2 u˙ ν /c2 ξ ν .

(C.6)

1 3

Λ ξµ (C.7)

Here, again ξ denotes the flow lines’ (spacelike) orthogonal separation.

C.2.1

Lorentz transformations

Without lack of generality it is assumed that the new 1 + 3 ONF { ˜ ea } moves with (constant) relative velocity (v/c) along the spatial e1 -direction of the 1+3 ONF { ea }. By a cyclic permutation of indices 1 → 2 → 3 → 1 the following transformation rules can be easily adapted to either of the spatial e2 or e3 -directions as well. 1 This feature can be easily verified for a de Sitter spacetime geometry. It follows on using the components of the Riemann curvature as expressed in Eqs. (C.3) and applying the Lorentz transformation rules given in Eqs. (C.10).

153

C.3. WEYL CURVATURE COMPONENTS RELATING TO NULL FRAMES e1 –boost 

γ v  − γ c Λa˜ b (v/c) =   0 0

−γ γ 0 0

v c

 0 0 0 0   1 0  0 1



 γ Λa ˜b (v/c) =  

γ γ v c

0 0

v c

γ 0 0

 0 0 0 0   , (C.8) 1 0  0 1

where γ := [ 1 − (v/c)2 ]−1/2 . Then, from ˜ abcd = Λe a˜ Λf ˜ Λg c˜ Λh ˜ Ref gh , R b d

(C.9)

one obtains ˜ 0101 R

=

˜ 0102 R

=

γ

R0102 +

v c

R0112

˜ 0103 R

=

γ

R0103 −

v c

R0131

˜ 0202 R

=

˜ 0203 R

= γ2

˜ 0303 R

=

˜ 2323 R

=

˜ 2331 R

=

γ

R2331 −

v c

R0323

˜ 2312 R

=

γ

R2312 +

v c

R0223

˜ 3131 R

=

˜ 3112 R

= γ2

˜ 1212 R

=

R0101

h

γ2 h

R0202 + 2 v c

R0203 − h

γ2

v c

v 2 c

R0212 +

v 2 c

( R0231 − R0312 ) −

R0303 − 2

v c

v 2 c

R0331 +

R1212

i

R3112

R3131

i

i (C.10)

R2323

h

γ2 h

R3131 − 2

R3112 +

γ2

h

v c

v c

v 2 c

R0331 +

R0303

( R0231 − R0312 ) −

v 2 c

v 2 c

R0202

R1212 + 2

v c

R0212 +

i

R0203 i

i

.

In this set of equations only those components of the Riemann curvature tensor occur on the lefthand ˜αβ of the “electric” part of the Weyl curvature tensor as measured sides, which contain components E by an observer comoving with { ˜ ea }.

C.3

Weyl curvature components relating to null frames

The 1 + 3 ONF spanning basis fields ea can be combined to construct a complex null frame of the form ( see Kramer et al (1980) [88] ) ka

:=

a

:=

ma

:=

l

a

m ¯

:=

√1 2 √1 2 √1 2 √1 2

( ua /c + e1 a ) a

a

(C.11)

( u /c − e1 )

(C.12)

( e2 a /c − i e3 a )

(C.13)

a

a

( e2 /c + i e3 ) .

(C.14)


154

The non-zero contractions between the null frame fields are ka la = − 1 = − ma m ¯ a , while all other contractions vanish. From the null frame fields a self-dual complex bivector basis is defined by U[ab]

:= 2 m ¯ [a lb]

(C.15)

V[ab]

:= 2 k[a mb]

(C.16)

:= 2 m[a m ¯ b] − 2 k[a lb] ,

(C.17)

W[ab]

where the only non-zero contractions are given by Uab V ab = 2, Wab W ab = − 4. Then it follows that the self-dual complex Weyl curvature tensor, which is defined by ∗ Cabcd := Cabcd −

i 2 abef

C ef cd := Cabcd + i ∗Cabcd ,

(C.18)

can be expressed in terms of this self-dual complex bivector basis by ( see Eq. (3.58) in Kramer et al (1980) [88] ) 1 2

∗ Cabcd

=

Ψ0 ( Uab Ucd ) + Ψ1 ( Uab Wcd + Wab Ucd ) + Ψ2 ( Uab Vcd + Vab Ucd + Wab Wcd ) + Ψ3 ( Vab Wcd + Wab Vcd ) + Ψ4 ( Vab Vcd ) ,

(C.19)

where the complex Newman–Penrose components of the Weyl curvature tensor are defined by 1 8

∗ Cabcd V ab V cd = Cabcd la m ¯ b lc m ¯d

Ψ0

:=

Ψ1

∗ 1 := − 16 Cabcd V ab W cd = Cabcd la m ¯ b k c ld 1 8

∗ Cabcd

a

b

c

d

:=

Ψ3

∗ 1 := − 16 Cabcd U ab W cd = Cabcd k a mb k c ld

Ψ4

:=

∗ Cabcd

U

ab

V

cd

Ψ2

1 8

U

ab

(C.20)

U

cd

= Cabcd l m ¯ k m a

b

c

d

= Cabcd k m k m .

(C.21) (C.22) (C.23) (C.24)

For fluid spacetime geometries, for which a 1 + 3 ONF is generally chosen to be aligned with the normalised tangent u/c of the matter flow, on defining ∗ Ccedf hca ue /c hdb uf /c := Eab (u/c) + i Hab (u/c) ,

(C.25)

Barnes and Rowlingson (1989) [5] employed Eq. (C.19) to obtain the following expressions for the frame components of the “electric” and “magnetic” parts of the Weyl curvature tensor:2 Eab (u/c)

=

1 2

Re (Ψ0 + Ψ4 ) ( e2 a e2 b − e3 a e3 b ) −

1 2

Im (Ψ0 − Ψ4 ) ( e2 a e3 b + e3 a e2 b )

− Re Ψ2 ( e2 a e2 b + e3 a e3 b − 2 e1 a e1 b ) + Im (Ψ1 + Ψ3 ) ( e3 a e1 b + e1 a e3 b ) − Re (Ψ1 − Ψ3 ) ( e1 a e2 b + e2 a e1 b )

Hab (u/c)

=

1 2

(C.26)

Im (Ψ0 + Ψ4 ) ( e2 a e2 b − e3 a e3 b ) +

1 2

Re (Ψ0 − Ψ4 ) ( e2 a e3 b + e3 a e2 b )

− Im Ψ2 ( e2 a e2 b + e3 a e3 b − 2 e1 a e1 b ) − Re (Ψ1 + Ψ3 ) ( e3 a e1 b + e1 a e3 b ) − Im (Ψ1 − Ψ3 ) ( e1 a e2 b + e2 a e1 b ) . 2 These

relations were earlier obtained by MacCallum ( see MacCallum, Private Notes, Ref. [103] ).

(C.27)

155

C.3. WEYL CURVATURE COMPONENTS RELATING TO NULL FRAMES

Ending this appendix, for the sake of completeness the four algebraically independent invariant spacetime scalars related to the Weyl curvature tensor Cabcd and its dual ∗Cabcd are given in terms of the “electric” and “magnetic” parts. These are C ab cd C cd ab ∗C

ab

cd

C

cd

ab

C ab cd C cd ef C ef ab ∗C

ab

cd

∗C

cd

ef

∗C

ef

ab

= 8 [ E ab E ba − H ab H ba ] =

− 16 E ab

(C.28)

H ba

(C.29)

= − 16 [ E ab E bc E ca − 3 E ab H bc H ca ] = 16 [

H ab

H bc

H ca

−

3 H ab

E bc

E ca

].

(C.30) (C.31)

156


Appendix D

Extended 1 + 3 ONF Equations in Dimensionless Formulation Wainwright and collaborators have introduced the concept of “expansion-normalised” dimensionless variables into the relativistic modelling of spatially homogeneous and spatially inhomogeneous spacetime geometries, where the latter were assumed to contain two commuting spacelike Killing vector fields ( see, e.g., Wainwright and Ellis (1996) [160] ). Expansion-normalised dimensionless variables lead to relatively simple equations and provide a natural choice for the description of cosmological or gravitational collapse scenarios. On the other hand, in other situations, like, e.g, for static models or for maximal slicings, one is forced to consider other possibilities. Here the work by Hewitt and Wainwright ( see Ref. [160] ) on comoving perfect fluids in spacetime geometries with two commuting spacelike Killing vector fields is extended to the case of a general fluid matter source, where no isometries are assumed to exist on ( M, g, u/c ). That is, in the following the extended 1 + 3 ONF dynamical equations of Chapter 4 are given in expansion-normalised form. For a start, one first introduces expansion-normalised dimensionless differential operators defined by ∂ a :=

3 ea , (Θ/c)

(D.1)

then expansion-normalised dimensionless commutation functions defined by n

U˙ α , Σαβ , Wα , Rα , Aα , Nαβ

o

:=

u˙ α /c2 , σαβ /c, ωα /c, Ωα /c, aα , nαβ

/ (Θ/c) ,

(D.2)

and, finally, expansion-normalised dimensionless fluid matter and Weyl curvature variables, as well as a dimensionless variable related to the cosmological constant. These are defined by 2

{ Ω, P, Qα , Παβ } := 6 4πG c4 { µ, p, qα /c, παβ } / (Θ/c) 2

{ Eαβ , Hαβ , ΩΛ } := 3 { Eαβ , Hαβ , Λ } / (Θ/c) ,

(D.3) (D.4)

respectively. For the tracefree-adapted irreducible frame components of the fluid rate of shear tensor, Eq. (4.54) in Section 4.6 of Chapter 4 as applied to σαβ /c, one can use the definition { Σ± , Σi } := { σ± /c, σi /c } / (Θ/c) , 157

(D.5)

APPENDIX D. 1 + 3 ONF EQUATIONS IN DIMENSIONLESS FORMULATION

158

( where i = 1, 2, 3 ) which is aimed at simplifying the generalised Friedmann equation arising from Eq. (4.16) ( cf. Eq. (D.15) below ). Then it follows that Σ2 :=

3 2

Σαβ Σαβ = (Σ+ )2 + (Σ− )2 + (Σ1 )2 + (Σ2 )2 + (Σ3 )2 .

(D.6)

A similar transformation can be obtained for the remaining rank 2 symmetric tracefree spacelike tensor field orthogonal to u/c: { Π± , Πi , E± , Ei , H± , Hi } := 3

4πG 2 4πG c4 π± , 2 c4 πi , E± , Ei , H± , Hi

2

/ (Θ/c) .

(D.7)

Since (Θ/c) is the only variable that continues to carry a physical dimension, which is [ length ]−1 , the equations associated with this variable will decouple from the remaining ones. The decoupled equations are given by Decoupled equations ∂ 0 (Θ/c)

:= − (1 + q) (Θ/c)

(D.8)

∂ α (Θ/c)

:= − rα (Θ/c) ,

(D.9)

where q is the dimensionless cosmological deceleration parameter defined in Eq. (7.20) of Chapter 7. Using the new set of expansion-normalised dimensionless variables, Eqs. (D.2) - (D.4), and employing the related derivative operators of Eq. (D.1), one obtains analogues for the extended 1 + 3 ONF dynamical equations of Chapter 4 — the commutators, Eqs. (4.8) - (4.9), the curvature equations, Eqs. (4.14) - (4.30), and the second Bianchi identities, Eqs. (4.31) - (4.36). These expansion-normalised dimensionless analogues are given in the following.

D.1

The commutators [ ∂ 0, ∂ α ]

= − [ rα − 3 U˙ α ] ∂ 0 + 3

[ ∂ α, ∂ β ]

= 6 αβγ W γ ∂ 0 + 2 (r[α + 3 A[α ) δ γβ] + 3 αβδ N δγ ∂ γ .

D.2

The curvature

D.2.1

Einstein field equations q

1 3

q δ βα − Σβα + βαγ ( W γ + Rγ )

∂ α − rα + 3 U˙ α − 6 Aα ) U˙ α + 2 Σ2 − 6 W 2 + = − (∂

1 2

(Ω + 3P ) − ΩΛ

∂β

(D.10)

(D.11)

(D.12)

∂ 0 Σαβ

=

(q − 2) Σαβ + (δ γ(α ∂ γ − r(α + 3 U˙ (α + 3 A(α ) U˙ β) − 6 W (α Rβ) + Παβ − S αβ h i ∂ γ − rγ + 3 U˙ γ + 3 Aγ ) U˙ γ − 6 Wγ Rγ − 31 δ αβ (∂ h i + 3 γδ(α 2 Rγ Σβ)δ − N β)γ U˙ δ (D.13)

∂ 0 ΩΛ

=

2 (q + 1) ΩΛ

(D.14)

159

D.2. THE CURVATURE

0

=

1 − Σ 2 + 3 W 2 − 6 W α R α − Ω − K − ΩΛ

(D.15)

0

=

∂ β − rβ − 9 Aβ ) Σαβ + 23 rα − 3 N αβ W β + Qα (∂ h i ∂ β − rβ + 6 U˙ β − 3 Aβ ) Wγ − 3 Nβδ Σδγ + αβγ (∂

(D.16)

∂ α − 2 rα ) ΩΛ , (∂

(D.17)

0

=

where Sαβ

2

:= 3 ∗Sαβ / (Θ/c)

∂ (α − r(α ) Aβ) + 3 Bαβ − (∂

=

1 3

∂ γ − rγ ) Aγ + 3 B γγ ] δαβ [ (∂

∂ |γ| − r|γ| − 6 A|γ| ) Nβ)δ − γδ(α (∂

K

Bαβ

D.2.2

2

:= − ( 3 ∗R ) / ( 2 (Θ/c) ) = − (2 ∂ α − 2 rα − 9 Aα ) Aα +

2 Nαγ N γβ − N γγ Nαβ ,

=

(D.18)

3 4

B αα

W 2 := Wα W α .

(D.19)

(D.20)

Jacobi identities ∂ 0 Aα

= q Aα + +

∂ 0 N αβ

1 3

rα − U˙ α +

1 αβγ 2

1 2

∂ β − rβ + 3 U˙ β − 6 Aβ ) Σαβ (∂

∂ β − rβ + 3 U˙ β − 6 Aβ ) (Wγ + Rγ ) (∂

= q N αβ + (δ γ(α ∂ γ − r(α + 3 U˙ (α ) (W β) + Rβ) ) + 6 Σ(αγ N β)γ ∂ γ − rγ + 3 U˙ γ ) (W γ + Rγ ) − δ αβ (∂ h i ∂ γ − rγ + 3 U˙ γ ) Σβ)δ + 6 N β)γ (Wδ + Rδ ) − γδ(α (∂

∂ 0W α

0

=

=

(q − 1) W α + 3 Σαβ W β − 32 N αβ U˙ β i h ∂ β − rβ − 3 Aβ ) U˙ γ − 3 Wβ Rγ + αβγ 12 (∂

=

(D.22)

(D.23)

∂ β − rβ − 6 Aβ ) N αβ + 2 W α + 6 Σαβ W β (∂ ∂ β − rβ ) Aγ − 6 Wβ Rγ ] + αβγ [ (∂

0

(D.21)

∂ α − rα − 3 U˙ α − 6 Aα ) W α . (∂

(D.24)

(D.25)


160

D.2.3

“Electric” and “magnetic” parts of the Weyl curvature tensor ∂ 0 − q + 1) Σαβ + (∂ ∂ (α − r(α + 3 U˙ (α + 3 A(α ) U˙ β) − 3 Σαγ Σγβ = − (∂

Eαβ

− 3 Wα Wβ + 21 Παβ h i ∂ γ − rγ + 3 U˙ γ + 3 Aγ ) U˙ γ − 6 Σ2 − 3 W 2 − 13 δαβ (∂ h i + 3 γδ(α 2 R|γ| Σβ)δ − Nβ)γ U˙ δ .

(D.26)

The field equation (D.13) combined with the above expression (D.26) for Eαβ leads to (Eαβ +

1 2

Παβ )

=

Σαβ − 3 Σαγ Σγβ − 3 Wα Wβ + 6 W(α Rβ) + 13 δαβ 2 Σ2 + 3 W 2 − 6 Wγ Rγ + Sαβ .

(D.27)

The dimensionless version of Eq. (4.28) is Hαβ

D.2.4

∂ (α − r(α + 6 U˙ (α + 3 A(α ) Wβ) + 32 N γγ Σαβ − 9 N γ(α Σβ)γ = − (∂ h i ∂ γ − rγ + 6 U˙ γ + 3 Aγ ) W γ + 9 Nγδ Σγδ + 13 δαβ (∂ ∂ |γ| − r|γ| − 3 A|γ| ) Σβ)δ + 3 Nβ)γ Wδ . + γδ(α (∂

(D.28)

Conformal 3-Cotton–York tensor Cαβ

3

:= 9 ∗Cαβ / (Θ/c) =

∂ |γ| − 2 r|γ| − 3 A|γ| ) Sβ)δ − 9 N γ(α Sβ)γ + γδ(α (∂

3 2

N γγ Sαβ

+ 3 δαβ Nγδ S γδ =

(D.29)

∂ 0 − 2 q + 2) Hαβ + 9 Σγ(α Hβ)γ − 27 N γ(α Σδβ) Σγδ + − (∂

9 2

N γγ Σαδ Σδβ

+ 6 Nαβ Σ2 − 9 N γ(α Πβ)γ + 32 N γγ Παβ − 3 δαβ Σγδ Hγδ − 3 N γδ Σδ Σγ + N γγ Σ2 − 3 Nγδ Πγδ ∂ |γ| − 2 r|γ| − 3 A|γ| ) (Σβ) Σδ + 13 Πβ)δ ) + r|γ| Σβ)δ + 3 γδ(α (∂ i − 2 U˙ |γ| Eβ)δ + 12 Σβ)γ Qδ + 2 R|γ| Hβ)δ .

D.3


D.3.1

Bianchi identities for the Weyl curvature tensor

∂ 0 (E αβ +

1 2

Παβ )

=

(2 q − 1) E αβ + q + −

1 2

1 2

Παβ −

3 2

(D.30)

(Ω + P ) Σαβ

(δ γ(α ∂ γ − 2 r(α + 6 U˙ (α + 3 A(α ) Qβ)

+ 9 Σ(αγ (E β)γ − 16 Πβ)γ ) + 32 N γγ Hαβ − 9 N (αγ Hβ)γ h ∂ γ − 2 rγ + 6 U˙ γ + 3 Aγ ) Qγ − 9 Σγδ (E γδ − + 13 δ αβ 12 (∂ + 9 Nγδ Hγδ h ∂ γ − 2 rγ + 6 U˙ γ − 3 Aγ ) Hβ)δ + γδ(α (∂ + 3 (Wγ + 2 Rγ ) (E β)δ +

1 2

Πβ)δ )

1 6

Πγδ )

161

D.3. THE SECOND BIANCHI IDENTITIES +

∂ 0 H αβ

=

3 2

N β)γ Qδ

(2 q − 1) Hαβ + 9 Σ(αγ Hβ)γ +

i 9 2

(D.31) W (α Qβ) −

3 2

N γγ (E αβ −

1 2

Παβ )

+ 9 N (αγ (E β)γ − 12 Πβ)γ ) − 3 δ αβ Σγδ Hγδ + 12 Wγ Qγ + Nγδ (E γδ − 12 Πγδ ) h ∂ γ − 2 rγ − 3 Aγ ) (E β)δ − 12 Πβ)δ ) + 6 U˙ γ E β)δ − γδ(α (∂ i − 32 Σβ)γ Qδ − 3 (Wγ + 2 Rγ ) Hβ)δ 0

=

∂ β − 2 rβ − 9 Aβ ) (E αβ + (∂ + Qα − − 3 αβγ

0

=

3 2

1 2

Παβ ) −

1 3

∂ β − 2 rβ ) Ω δ αβ (∂

Σαβ Qβ − 9 Wβ Hαβ Σβδ Hδγ −

3 2

Wβ Qγ + Nβδ (E δγ +

1 2

Πδγ )

N αβ Qβ ∂ β − 2 rβ − 3 Aβ ) Qγ + Σβδ (E δγ + + 3 αβγ 16 (∂ − Nβδ Hδγ .

D.3.2

∂ β − 2 rβ − 9 Aβ ) Hαβ + 3 (Ω + P ) W α + 9 Wβ (E αβ − (∂ −

(D.32)

(D.33)

1 6

Παβ )

3 2

1 2

Πδγ ) (D.34)

Bianchi identities for the source terms ∂ 0Ω

=

∂ α − 2 rα + 6 U˙ α − 6 Aα ) Qα − 3 Σαβ Παβ (2 q − 1) Ω − 3 P − (∂

∂ 0 Qα

=

∂ β − 2 rβ ) P − 3 (Ω + P ) U˙ α 2 (q − 1) Qα − δ αβ (∂ ∂ β − 2 rβ + 3 U˙ β − 9 Aβ ) Παβ − 3 Σαβ Qβ − (∂ − 3 αβγ (Wβ − Rβ ) Qγ − Nβδ Πδγ .

(D.35)

(D.36)

162


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Thank You, . . . Danke, Euch . . . • Reza Tavakol (QMW London, UK, EU), whose generosity and determination to give support (wherever possible) cannot be described in terms of words. And Emma Jones. • Anne & Gerd van Elst (Wörth/Rhein, FRG, EU), meinen Eltern. • Ulrich und Ansgar van Elst (Wörth/Rhein, FRG, EU), meinen Brüdern. • George F R Ellis (UCT Cape Town, RSA). It’s been a great pleasure to work with him. Most impressive is his enthusiasm for the subject. In his way of giving talks he reminds me of a particular famous rock star as seen on stage. • . . . to my dear relativity pals (and, touching the issue of priorities, far beyond) Marco Bruni (Trieste, I, EU) and Claes Uggla (Stockholm, S, EU). • Ian W Roxburgh (QMW London, UK, EU), for his concern about maintaining a strong European — well, cosmopolitan — atmosphere within the QMW Astronomy Unit. • The Drapers’ Society at Queen Mary & Westfield College for essential financial support. • Irmgard & Albert Freiberg (Coburg, FRG, EU), meinen Großeltern. • Gisela & Rudolf Ernst (Köln, FRG, EU). • Sibylle & Jochen Hoffmann (Uhingen–Nassachmühle, FRG, EU). • Edeltraut van Elst (Herne, FRG, EU). • Roustam M Zalaletdinov (University of Tashkent, Uzbekistan), who dared speaking out what I was only thinking; that “Einstein’s field equations are erotic”. • John Wainwright (University of Waterloo, CDN). • Roy Maartens (University of Portsmouth, UK, EU). • Malcolm A H MacCallum (QMW London, UK, EU), for a very warm welcome at the beginning of my studies, which made me feel as if I had always been a member of the QMW Relativity Group. 173

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ACKNOWLEDGEMENTS

• . . . to that funky little man from Minneapolis, Minnesota, USA, known until Mon, June 7, 1993 as Prince Rogers Nelson, whose music has constantly been of incredibly revitalising power ever since I became an addict (a long time ago). • . . . to the heart and soul of the QMW Maths Department, the secretarial staff Ms Ann Cook, Ms Claire Bald, Ms Shirley Platt, Ms Ruth Anderson, Ms Angie Waidson, and the tea lady Ms Pat Kendal. • Gunda Metzler-Ruder und Dieter Ruder (Jena, FRG, EU). • Claudia Metzler (Jena, FRG, EU). • Dietrich Kramer, Hans Stephani und Gernot Neugebauer (FSU Jena, FRG, EU). • Bernard J Carr (QMW London, UK, EU), for his lectures on “Relativity and Cosmology”, held between January and May 1989. • Thomas Wolf (QMW London, UK, EU). • William B Bonnor (QMW London, UK, EU), for establishing contacts with relativity researchers in Jena, FRG, EU. • . . . all members of the QMW Relativity Group. • Friedrich Herrmann (Universität Karlsruhe, FRG, EU). • Susan Shea (Omaha, NE, USA). • Alison Fiddian-Green (London, UK, EU). • . . . dem Jugendzentrum Wörth (Rhein), und der Wörth (Rhein)-Basis, mit Anhang: Marco Kattannek und Mari Fernandez, Sebi G Meyer und Petra Wolff, Thomas “Joe” Hänig und Silvia Fleischer, und Paisley Park Soulmate Roland Gimmler. — “Rock ‘n’ Roll.”

Curriculum Vitae • Name: Henk van Elst. • Born: July 3, 1964 in Essen (Bundesland: Nordrhein–Westfalen), Federal Republic of Germany, EU. • Education: August 1970 - August 1974: Grundschule Dorschbergschule Wörth (Rhein) (Bundesland: Rheinland–Pfalz), FRG, EU. August 1974 - June 1983: Europa–Gymnasium Wörth (Rhein). Abitur degree taken in Mathematics, Physics and English. • October 1983 - December 1984: National Service in the German Bundeswehr. • April 1986: Official confirmation of the status of a Conscientious Objector received. • October 1985 - August 1992: Studies towards a Diplom degree in Physics at the Universität Karlsruhe (Bundesland: Baden–Württemberg), FRG, EU. • October 1988 - June 1989: Associate Student within the Physics Department of Queen Mary College, London, United Kingdom, EU. • November 1990 - December 1991: Diplomarbeit at the Theoretisch–Physikalisches Institut of the Friedrich–Schiller–Universität Jena (Bundesland: Thüringen), FRG, EU. • October 1992 - September 1996: Studies towards a Ph.D. degree in Relativity and Cosmology within the Astronomy Unit of the School of Mathematical Sciences at Queen Mary & Westfield College, London, UK, EU.

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