Symmetries of spacetimes embedded with an Electromagnetic String Fluid
arXiv:1809.00293v1 [gr-qc] 2 Sep 2018
Michael Tsamparlis∗ Faculty of Physics, Department of Astronomy-Astrophysics-Mechanics, University of Athens, Panepistemiopolis, Athens 157 83, Greece. Antonios Mitsopoulos Faculty of Physics, Department of Astronomy-Astrophysics-Mechanics, University of Athens, Panepistemiopolis, Athens 157 83, Greece. Andronikos Paliathanasis† Instituto de Ciencias F´ısicas y Matem´aticas, Universidad Austral de Chile, Valdivia, Chile. Department of Mathematics and Natural Sciences, Core Curriculum Program, Prince Mohammad Bin Fahd University, Al Khobar 31952, Kingdom of Saudi Arabia. Institute of Systems Science, Durban University of Technology Durban 4000, Republic of South Africa.
Abstract The electromagnetic string fluid (EMSF) is an anisotropic charged string fluid interacting with a strong magnetic field. In this fluid we consider the double congruence defined by the 4-velocity of the fluid ua and the unit vector na along the magnetic field. Using the standard 1+3 decomposition defined by the vector ua and the 1+1+2 decomposition defined by the double congruence ua , na we determine the kinematic and the dynamic quantities of an EM string fluid in both decompositions. In order to solve the resulting field equations we consider simplifying assumptions in the form of collineations. We decompose the generic quantity LX gab in a trace ψ and and a traceless part Hab . Because all collineations are expressible in terms of the quantity LX gab it is possible to compute the Lie derivative of all tensors defined by the metric i.e. the Ricci tensor, the Weyl tensor etc. This makes possible the effects of any assumed collineation on the gravitational field equations. This is done as follows. Using relevant identities of Differential Geometry we express the quantity LX Rab where Rab is the Ricci tensor in terms of the two irreducible parts ψ, Hab . Subsequently using the gravitational field equations we compute the same quantity LX Rab in terms of the Lie derivative of the dynamic variables. We equate the two results and find the field equations in the form LX Dynamic variable = F (ψ, Hab , dynamic variables). This result is general and holds for all gravitational systems and in particular for the EMSF. Subsequently we specialize our study at two levels. We consider the case of a Conformal Killing Vector (CKV) parallel to ua and a CKV parallel to na . Finally we solve the resulting field equations in the first case to the Friedman Robertson Walker (FRW) spacetime and in the second case for the Bianchi I spacetime. In the latter case we find a new solution of the gravitational field equations.
1
Introduction
When a charged plasma enters a strong magnetic field it is possible the pressures along the magnetic field and perpendicular to the magnetic field to be unequal. This results in a physical system which we call an anisotropic ∗ Email: † Email:
[email protected] [email protected]
1
electromagnetic string fluid (EMSF). In this fluid one is possible to define a double congruence consisting of the 4-velocity ua of the plasma fluid and the and the spacelike unit vector na = H a /H defined by the magnetic field. This makes possible the development of two decompositions of tensor fields over the spacetime manifold that is, the standard 1+3 decomposition defined by ua and the 1+1+2 decomposition defined by the double congruence (ua , na ). Each decomposition defines kinematic quantities and dynamic variables. The first are defined by the decomposition of the covariant derivatives ua;b and na;b and satisfy the propagation and the constraint equations which result form the decomposition of the Ricci identity for the vectors (ua , na ). The second result from the decomposition of the energy momentum tensor Tab and the conservation equations T;bab = 0. Both types of quantities are related by the field equations of the electromagnetic field in the RMHD approximation (with infinite conductivity and vanishing electric field) and the Einstein’s field equations. In order to solve the resulting field equations one has to make simplifying assumptions. At the geometrical level these assumptions are mainly collineations (or symmetries) of geometric objects A defined by the metric and are expressed by equations of the form LX A = B where B is a tensor field with the same number and the same symmetries of indices as A. X is the vector generating the symmetry. At the level of Physics the simplifying assumptions are mainly equations of state which are expressed by relations among the dynamic variables introduced either by the 1+3 or the 1+1+2 decomposition. In this work we restrict our considerations to the first type of simplifying assumptions, that is the symmetries. In order to approach the subject from a general point of view we decompose the quantity LX gab in a trace and a traceless part. The reason for doing this is because all collineations can be expressed in terms of the quantity LX gab therefore it is possible to compute easily the effects of a symmetry. Subsequently in order to find the effects of an assumed collineation on the field equations we compute the quantity LX Rab in terms of these two irreducible parts. We specialize the general result to the cases the EMSF admits a timelike conformal Killing vector (CKV) parallel to the four velocity ua and a CKV parallel to the magnetic field H a . Still this is quite general and the equations we obtain are independent of the particular spacetime metric considered. We specialize further the results to the cases of the Friedmann Robertson Walker (FRW) spacetime and the case of the Bianchi I spacetime. In the latter case we find a new solution of Einstein’s equations. The structure of the paper is as follows. In Section 2 for the convenience of the reader we present briefly the basic properties of the 1+3 and 1+1+2 decompositions. The EMSF is defined in Section 3 where we discuss the 1+1+2 decomposition of Maxwell equations. Furthermore, the gravitational field equations for the model are presented in Section 4. In Sections 5 and 6 we find the kinematic and the dynamic conditions for the gravitational field equations when the spacetime admits a timelike or a spacelike CKV. We demonstrate our results in Section 6.3 where we apply the constraint conditions in the case of Bianchi I spacetime. Finally, in Section 7 we draw our conclusions.
2
The 1+3 and the 1+1+2 decompositions
In order to make the material to follow tangible it is necessary that we refer briefly the 1+3 decomposition and the 1+1+2 decomposition of geometrical objects in spacetime manifolds M with metric gab . Details on these decompositions can be found in various sources e.g. [14].
2.1
The 1+3 decomposition
Consider on a spacetime manifold M a class of observers with four-velocity ua (ua ua = −1). ua defines the projection operator hab = gab + ua ub with respect to which all geometric objects defined on M can be 1 + 3 decomposed. The first application of the 1+3 decomposition is at the level of kinematics where one defines the kinematical variables associated to the observers ua as follows 1 ua;b = −u˙ a ub + ω ab + σ ab + θhab | {z3 } =θab
In this decomposition ωab = hca hdb u[c;d] is called the vorticity tensor,h θ ab = hca hdb u(c;d) ,iθ = θ aa = hab ua;b = ua ;a
is called the expansion (isotropic strain) and σ ab = θab − 31 θhab = hc(a hdb) − 31 hcd hab uc;d is called the shear 2
stress tensor or simply the shear. From the vorticity tensor one defines the vorticity vector ωa =
1 abcd η ub;c ud . 2
At the level of dynamics the 1+3 decomposition concerns the energy momentum tensor Tab of a fluid in M and defines the physical quantities of the fluid as observed by the observers ua as follows Tab = µua ub + phab + 2q(a ub) + π ab .
(1)
The variables µ, p, q a , π ab have the following physical interpretation: a. The scalars µ = Tab ua ub and p = 31 hab Tab correspond respectively to the energy (mass) density and the isotropic pressure of the fluid. b. The spacelike vector qa = −hda Tdc uc is the energy (heat) flux in the three space defined by the projection tensor hab . c. π ab = hca hdb − 31 hcd hab Tcd is the traceless (π aa = 0) stress tensor (measures the anisotropy). The fluids are classified according to the dynamical variables in dust (p = qa = π ab = 0), perfect fluid (qa = π ab = 0), heat conducting fluid (qa 6= 0) and anisotropic fluid (π ab 6= 0). At the same level one considers the 1+3 decomposition of the conservation equation T ab;b = 0 which leads to the following two equations ( see [14]) .
.
a µ + (µ + p)θ + π ab σ ab + q;a + q a ua 4 . . (µ + p)ua + hca (p,c + π bc;b + q c ) + ω ac + σ ac + θhac q c 3
=
0
(2)
=
0.
(3)
Finally the 1+3 decomposition of the Ricci identity ua;bc − ua;cb = Rd abc ud leads to two sets of equations which are called the propagation and the constraint equations (see [23]).
2.2
The 1+1+2 decomposition
Assume that in addition to the four-velocity vector ua there exists in M a second unit spacelike vector field na (na na = 1, ua na = 0). The pair of vectors ua , na constitutes a double congruence and defines the new projection operator pab = hab − na nb which projects normal to both vectors ua , na on the 2-dimensional screen space. Using the operator pab one is possible to 1+1+2 decompose a general tensor field on M along the vectors ua , na and on the screen space (see [14], [18]). At the kinematic level one has the 1+1+2 decomposition of the tensor na;b which defines the “kinematic” variables of the spacelike vector field na . This decomposition is (see [12], [14] and references cited therein): i h ∗ ∗ (4) na;b = Aab + na nb − n˙ a ub + ua nc uc;b + (nc u˙ c )ub − (nc uc )nb ∗
where s˙ ≡ s...;a ua , s ≡ s...;a na and Aab = pca pdb nc;d . We decompose the screen tensor Aab into its irreducible parts (the kinematic variables of na ): 1 Aab = Sab + Rab + Epab (5) 2 where Sab = Sba , S bb = 0 is the traceless part (screen shear), Rab = −Rba is the antisymmetric part (screen rotation) and E is the trace (screen expansion). We have the defining relations: 1 cd c d (6) Sab = pa pb − p pab n(c;d) 2 Rab = pca pdb n[c;d] cd
E=p nc;d = n 3
(7)
c
;c
c
+ n˙ uc .
(8)
One can define also the screen rotation vector Ra = 21 η abcd Rbc ud . The ua −term in (4) can be written: . − Nb + 2ωcb nc + pcb nc where
∗ Nb = pbc n˙ c − uc = pbc Lu nc
(9) (10)
is called the Greenberg vector. This vector is important because it vanishes iff the vector fields ua , na are surface forming (that is iff Lu nb = Aub + Bnb ). From the kinematics point of view the vector Na vanishes iff the vector field na is “frozen in” along the observers ua . Using the Greenberg vector, identity (4) is written: . ∗ . na;b = Aab + na nb − na ub + ua pcb nc + 2ω dc nd − Nc . At the level of dynamics the energy momentum tensor is 1+1+2 decomposed as follows
1 Tab = µua ub + 2κu(a nb) + 2Q(a ub) + γna nb + 2P(a nb) + αpab + Dab 2 1 = µua ub + phab + 2κu(a nb) + 2Q(a ub) + γ¯ na nb − pab + 2P(a nb) + Dab 2 which introduces the new dynamical variables κ, γ, α, γ¯ , Qa , Pa , Dab which concern physical quantities observed by the observers ua . They are given by the formulae κ = −Tab ua nb a b
γ = Tab n n ,
(11) a b
γ¯ = π ab n n = γ − p
ab
α = p Tab = 3p − γ Qa = Dab
−Tbc ub pca ,
(12) (13)
Tbc nb pca ,
Pa = 1 = pca pdb − pcd pab Tcd . 2
qa = κna + Qa
(14) (15)
The physical meaning of each of the new physical variables is assumed to be the following a. The scalar κ and the screen vector Qa are related to the heat conduction of the fluid b. The scalar γ, the screen vector Pa , and the traceless screen tensor Dab have to do with the anisotropy of the fluid. The 1+1+2 dynamical variables are related to the 1+3 dynamical variables as follows q a = κna + Qa 1 π ab = γ¯ na nb − pab + 2P(a nb) + Dab . 2
(16) (17)
The above describe briefly the tools introduced by the 1+3 and the 1+1+2 decompositions. The former has been considered since many years in the study of relativistic Physics whereas the latter is relatively more recent and less known. However both decompositions are important because they allow one to classify the fluids in a covariant manner in various classes and makes possible the separate study of the elements in each class.
2.3
The string fluid defined by the electromagnetic field
Using the 1+1+2 decomposition one is possible to classify again the relativistic fluids. However this is not usually done and people have considered only special cases. One such case concerns the string fluid. Physically this fluid is considered to be a combination of geometric strings with particles attached to them so that both have the same four velocity [3],[4],[5]. The energy momentum tensor of a string fluid is considered to be Tab = ρ(ua ub − na nb ) + qpab where: 4
(18)
- ρ = ρp + ρs is the sum of the mass density of the strings (ρs ) and the mass density of the particles (ρp ), - ua is the common four velocity (ua ua = −1) of the string and the attached particle - na is a unit spacelike vector (na na = 1) normal to ua (ua na = 0), which specifies the direction of the string (and the direction of anisotropy of the string fluid) - q is a parameter contributing to the dynamic and kinematic properties of the string - pab = hab − na nb is the screen projection operator defined by the vectors ua , na . By rewriting the energy momentum tensor as: 1 1 Tab = ρua ub + (2q − ρ)hab + (q + ρ) (19) hab − na nb 3 3 (or otherwise) we find that in the 1+3 classification the dynamical variables of a string fluid are as follows 1 1 a µ = ρ, p = (2q − ρ), q = 0, π ab = (q + ρ) (20) hab − na nb . 3 3 It follows that in the 1+3 classification the string fluid is an anisotropic fluid with vanishing heat flux. Furthermore we note that na is an eigenvector of the anisotropic stress tensor π ab with eigenvalue − 23 (q + ρ). We assume q + ρ 6= 0 otherwise the string fluid reduces to a perfect fluid with energy momentum tensor Tab = qgab . This latter fluid has the unphysical equation of state µ + p = 0. We consider now an anisotropic fluid defined by the requirement Tab = µua ub + pk na nb + p⊥ pab .
(21)
In order to relate the 1+1+2 dynamical variables of the string fluid with the dynamical variables of the 1+3 classification we rewrite (21) as follows 1 1 (22) hab − na nb . Tab = µua ub + (pk + 2p⊥ )hab + (p⊥ − pk ) 3 3 from which follows p = π ab
=
qa
=
1 (pk + 2p⊥ ), 3 1 (p⊥ − pk ) hab − na nb , 3 0.
Furthermore, in the 1+1+2 decomposition the tensor π ab is written as 2 1 π ab = − (p⊥ − pk ) na nb − pab , 3 2
(23) (24) (25)
(26)
from which it follows that the only nonvanishing irreducible part of π ab is: 2 γ¯ = − (p⊥ − pk ). 3
(27)
Hence, we may say that the string fluid defined by (21) is the “simplest” anisotropic fluid. An important example of a string fluid is the electromagnetic field in the RMHD approximation with infinite conductivity and vanishing electric field. Indeed in this approximation the electromagnetic tensor Fab is given by the expression Fab = η abcd uc H d where H a is the magnetic field. ab The Minkowski energy momentum tensor of the electromagnetic field TEM is given by 1 ab = λ F ac F b c − g ab Fcd F cd EM T 4 5
(28)
(29)
where λ is a constant. Using Maxwell equations one shows that EM T
ab
;b
= −F ab Jb .
(30)
Replacing in (29) Fab from (28) we find EM Tab
=
1 1 λH 2 ua ub + λH 2 hab + λH 2 2 6
1 hab − na nb , 3
(31)
where na = H a /H, that is, na is the unit vector in the direction of the magnetic field. To write the tensor EM Tab in 1+1+2 decomposition wrt the vectors ua , na we replace hab = pab + na nb and find 1 1 2 2 (32) EM Tab = λH (ua ub − na nb ) + λH pab 2 2 which is a string fluid energy momentum tensor with −µ = pk = −p⊥ = − 12 λH 2 .
3
The electromagnetic string fluid (EMSF)
We consider the dynamical system consisting of a charged perfect fluid with isotropic pressure p and energy density ρ which interacts with the electromagnetic field in the RMHD approximation with infinite conductivity and vanishing electric field. Physically this situation is considered to be the case in various plasmas [22]. Due to the interaction of the fluid with the electromagnetic filed it is possible that the magnetic field produces a different fluid pressure perpendicular and parallel to the magnetic field therefore the perfect fluid becomes an anisotropic fluid with pressure distribution pk na nb + p⊥ pab . The energy momentum tensor of the interacting fluid is then 1 1 1 2 2 2 (33) Tab = µ + λH ua ub + pk − λH na nb + p⊥ + λH pab . 2 2 2 To find the 1+3 dynamical variables of the interacting fluid we replace pab = hab − na nb and find: 2 1 1 1 1 p⊥ + pk + λH 2 hab − (pk − p⊥ − λH 2 ) hab − na nb . Tab = µ + λH 2 ua ub + 2 3 3 6 3
(34)
from which follows ρ p π ab qa
1 = µ + λH 2 , 2 2 1 1 = p⊥ + pk + λH 2 , 3 3 6 1 = −(pk − p⊥ − λH 2 ) hab − na nb , 3 = 0
(35) (36) (37) (38)
that is we find a non-conducting anisotropic fluid. The interacting fluid is not a string fluid (EMSF). For this to be the case the energy momentum tensor (34) must agree with (18) which leads to the following condition 1 1 2 2 µ + λH = − pk − λH ⇒ µ = −pk . (39) 2 2 We note that this condition is on the perfect fluid and not on the magnetic field which is already a string fluid. With this condition assumed the energy momentum of the interacting fluid becomes 1 1 1 2 2 2 Tab = µ + λH ua ub − µ + λH na nb + p⊥ + λH pab . 2 2 2
6
Relations (35) - (38) become ρ
=
p = qa
=
π ab
=
1 µ + λH 2 , 2 1 1 2p⊥ − µ + λH 2 , 3 2 0, 1 2 (µ + p⊥ + λH ) hab − na nb 3
(40) (41) (42) (43)
from which follows that the EMSF is a non-conducting charged viscous fluid. We note that in terms of the dynamical variables the interacting string fluid gives 1 1 µ′ = µ + λH 2 , κ = 0, Qa = 0, γ = −µ − λH 2 , Pa = 0, α = 2p⊥ + λH 2 , Dab = 0. 2 2 But
2 α = 3p − γ =⇒ p = 3
therefore γ¯ = γ − p =⇒ γ¯ = − Finally we note the relation:
1 γ 2 p⊥ + λH + 2 3 2 µ + p⊥ + λH 2 . 3
ρ + q = µ + p⊥ + λH 2 which is useful in the calculations. One direction in which the string fluids have been studied is the simplification of the field equations for various types of collineations of spacetime [6], [7], [9], [8], [10], [11] [20]. In the next sections we extend these studies to the case of the EMSF.
3.1
The Ricci tensor of the EMSF
We consider Einstein field equations in the form Rab = Tab + Λ − 21 T gab where T ≡ Taa and compute Rab in terms of the string fluid variables: ρ
=
q
=
1 µ + λH 2 2 1 p⊥ + λH 2 . 2
(44) (45)
We find: Rab
= (q − Λ)(ua ub − na nb ) + (ρ + Λ)pab 1 1 2 2 = p⊥ + λH − Λ (ua ub − na nb ) + µ + λH + Λ pab . 2 2
(46) (47)
We note immediately that Rab is found from Tab if we interchange ρ ↔ q − Λ, q ↔ ρ+ Λ and vice versa. This is a useful observation because it allows us to compute various results for Rab /Tab and write down the answer for the corresponding quantities for Tab /Rab by interchanging the string variables as indicated above. For example the 1+3 decomposition of Rab is written directly from (19) as follows: 1 1 1 1 2 2 2 Rab = p⊥ + λH − Λ ua ub + 2µ − p⊥ + λH + 3Λ hab + (µ + p⊥ + λH ) hab − na nb . (48) 2 3 2 3
7
3.2
The conservation equations for the EMSF in the 1+1+2 decomposition
In the case of the EMSF the conservation equations (2),(3) are simplified as follows 2 . a b ρ + (ρ + q) = 0 θ − σ ab n n 3 i h. ∗ ∗ . (ρ + q) ua − (E − nb ub )na − hba nb + pba q,b − ρna = 0.
(49) (50)
Furthermore, by projecting the second equation along na and using of the tensor pab , we get the two equations ∗
ρ + (ρ + q)E h i . ∗ pba q,b + (ρ + q)(ub − nb )
= 0
(51)
= 0.
(52)
Replacing the energy density ρ and the heat coefficient q from expressions (44),(45) we find that the 1+1+2 decomposition of the conservation equation for an EMSF are . 2 . a b 2 = 0 (53) θ − σ ab n n µ + λH H + (µ + p⊥ + λH ) 3 ∗
∗
µ + λH H + (µ + p⊥ + λH 2 )E h i . ∗ pba p⊥,b + λHH,b + (µ + p⊥ + λH 2 )(ub − nb )
= 0
(54)
= 0.
(55)
These equations are independent of any other assumptions which one might do concerning the fluid, including the symmetries. Therefore for each additional assumption (including the symmetry assumptions) the conservation equations take a different form and in that form supplement the rest of the field equations as constraint equations. We continue our analysis with the 1+1+2 decomposition of Maxwell equations.
3.3
Maxwell equations in the 1+3 and the 1+1+2 formalisms
Maxwell equations are F[ab;c] = 0, F ab;b = J a
(56)
where F ab is the electromagnetic field tensor and J a is the 4-current. The 4-current and the electromagnetic field tensor in the 1+3 decomposition are decomposed as follows Ja
= eua + j a
ab
a
F
b
(57) b
a
= u E −u E +η
abcd
Hc u d
(58)
where η abcd is the alternating tensor1 and the various physical quantities introduced are (a) e the charge density (b) j a the conduction current (c) E a the electric field and (d) H a the magnetic field, all these quantities 1 In
Minkowski spacetime the alternating tensor is defined as follows: η abcd = η[abcd] , η0123 = (−g)−1/2
where g = det(gab ).It satisfies the properties: [b
d]
[c d]
η abcd η arst = −3!δr δ cs δ t , η abcd η abst = −4δ s δ t In the Euclidian 3-d space the alternating tensor is defined as follows: ηµνρ = η [µνρ] , η 123 = h−1/2 where h = det(hµν ). It satisfies the properties: [ν ρ]
η µνρ η µστ = 2δ σ δ τ , η µνρ η µντ = 2δ ρτ .
8
measured by the observer ua . Inverting (57), (58) we find
E
e
=
a
=
−ua Ja , j a = hab J b 1 F ab ub , H a = η abcd Fbc ud . 2
(59) (60)
Taking into account the 1+3 kinematic variables Maxwell equations are 1+3 decomposed wrt the observer ua into the following constraint and propagation equations: b hab H;a b hab E;a . b hab H . b hab E
= 2ω a Ea
(61) a
= e − 2ω Ha
(62)
= ua;b H b − θH a − I a (E)
(63)
= ua;b E b − θE a + I a (H) − j a
(64)
where: I a (E) = I a (H) =
.
η abcd ub (uc Ed − Ec;d ) . η abcd ub (uc Hd − Hc;d )
(65) (66)
and ω a = 12 η abcd ub;c ud and θ = ua;a are the the vorticity vector and the expansion of the fluid as measured by the observers ua . A dot over a symbol denotes covariant differentiation wrt ua (i.e. along the fluid particle world line). If we operate on (65) and (66) with η abcd ud then a direct calculation yields the following two mathematical identities: E[r;s] H[r;s]
. 1 . = u[r E s] + u[r Es] + ut Et;[r us] + η rstm ut I m (E) 2 . 1 . = u[r H s] + u[r Hs] + ut Ht;[r us] + η rstm ut I m (H). 2
(67) (68)
From the identity (68) one computes the screen rotation of the magnetic field lines. The result is: Rab
= pca pdb n[c;d] = = −
1 c d p p H[c;d] H a b
1 c d p p η I r (H)us . 2H a b cdrs
(69)
Proposition 1 The screen rotation vector of the magnetic field lines is proportional to the magnetic field as follows: Hc I c (H) a H (70) Ra = − 2H 3 Proof Expanding pca , pdb in (69) we get: Rab = −
1 1 η abrs I r (H)us − 3 H[a η b]crs H c I r (H)us 2H H
We operate with η abpq on both sides and find: 1 abpq η η abrs I r (H)us 2H
=
1 abpq η H[a η b]crs H c I r (H)us H3
=
− −
1 p [I (H)uq − I q (H)up ] H 1 Hc I c (H) p q [H u − H q up ] . [−I p (H)uq + I q (H)up ] + H H3
Thus: η abpq Rab =
Hc I c (H) p q [H u − H q up ] . H3 9
(71)
In terms of the screen rotation vector Ra = 12 η abcd ub Rcd equation (71) is written: Ra = −
Hc I c (H) a H 2H 3
which completes the proof. From Proposition 1 we infer that the screen rotation of the magnetic field congruence vanishes iff Hc I c (H) = 0.
3.4
Maxwell equations in the RMHD approximation
In the RMHD approximation with infinite electrical conductivity and vanishing electric field Maxwell equations become: b hab H;a
= 0
(72) a
. b
hab H I a (H)
= 2ω Ha
(73)
= ua;b H b − θH a = ja.
(74) (75)
Let na = H a /H be the unit vector in the direction of the magnetic field. Geometrically na is the unit tangent to the spacelike magnetic field lines. The pair (ua , na ) forms a double congruence. Maxwell equations in terms of the irreducible parts defined by this double congruence take a geometric form. The constraint equation (72) for the magnetic field gives: Hhab nb;a + H,a na = 0. But hab nb;a = pab nb;a = E where E is the screen expansion of the magnetic field lines. Therefore E = −(ln H)∗
(76)
where a “*” over a symbol means covariant differentiation wrt na . From this equation we infer that the stronger the magnetic field the denser the magnetic field lines on the screen space, that is the greater is the magnetic flux through the screen space (as expected). We examine now the propagation equation (74) of the magnetic field. We have: H˙ a .b n + hab n = ua;b nb − θna . H Contracting with na and projecting with pab we get the pair of equations: (ln H)· Na
2 = σ ab na nb − θ 3 ≡ pab Lu nb = 0.
(77) (78)
The first equation involves the change of the strength of the magnetic field along the flow lines of the fluid i.e. the field ua . A kinematic interpretation is that the vector na is an eigenvector of the shear with eigenvalue (ln H)· + 32 θ. The second is the geometric condition that the magnetic field lines are material lines in the fluid and correspond to the statement that the magnetic field is “frozen” along the fluid. Physically this means that each particle of the fluid moves always on the same magnetic field line. Relation (75) due to (70) it is written as: Ra = −
Hc j c a H 2H 3
(79)
therefore in the RMHD approximation the screen rotation of the magnetic field lines vanishes iff the conduction current j a is normal to the magnetic field. 10
Ohm’s Law in its generalized form which includes the Hall current is written (see [21]): J a = ρua +
a 1 kE + λkη abcd Eb uc Bd + λ2 k(E c Bc )B a . c (1 + λ B Bc ) 2
(80)
In the RMHD approximation we have that the spatial part j a of the 4-current J a vanishes, therefore Ra = 0. Hence in a perfectly conducting fluid for which generalized Ohm’s law applies, the magnetic field lines have zero rotation as measured by ua . Because (80) is not the most general form of Ohm’s Law we shall assume in the following Ra to be given by (79). Summarizing we have that in the RMHD approximation Maxwell equations are: E Ra e (ln H)· Na
= −(ln H)∗ Hc j c a = − H 2H 3 a = 2ω Ha
(81) (82) (83)
2 = σ ab na nb − θ 3 ≡ pab Lu nb = 0.
(84) (85)
Note that equation (85) can be written in the equivalent form: .b
pab n = (pa.c σ cb + ω a.b ) nb .
(86)
These equations are general and independent of further simplifying assumptions (e.g. symmetry assumptions) we might do, and hold in all cases.
4
The field equations for the EMSF
The EMSF must satisfy three sets of equations: a) Maxwell equations, b) Conservation laws and c) Einstein field equations. We have already given Maxwell equations and the conservation equations. Concerning Einstein field equations we shall consider their Lie derivative along some characteristic direction of the EM fluid. The reason for this is that we want to employ symmetry assumptions, that is equations of the form Lξ Mab = Aab where Mab is a tensor computed in terms of the metric (or the metric itself) and Aab is an arbitrary tensor having the same symmetries as the Mab . Due to the form of Einstein field equations we compute Lξ Rab in terms of Lξ gab using various identities of Riemannian Geometry. Then we impose the symmetry assumption by choosing a specific form for Aab . For example for a CKV Mab = gab and Aab = 2ψgab where ψ(xa ) is the conformal factor. Then we replace Lξ Rab in the Lie derivative of the field equations and we find the field equations in a form that incorporates already the imposed geometric symmetry assumption. In a previous work on string fluids [20] we have computed Einstein equations for a general string fluid and many types of symmetries in the cases that the symmetry vector is either ξ a = ξua or ξ a = ξna . Therefore we could write straight away the field equations in the case of an EMSF by simply specifying na = H a /H. Of course in this case the resulting equations will be supplemented by Maxwell equations. In the following we shall recall briefly some important intermediate steps in order to make the paper more readable and self contained. Details can be found in [20] The Lie derivative of the Ricci tensor wrt a general time-like vector ξ a = ξua has been computed (see equation (3.9) in [14]) in terms of the standard dynamic variables µ, p, qa , π ab . By using the general expression, Lξ Rab can be written in terms of the string fluid parameters ρ, q. In a similar way, the Lie derivative of the Ricci tensor along the spacelike vector ξ a = ξna can be expressed in terms of the 1+1+2 dynamic quantities. Using Maxwell equations we show easily that the conservation equations (53), (54) and (55) (which must also be satisfied in all cases) are simplified as follows: .
µ − (µ + p⊥ )(ln H)· ∗
pba
h
∗
µ − (µ + p⊥ )(ln H) i . ∗ p⊥,b + λHH,b + (µ + p⊥ + λH 2 )(ub − nb ) 11
= 0
(87)
= 0
(88)
= 0.
(89)
Concerning the Einstein filed equations we have from[14] and [20] the results: 1 Lξ Rab ξ
. . q + 2(q − Λ)(ln ξ)· ua ub + 2(q − Λ) uc − (ln ξ),c u(a hcb) + 1 . . 2 + 2ρ − q + (2ρ − q + 3Λ)θ hab + 3 3 2 2 1 . . + ρ + q + (ρ + q)θ hab − na nb + (2ρ − q + 3Λ)σ ab + 3 3 3 1 +2(ρ + q) hcd − nc nd δ d(a (ω c.b) + σ c.b) ) − 3
=
.
−2(ρ + q)nd hd(a nb) .
1 Lξ Rab ξ
h∗ i h∗ i . q + 2(q − Λ)uc nc ua ub − 2(q − Λ) uc nc − (ln ξ)· u(a nb) − i h∗ − q + 2(q − Λ)(ln ξ)∗ na nb − −2 (ρ + Λ)Nc + 2(q − Λ)ω dc nd u(a pcb) − h∗ i h∗ i −2(q − Λ)pdc nd + (ln ξ),d n(a pcb) + ρ + (ρ + Λ)E pab +
=
+2(ρ + Λ)Sab .
(90)
(91)
Expressions (90) and (91) are general and hold for all collineations and all string fluids. For each type of collineation the lhs of the expressions (90) and (91) simplifies accordingly and for each specific string fluid the rhs is simplified the same. Equating the two parts one finds Einstein field equations for the specific string fluid considered and the specific symmetry assumed. In the case of the EMSF one computes the quantities ρ, q in terms of the physical variables of the EMSF, . ∗ that is the quantities µ, p⊥ , H using equations (44), (45) and the kinematic quantities ub , nb etc. The result applies to all collineations concerning the EMSF. For easy reference we collect below the results of the calculations : Maxwell equations: .b
Na E 2 σ ab na nb − θ 3
= =
0 ⇐⇒ pab n = (pa.c σ cb + ωa.b ) nb −(ln H)∗
=
(ln H)
e
=
2ωa Ha ,
·
(92) (93) (94)
Ra = −
Hc j c a H 2H 3
(95)
Conservation equations: .
µ − (µ + p⊥ )(ln H)· ∗
pba Einstein filed equations:
h
∗
µ − (µ + p⊥ )(ln H) i . ∗ p⊥,b + λHH,b + (µ + p⊥ + λH 2 )(ub − nb )
12
= 0
(96)
= 0
(97)
= 0.
(98)
1 Lξ Rab ξ
=
. . 1 1 . p⊥ + λH H + 2 p⊥ + λH 2 − Λ (ln ξ)· ua ub + 2 p⊥ + λH 2 − Λ uc − (ln ξ),c u(a hcb) + 2 2 . 2 1 1 . . 2µ − p⊥ + λH 2 + 3Λ θ hab + + 2µ − p⊥ + λH H + 3 3 2 . 2 1 1 2 . . + µ + p⊥ + 2λH H + (µ + p⊥ + λH 2 )θ hab − na nb + 2µ − p⊥ + λH 2 + 3Λ σ ab + 3 3 3 2 1 hcd − nc nd δ d(a (ωc.b) + σ c.b) ) − +2(µ + p⊥ + λH 2 ) 3
.
−2(µ + p⊥ + λH 2 )nd hd(a nb) .
1 Lξ Rab ξ
=
(99)
h∗ i 1 1 . 2 c + λH H + 2 p⊥ + λH − Λ uc n ua ub − 2(p⊥ + λH 2 − Λ) uc nc − (ln ξ)· u(a nb) − 2 2 ∗ 1 ∗ − p⊥ + λH H + 2 p⊥ + λH 2 − Λ (ln ξ)∗ na nb − 2 1 −4 p⊥ + λH 2 − Λ ω dc nd u(a pcb) − 2 h i ∗ 1 1 ∗ c 2 d ∗ 2 −2 p⊥ + λH − Λ pc nd + (ln ξ),d n(a pb) + µ + λH H + µ + λH + Λ E pab + 2 2 1 (100) +2 µ + λH 2 + Λ Sab . 2 ∗ p⊥
∗
The EMSF admitting a timelike CKV ξ a = ξua (ξ > 0)
5
As it has mentioned there are two types of equations constraining the evolutions of a gravitational system which admits a symmetry, that is the kinematic conditions and dynamic equations. The kinematic conditions are equations among the kinematic variables of the gravitational system which result from geometric identities and additional geometric assumptions (such as symmetries). The dynamic equations do not necessarily inherit the kinematic symmetries of the system. In the following we consider two types of symmetries (a) CKVs defined by timelike vectors ξ a = ξua (ξ > 0); and (b) CKVs defined by the spacelike vectors ξ a = ξna (ξ > 0).
The case of a timelike CKV ξ a = ξua (ξ > 0)
5.1
We look first on the kinematic implications of the assumed symmetry and then on the dynamical ones.
5.2
The kinematic implications
From previous works we have the following kinematic conditions for a CKV ξ a = ξua [15]. Proposition 2 A fluid space-time ua admits a CKV ξ a = ξua iff: 1. σ ab = 0 .
.a
2. ua = (ln ξ),a + 13 θua where σ ab , θ and u are, respectively, the shear, expansion and acceleration of the ˙ timelike congruence generated by ua . The conformal factor ψ = 13 ξθ = ξ. The conditions imposed by Proposition 2 supplement Maxwell equations and simplify the conservation equations. Because σ ab = 0 the “energy” conservation equation (87) gives: 2 . µ + (µ + p⊥ )θ = 0. 3 13
(101)
Equation (88) remains the same and equation (89) becomes: h i ∗ pba p⊥,b + λHH,b + (µ + p⊥ + λH 2 )((ln ξ),b − nb ) = 0.
(102)
Eventually the conservation equations are equations (88), (101), (102).
5.3
The dynamic implications
For a CKV we have the identity: Lξ Rab = −2ψ;ab − gab ψ. The 1+3 decomposition of ψ ;ab wrt ua is (note that ψ ;ab = ψ ;ba ): ψ ;ab = λψ ua ub + pψ hab + 2qψ(a ub) + π ψab where:
1 λψ = ψ ;ab u u , pψ = ψ ;ab hab , qψa = −ψ;bc hba uc , π ψab = 3 a b
1 r s rs ψ ;rs . ha hb − hab h 3
(103) (104)
We also compute: ψ = ψ ;ab g ab = −λψ + 3pψ . a
(105)
a
Therefore for a CKV ξ = ξu we have that: Lξ Rab = − 3(λψ − pψ )ua ub − (λψ − 5pψ )hab + 4qψ(a ub) + 2π ψab .
(106)
Using the kinematic conditions and the conservation equations, the rhs of equation (99) simplifies as follows2 : 1 Lξ Rab ξ
=
. . 1 1 1 1 . . p⊥ + λH H + 2(p⊥ − Λ) θ ua ub − p + 2(p⊥ − Λ)θ − λH H hab 2 3 3 ⊥ 2 h. . i 1 p⊥ + λH H hab − na nb . 3
(107)
From (106) and (107) we find that the field equations for an EM string fluid admitting the CKV ξ a = ξua are . . 1 1 1 1 . p + 2(p⊥ − Λ)θ − λH H hab + λH H + 2(p⊥ − Λ) θ ua ub − 2 3 3 ⊥ 2 h. . i 1 + p⊥ + λH H hab − na nb 3 1 = − 3(λψ − pψ )ua ub − (λψ − 5pψ )hab + 4qψ(a ub) + 2πψab ξ
. p⊥
This relation implies the field equations3 : 2 It
.
is easy to show (use Maxwell equations in RMHD approximation) that nd hd(a nb) + nc nd δ d(a ω c.b) = 0 same equations are found from [20] where the field equations were:
3 The
.
3(pψ + λψ )
q
=
(q − Λ)θ 1 hab − na nb 3 a qψ
=
14
3 − (pψ + λψ ) ξ 9 pψ ξ
=
2π ψab .
=
0
.
.
p⊥ + λH H 1 2 p⊥ + λH − Λ θ 2 1 3(pψ + λψ ) hab − na nb 3 qψa
= =
3 − (pψ + λψ ) ξ 9 pψ ξ
(108) (109)
=
2πψab
(110)
=
0.
(111)
Eqn (108) using also (lnξ)· = 31 θ can be written : · 1 1 2 2 = − p⊥ + λH − Λ (lnξ)· − 3λψ =⇒ p⊥ + λH − Λ 2 2 · 1 1 2 λψ = − p⊥ + λH − Λ ξ 3 2
(112)
and (109):
1 1 ˙ p⊥ + λH 2 − Λ ξ. (113) 3 2 The final set of equations which results from the assumption that the EMSF admits the CKV ξ a = ξua is the following: Geometric implications: pψ =
σ ab .
ua
= 0
(114)
1 = (ln ξ),a + θua 3
(115)
Maxwell equations: Na E .b
pab n 2 − θ 3
= =
0 −(ln H)∗
(116) (117)
=
ωa.b nb
(118)
=
(ln H)·
(119)
Conservation equations: .
µ − (µ + p⊥ )(ln H)· ∗
∗
µ − (µ + p⊥ )(ln H) h i ∗ pba p⊥,b + λHH,b + (µ + p⊥ + λH 2 )((ln ξ),b − nb )
= 0
(120)
= 0
(121)
= 0.
(122)
Gravitational Field equations:
λψ
=
pψ
=
2π ψab
=
qψa
=
· 1 1 2 p⊥ + λH − Λ ξ − 3 2 1 1 2 p⊥ + λH − Λ ξ˙ 3 2 · 1 1 hab − na nb −ξ p⊥ + λH 2 − Λ 2 3 0.
(123) (124) (125) (126)
This is the complete set of equations which must be satisfied by the various variables (geometric, kinematic and dynamic) of an EMSF which admits the CKV ξ a = ξua . 15
5.4
The case of an EMSF admitting a timelike CKV ξ a = ξua in the FRW spacetime
We apply the results of the last section to the case of the FRW spacetime. The FRW spacetime has metric (in conformal coordinates): (127) ds2 = R2 (τ ) −dτ 2 + U 2 (xµ )dσ 2E −1 and k = 0, ±1. This metric admits the gradient CKV ∂τ whose where the function U 2 (xµ ) = 1 + k4 x · x dR conformal factor is ψ = dτ . We define the timelike unit vector ua = R1 ∂τ . If we define the new coordinate t by the requirement: 1 dτ = dt (128) R(t) then the metric is written: ds2 = −dt2 + R2 (t)U 2 (xµ )dσ 2E
(129)
a
and the unit vector u = ∂t . The conformal factor becomes: ψ= We compute:
. dR ≡ R(t) dt
(130)
...
ψ ;ab = Rδ ta δ tb which implies:
...
λψ = R, pψ = 0, qψa , π ψab = 0.
(131)
Field equation (113) gives (as ψ 6= const and R˙ = 6 0): 1 p⊥ = − λH 2 + Λ 2 ...
and (112) λψ = R = 0. It follows that: 1) ∂τ = R(t)ua is a special CKV or one of its specializations (If ψ = 0 ⇒ R(t) = const the spacetime reduces to a Einstein space); 2) The conformal factor is ψ = bt + c; 3) The spacetime admits the special gradient CKV ψ ,a = bδ ta ; 4) The scale factor R(t) = 12 bt2 + ct + d. We work now with the rest of the conservation equations. Equation (120) using p⊥ = − 21 λH 2 + Λ gives 1 1 . . 2 µ − µ − λH + Λ (ln H)· = 0 =⇒ µ − (µ + Λ)(ln H)· + λH 2 (ln H)· = 0 =⇒ 2 2 . · . . µ 1 H 1 H µ+Λ H . + λ − + λ H = 0 =⇒ ln H = 0 =⇒ µ+Λ H 2 µ+Λ H 2 µ+Λ · 1 . µ+Λ + λH = 0 =⇒ H 2 · 1 = 0 (132) µ + λH 2 2 This equation means that the energy of the EM string fluid is constant along the flow lines of the observers. Working similarly we show that equation (121) becomes: ∗ 1 2 =0 (133) µ + λH 2 and implies that the total energy of the magnetofluid is conserved along the magnetic field lines. Both these results are compatible with: 16
a. The fact that the magnetic field lines are frozen along the flow lines of the fluid (there is no relative motion of the two sets of lines) due to the condition N a = 0 b. The dynamic equation qψa = 0 i.e. there is no heat flux wrt the observers ua . There remains equation (122). Taking into account the fact that ξ(t) (hence pab ξ ,a = 0) we find that this equation becomes: ∗ 1 µ + Λ + λH 2 pba na = 0. 2 Because the total energy of the fluid (including the cosmological constant) is considered to be positive this equation gives the condition: ∗
pba na = 0. This is a dynamical equation which involves the magnetic field only.
∗
∗
∗
However we also have na ua = ∗
−σ ab na nb − 3θ = 0. But since σ ab = 0 we find that na ua = − θ3 which from pba na = 0 gives na = 31 θua = (lnR)· ua . Eventually we have that the magnetic field lines are carried along with the fluid so that the total energy density (that is fluid energy and magnetic field energy) remains constant. Furthermore the fluid does not heat. The magnetic field lines are coplanar with the fluid lines but they are not Lie transported along these lines except in the case of Minkowski spacetime. Indeed from the condition N a = 0 we have Lu na = aua + bna where .a
∗
a, b are quantities which have to be computed. From the definition of the Lie derivative we have Lu na = n − ua therefore we have: ∗ .a n − ua = aua + bna . Contracting in turn with ua , na we find a = (ln ξ)∗ , b = − 31 θ therefore: 1 Lu na = (ln ξ)∗ ua − θna 3
(134)
which proves our assertion. From this relation it follows that pab Lu na = 0, that is N a = 0. Concerning the magnetic field we have . a
∗
.
Lu H a = H − ua H = Hna + HLu na = (ln ξ),b H b ua − θH a .
(135)
The EMSF in spacetimes admitting a spacelike CKV ξ a = ξna
6
We derive again the kinematic and the dynamic equations as we did for the case of ξ a = ξua .
6.1
The Kinematic conditions of a spacelike CKV ξ a = ξna .
For a double congruence ua , na one has the kinematic quantities σ ab, ω ab , θ, ua for the timelike congruence . ∗ ua and the kinematic quantities Sab , Rab , E na , ua for the spacelike congruence na . Therefore the kinematic restrictions in this case involve in general all nine quantities plus the parameters ψ and Hab and their derivatives. To find the kinematic conditions resulting from a collineation relative to a double congruence we need the 1+1+2 decomposition of Hab . To do that we consider the symmetry defining equation and contract it to get: ψ=
ξ .c E + (ln ξ)∗ − n uc 4
For the case of a CKV ξ a = ξna we find the following kinematic conditions [18]:
17
Proposition 3 A fluid spacetime ua with a spacelike congruence na (ua na = 0) admits the spacelike CKV4 ξ a = ξna (ξ > 0) iff: Sab
=
0
(136)
na u a
=
1 − E 2
(137)
=
(ln ξ)· ua − pab (ln ξ),b
.
∗
na Na
=
(138)
b
−2ωab n .
(139)
The conformal factor ψ satisfies:
∗ 1 ξE =ξ. 2 Furthermore we can show that the Lie derivatives [13]:
ψ=
(140)
L ξ na
=
−ψna
(141)
a
=
= −ψua − ξN a .
(142)
Lξ u We note that:
E =(ln ξ 2 )∗ .
(143)
∗ a
Also n is the principal normal to the magnetic field lines. We note that in general these lines are not straight lines. The main results on the kinematics of a CKV ξ a = ξna are given in the following Proposition (see Theorem 4.1. of [14]): Proposition 4 Let ξ a = ξna be a spatial conformal Killing vector orthogonal to ua . Then Lξ na = ψna . Furthermore the following statements are equivalent: 1. N a = 0 2. ωa k ξ a or ωa = 0 3. Lξ ua = ψua 4. Lξ ω ab = ψω ab 5. Lξ σ ab = ψσ ab .
.
6. Lξ ua = ψ,a +ψua .
7. Lξ θ = −ψθ + 3ψ We have the obvious identity:
.a
.
.b
n = −(nb ub )ua + pab n . We also have: Na
.b
pab n
.b
∗b
.b
∗b
= pab (Lu nb ) = pab (n − u ) = pab n − pab u .b .b = pab n − pab σ bc + ωbc nc = pab n − pab σ bc nc − ω ac nc 1 .b = pab n − pab σ bc nc + N a ⇒ 2 1 a a b c = pb σ c n + N . 2
(144)
Using the symmetry equation we find: .a
n = 4ξ
1 1 a Eu + pab σ bc nc + N a . 2 2
is not necessarily equal to H!
18
(145)
6.2
The dynamic conditions of a spacelike CKV ξ a = ξna
We have to consider three sets of equations i.e. Maxwell equations, the conservation equations and the gravitational field equations. 6.2.1
Maxwell equations
The above results hold for any spacelike CKV and any string fluid. For the particular case of the EMSF we have to supplement these equations with Maxwell equations which are Na E 2 σ ab na nb − θ 3 e Using E= − (ln H)∗ and (143) we find:
.b
= 0 ⇐⇒ pab n = pa.c σ cb nb = −(ln H)∗ ·
= (ln H)
= 2ωa Ha , Ra = −
(146) (147) (148)
j b Hb a H . 2H 3
(149)
(ξ 2 H)∗ = 0
that is the quantity ξ 2 H is constant along the magnetic field lines. We also conclude that ω a k H a that is the magnetic field congruence coincides with the vorticity congruence. Using Maxwell equations we show the following important Proposition. Proposition 5 ξ a is a CKV of the screen metric pab as well as of the 3-metric hab with conformal factor ψ = 21 ξE (the same for both metrics). Proof In [20] it has been shown (see relations (26),(27)) that the following general relations/identities hold for the Lie derivatives of the projection tensors hab and pab : 1 1 (150) Lξ pab = 2 Sab + Epab − 2u(a Nb) ξ 2 1 1 ∗ Lξ hab = 2 Sab + Epab − 2u(a Nb) + 2(ln ξ),(a nb) + 2n(a nb) . (151) ξ 2 From equation (146) and the kinematic condition (136) these equations reduce as follows: 1 Lξ pab ξ 1 Lξ hab ξ
= Epab
(152) ∗
= Epab + 2(ln ξ),(a nb) + 2n(a nb) .
(153)
It follows immediately that ξ a is a CKV for the 2-metric pab in the screen space with conformal factor 12 ξE. To show that ξ a is a CKV for the 3-metric hab we 1+1+2 decompose (ln ξ),a in terms of the vectors ua , na and find: (ln ξ),a = −(ln ξ)· ua + (ln ξ)∗ na + pca (ln ξ),c . From (153) and (138)
we have then:
1 Lξ hab ξ
= = =
∗
Epab + 2(ln ξ),(a nb) + 2n(a nb) h∗ i Epab + 2 nd − (ln ξ)· ud + (ln ξ)∗ nd + pcd (ln ξ),c δ d(a nb) Epab + 2(ln ξ)∗ na nb .
But from (140) we have that E =2(ln ξ)∗ therefore: 1 Lξ hab = E(pab + na nb ) = Ehab ξ 19
from which it follows that ξ a is a CKV for the 3-metric hab with conformal factor 12 ξE. ⊡ From (145) we also have5 : 1 .a n = − (ln H)∗ ua + pab σ bc nc . 2 6.2.2
Conservation equations
These equations are the same as before, that is we have: .
µ − (µ + p⊥ )(ln H)· ∗
pba 6.2.3
h
∗
µ − (µ + p⊥ )(ln H) i . ∗ p⊥,b + λHH,b + (µ + p⊥ + λH 2 )(ub − nb )
= 0
(154)
= 0
(155)
= 0.
(156)
Gravitational field equations
We use (100) to compute these equations. Of course we can also take them directly from [20] but we prefer to derive them here in order to make clear the methods we follow. First we compute the Lξ Rab .We note that: pψ =
1 1 1 ψ hab = ψ ;ab (pab + na nb ) = (γ ψ + aψ ). 3 ;ab 3 3
(157)
We have: Lξ Rab
= −2ψ ;ab − gab ψ 1 = −2 λψ ua ub + 2kψ u(a nb) + 2Sψ(a ub) + γ ψ na nb + 2Pψ(a nb) + αψ pab + Dψab − 2 −(−λψ + 3pψ )(−ua ub + na nb + pab ) = 3(pψ − λψ )ua ub + (λψ − 3pψ − 2γ ψ )na nb + (λψ − 3pψ − aψ )pab + rest.
(158)
From (100) we get the following field equations (including equations kψ = 0, Sψa = 0, Pψa = 0, Dψab = 0 which result directly from the kinematic conditions over (100)): ∗ 1 1 1 ∗ E = 3(pψ − λψ ) p⊥ + λH H + 2 p⊥ + λH 2 − Λ 2 2 ξ ∗ 1 1 1 ∗ p⊥ + λH H+ 2 p⊥ + λH 2 − Λ E = − (λψ − 3pψ − 2γ ψ ) 2 2 ξ ∗ 1 1 ∗ (λψ − 3pψ − aψ ). µ + λH H + µ + λH 2 + Λ E = 2 ξ Using equation (147) to replace E in terms of (ln H)∗ we note that the first two equations have identical lhs and they result in the two equations: 1 1 ∗ 3(pψ − λψ ) p⊥ − p⊥ − λH 2 − Λ (ln H)∗ = 2 ξ λψ = −γ ψ . The last equation is written: 1 1 2 µ − µ − λH + Λ (ln H)∗ = (λψ − 3pψ − aψ ). 2 ξ ∗
5 One could possibly expect to get information on σ nb nc from this equation. But this is not so. Indeed by expanding pa we bc b find: 1 .a ∗ a a c b c a n = − (ln H) u + σ c n − (σ bc n n )n 2 and we get no information on σbc nb nc .
20
Using the conservation equation (155) and equation (157) we find: 2 1 p⊥ + λH 2 − Λ (ln H)∗ = − (γ ψ + aψ ). 2 ξ Finally we have that the field equations in the case of a spacelike vector ξ a = ξna are: ∗
∗
p⊥ + λH H 1 p⊥ + λH 2 − Λ (ln H)∗ 2
1 (2γ ψ − aψ ) ξ 2 = − (γ ψ + aψ ). ξ =
(159) (160)
where:
1 ψ ;ab = −γ ψ (ua ub − na nb ) + αψ pab . (161) 2 We see that ψ ;ab is the energy momentum tensor or equivalently the Ricci tensor of a string fluid with ρ = −γ ψ and q = 21 αψ or vice versa. Obviously one can make many scenarios with this observation. The result we found coincides with the one we found in [20] on strings. From equations (159) and (160) ones shows easily that: ∗ 1 3H p⊥ + λH 2 − Λ H = − αψ . (162) 2 ξ
This equation shows that if αψ = pab ψ ;ab = 0 then the quantity (p⊥ + 12 λH 2 − Λ)H is constant along the magnetic field lines. The constraint equation6 for a general anisotropic fluid of the form we consider is: (µ − 2p⊥ + pq + 2Λ)ψ = 2(2λψ − aψ ).
(163)
Setting µ = −pq and p⊥ = p⊥ + 12 λH 2 we obtain the EMSF. In this case equation (163) becomes: 1 (p⊥ + λH 2 − Λ)ψ = aψ + 2γ ψ . 2
(164)
But ψ = 21 ξE = − 21 ξ(ln H)∗ therefore we obtain the same result. We collect the above results in the following Proposition. Proposition 6 An EMSF spacetime admits a CKV of the form ξ a = ξna where na = H a /H iff the following 6 This
equation follows form the identity (Rab ξ b );a = −3ψ which holds for all CKVs.
21
system of equations is satisfied: .
µ − (µ + p⊥ )(ln H)·
=
0
(165)
µ − (µ + p⊥ )(ln H) = # " 1 ∗ 2 . 2 b + (µ + p⊥ + λH )(ub − nb ) = pa p⊥ + λH − Λ 2 ,b
0
(166)
0
(167)
∗
∗
ψ ;ab ∗ 1 p⊥ + λH 2 − Λ H 2 1 2 p⊥ + λH − Λ (ln H)∗ 2 Sab .a
n
∗a
n
Na 2 σ ab na nb − θ 3
= = = = =
1 −γ ψ (ua ub − na nb ) + αψ pab 2 3H αψ − ξ 2 − (γ ψ + aψ ) ξ 0, E = −(ln H)∗ 1 − (ln H)∗ ua + pab σ bc nc 2
(168) (169) (170) (171) (172)
=
(ln ξ)∗ na − (ln ξ),a
(173)
=
0
(174)
=
(ln H)
e =
·
2ωa Ha , Ra = −
(175) j b Hb a H . 2H 3
(176)
Furthermore the rotation ωa is either parallel to H a or vanishes and ψ = 12 ξE = − 12 ξ(ln H)∗ . One important result is that if the vorticity vanishes then the same must be true for the charge density and conversely. This is a restriction of physical nature resulting from a geometrical symmetry assumption. A CKV for which ψ ;ab = 0 is called a special CKV. Coley and Tupper [17] have shown that if an anisotropic fluid space-time admits a proper special CKV ξ a = ξna then (assuming Λ = 0): µ = −pk =
1 R, p⊥ = 0 2
(177)
where R is the Ricci scalar. From Einstein field equations it follows that for this case Tab is of the form: Tab =
1 R(ua ub − na nb ). 2
(178)
For the case of a string fluid this result gives ρ = 21 R and q = 0. Obviously R 6= 0 otherwise we do not have a fluid at all. Let us check if our results are compatible with this general result. ∗
From equation (170) assuming H 6= 0 we have: 1 p⊥ + λH 2 − Λ = 0 2 which gives immediately from (47): Rab from which follows:
(179)
1 2 = µ + λH + Λ pab 2
1 R = 2 µ + λH 2 + Λ . 2
Therefore: Rab =
R pab . 2
22
(180)
Consequently due to the symmetry between the Ricci tensor and the energy momentum tensor for a string fluid: Tab =
R (ua ub − na nb ) 2
which is in agreement with the quoted result. Proposition 7 Let ξ a = ξna be a proper special CKV in an EMSF space-time and let the total energy of the ∗
magnetofluid µ + 21 λH 2 + Λ 6= 0. Then for H 6= 0 we have the following (a) The Ricci tensor satisfies the property7 : R pab 2 (b) The quantity R/H is constant along the magnetic field lines and along the fluid lines and (c) The following equations hold: 1 ∗ 2 µ − µ − λH + Λ (ln H)∗ = 0 2 . µ − (µ + p⊥ )(ln H)· = 0 Rab =
together with equations (171) - (176). Proof The first part (a) has been shown above. Concerning (b) we note that (166) can be written: 1 ∗ µ − µ − λH 2 + Λ (ln H)∗ 2 R ∗ − λH 2 (ln H)∗ µ− 2 R ∗ µ + λHH ∗ − (ln H)∗ 2 R 1 2 ∗ (µ + λH + Λ) − (ln H)∗ 2 2 (ln R)∗ − (ln H)∗ ∗ R ln H
=
0 =⇒
=
0 =⇒
=
0 =⇒
=
0 =⇒
=
0 =⇒
=
0
(181)
(182)
from which follows that the quantity R/H is constant along the magnetic field lines. Working in exactly the same way we show that (ln(R/H))· = 0 from which follows that the quantity R/H is constant along the fluid flow line. Concerning the case of KVs we have the following result. Proposition 8 An EMSF spacetime admits a KV of the form ξ a = ξna where na = H a /H iff ∗
∗
(a) H = ξ = 0 (b) The following equations hold ∗
µ µ − (µ + p⊥ )(ln H)· .
Sab ∗a
n
.a
n 2 σ ab na nb − θ 3 Na 7 It
∗
= p⊥ = 0 = 0 = 0, Ra = −
(183) (184) j b Hb a H 2H 3
= −(ln ξ),a =
(186)
pab σ bc nc
(187)
·
= (ln H)
(188) a
= 0, e = 2ω Ha .
is not necessarily an Einstein space!
23
(185)
(189)
We conclude that when an EMSF admits the KV ξ a = ξna the following results hold: i) Because na = ω a /ω = H a /H (ω 6= 0) the string consists of the 2-dimensional timelike surface spanned by a u and the vorticity ω a (Nambu geometric string) or ωa = 0. ii) From equations (150), (151) it follows: Lξ pab = 0, Lξ hab = 0
(190)
that is, ξ a is also a KV of the metric hab of the 3-space normal to ua , and a KV of the screen space metric pab . iii) From Proposition 4 we have that the Killing symmetry is inherited by the geometric and the dynamic variables, that is: . (191) Lξ ua = Lξ na = 0, Lξ σ ab = 0, Lξ ω ab = 0, Lξ θ = 0, Lξ ua = 0. iv) If ω a 6= 0 then ua , na = ω a /ω must commute. Obviously these restrictions are severe and allow only few special choices for the string fluids in given spacetimes.
6.3
Application: EMSF in Bianchi I spacetime
The Bianchi I spacetime with metric ds2 = −dt2 + A21 (t)dx2 + A22 (t)dy 2 + A23 (t)dz 2 .
(192)
has been a platform for studying anisotropy and more specifically string fluids and electromagnetic fields. For example Letelier [3] studied string dust in Bianchi I spacetime whereas the electromagnetic field in the relativistic RMHD has been studied (among many others) in [19]. Following this line of research we shall use the results obtained in the last section to compute all possible Bianchi I spacetimes (if any), which carry a magnetic field satisfying the RMHD approximation and admit a spacelike CKV or a spacelike KV. In order to get comparable results with the literature we consider the comoving observers ua = (1, 0, 0, 0). This choice has a double effect. On the one hand gives that the vorticity ω a =0, therefore Maxwell equation e = 2ω a Ha implies that the charge density e = 0. This excludes all analytical solutions found in [19]. Secondly the geometric condition N a = 0 restricts heavily the possible symmetry vectors ξ a = ξna . All the CKVs of the Bianchi I spacetime have been found in [24]. We have checked that for this choice of ua none of these vectors satisfies the condition N a = 0. Therefore the only remaining choice is the KVs so that the system of equations we have to solve is the following ∗
∗
µ
=
p⊥ = 0
(193)
·
=
0
(194)
=
j b Hb a H 0, Ra = − 2H 3
(195)
=
−(ln ξ),a
(196)
=
pab σ bc nc
(197)
=
(ln H) , Na = 0, e = 2ωa Ha .
·
(198)
.
µ − (µ + p⊥ )(ln H)
Sab ∗a
n
.a
n 2 σ ab na nb − θ 3
∗
∗
Let us assume that ξ a = ξ(t)na where na = ∂z = (0, , 0, 0, 1/A3(t)). Equation (193) implies that µ, p⊥ are zero. Hence, we prove easily that equation (186) gives ξ = 1 therefore the KV is the ∂z . Equation (187) is satisfied identically, while equation (188) gives H(t) = (A1 (t)A2 (t))−1 . Therefore the magnetic field is given by H a = (A1 (t)A2 (t))−1 ∂z . It remains equation (184) which is written as .
·
µ + (µ + p⊥ ) ln [A1 (t)A2 (t)] = 0.
(199)
This equation can be solved if we assume an equation of state. The solution we have found appears to be a new one. 24
7
Conclusions
We have applied the 1+1+2 decomposition to the case of the EMSF in the RMHD approximation. We have shown that a geometric assumption in the form of a symmetry effects both the kinematics and the dynamics of the resulting EMSF. We have approached the problem in two steps a. In full generality independently of a particular symmetry and b. In the case of a CKV which is of the form ξ a = ξua and of the form ξ a = ξna with na = H a /H where ua is the four velocity of the fluid and H a is the magnetic field. We applied the results of the ξ a = ξua case in the FRW spacetime and the results of the case ξ a = ξna in the Bianchi I spacetime where. In the latter case we found new solutions for the gravitational field. It is apparent that the results stated in this work due to their generality can be used in many different situations involving the electromagnetic field and various types of symmetries.
Acknowledgement AP acknowledges the financial support of FONDECYT grant no. 3160121 and thanks the University of Athens for the hospitality provided.
References [1] Ray D (1978) “Some solutions for relativistic vortices interacting through a scalar field” Phys Rev D, 18, 3879 - 3880 [2] Lund F and Regge T (1976), “Unified approach to strings and vortices with soliton solutions” Phys Rev D, 14, 1524 - 1535 [3] Letelier P (1980), “Anisotropic fluids with two perfect fluid components” Phys Rev D 22, 807 - 813 [4] Letelier P (1981), “” Nuovo Cimento B63, 519 [5] Letelier P (1983), “String Cosmologies” Phys Rev D 28, 2414 - 2419 [6] I Yavuz and I Yilmaz ”Inheriting Conformal and Special Conformal Killing Vectors in String Cosmology” (1997) Gen Rel Grav 9, 1295 - 1307 [7] Yilmaz I (2001), “Timelike and Spacelike Ricci Collineation Vectors in String Cosmology” Inter Jour Modern Physics 10, 681 - 690 [8] Baysal H Yilmaz I (2002), “Spacelike Ricci Inheritance vectors in a model of string cloud and string fluid stress tensor” Class Quantum Grav 19, 6435 - 6443 [9] Baysal H, Camci U, Tarhan I and Yilmaz I (2002), “Conformal Collineations in String Cosmology” Inter Jour Modern Physics 11, 463 - 469 [10] Camci U (2002), “Conformal Collineations and Ricci Inheritance symmetry in String Cloud and String Fluids” Inter Jour Modern Physics 11, 353 - 366 [11] Sharif M Sheikh U (2005), “Timelike and Spacelike Matter Inheritance Vectors in Specific Forms of EnergyMomentum Tensor” Inter Jour Modern Physics (to appear), gr-qc/0504101 [12] Mason D P and Tsamparlis M 1985, “Spacelike Conformal Killing Vectors and Spacelike Congruences” J. Math. Phys. 26, 2881 - 2901 [13] Maartens R, Mason D P and Tsamparlis M 1986, “Kinematic and Dynamic Properties of Conformal Killing Vectors in Anisotropic Fluids” J. Math. Phys. 27, 2987 - 2994 [14] Saridakis E and Tsamparlis M 1991, “Symmetry Inheritance of Conformal Killing Vectors” J. Math. Phys. 32, 1541 - 1551 [15] Noris L K, Green P. and Davis W R (1977), J. Math. Phys. 18, 1305 25
[16] Tsamparlis M 1992, “Geometrization of a General Collineation” J. Math. Phys. 33, 1472 - 1479 [17] Coley A.A and Tupper B. O. (1989) J. Math. Phys. 30, 2616 [18] Tsamparlis M and Mason D P 1990, “Ricci Collineation Vectors in Fluid Space-times” J. Math. Phys. 31, 1707 - 1722 [19] Jacobs K (1969), “Cosmologies of Bianchi Type I with a Uniform Magnetic Field” Astro. Journal 155, 379 - 391 [20] Tsamparlis M (2006), “General symmetries of a string fluid spacetime” Gen Rel Grav (2006) [21] K.A. Dunn and B.O.J Tupper, Astroph. J. 235, 307 (1980) [22] V.N. Duarte and R.A. Clement, J. Phys. Conf. Ser. 511, 012015 (2014) [23] G.F.R. Ellis and H. van Elst and, Cosmological models, Carg`ese Lectures (1998) [24] M. Tsamparlis, A. Paliathanasis and L. Karpathopoulos, Gen. Relativ. Grav. 47, 15 (2015)
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