Weyl geometry

Weyl geometry James T. Wheeler∗

arXiv:1801.03178v1 [gr-qc] 9 Jan 2018

January 11, 2018

Abstract We develop the properties of Weyl geometry, beginning with a review of the conformal properties of Riemannian spacetimes. Decomposition of the Riemann curvature into trace and traceless parts allows an easy proof that the Weyl curvature tensor is the conformally invariant part of the Riemann curvature, and shows the explicit change in the Ricci and Schouten tensors required to insure conformal invariance. We include a proof of the well-known condition for the existence of a conformal transformation to a Ricci-flat spacetime. We generalize this to a derivation of the condition for the existence of a conformal transformation to a spacetime satisfying the Einstein equation with matter sources. Then, enlarging the symmetry from Poincaré to Weyl, we develop the Cartan structure equations of Weyl geometry, the form of the curvature tensor and its relationship to the Riemann curvature of the corresponding Riemannian geometry. We present a simple theory of Weyl-covariant gravity based on a curvature-linear action, and show that it is conformally equivalent to general relativity. This theory is invariant under dilatations, but not the full conformal group.

1

Introduction

In 1918, H. Weyl introduced an additional symmetry into Riemannian geometry in an attempt to unify electromagnetism with gravity as a fully geometric model [1, 2]. The idea was to allow both the orientation and the length of vectors to vary under parallel transport, instead of just the orientation as in Riemannian geometry. The resulting geometries are called Weyl geometries, and they form a completely consistent generalization of Riemannian geometries. However, Weyl’s attempt to identify the vector part of the connection associated with stretching and contraction with the vector potential of electromagnetism failed. As Einstein pointed out immediately following Weyl’s first paper on the subject [3], the identification implies that identical atoms which move in such a way as to enclose some electromagnetic flux would be different sizes after the motion. Different sized atoms would have different spectra, and it is easy to show that change in frequency resulting from the size change would be vastly inconsistent with the known precision of spectral lines. Many attempts were made to patch up the theory. In the end, following some notable work by London [4], Weyl showed that a satisfactory theory of electromagnetism is achieved if the scale factor is replaced by a complex phase. This is the origin of U (1) gauge theory. Many interesting details are discussed in O’Raifeartaigh [5]. In modern language, the new vector part of the connection introduced by Weyl is the dilatational gauge vector, often called the Weyl vector. When this vector is given by the gradient of a function, then there exists a scale transformation (understandable as a change of units, or a dilatation) that sets the vector to zero. When this is possible, the Weyl geometry is called trivial: there exists a subclass of global gauges in which the geometry is Riemannian. While Weyl’s theory of electromagnetism fails, Weyl geometry does not. Indeed, although no new physical predictions have emerged directly from its use, there are at least the following three considerations for seeking a deeper understanding of general relativity formulated within a trivial Weyl geometry: ∗ Utah

State University Dept of Physics email: [email protected]

1

1. General relativity is naturally invariant under global changes of units. By formulating general relativity in a trivial Weyl geometry, this scale invariance becomes local. We refer to this generalization as scale invariant general relativity. As soon as we make a definition of a fundamental standard of length – for example, as the distance light travels in one second1 – scale invariant general relativity reduces to general relativity. 2. In [6], Ehlers, Pirani and Schild make the following argument. First, the paths of light pulses may be used to determine a conformal connection on spacetime. Then, a projective connection is found by tracing trajectories of massive test particles (“dust”). Finally, requiring the two connections to approach one another in the limit of near-light velocities reduces the possible connection to that of a Weyl geometry. When this program is carried out with mathematical precision [7], the resulting geometry is a trivial Weyl geometry. 3. Deeper physical interest in Weyl geometry also arises in higher symmetry approaches to gravity. Gravitational theories based on the full conformal group ([1],[8]-[18]) often yield general relativity formulated on a trivial Weyl geometry rather than on a Riemannian one and are therefore equivalent to general relativity while providing additional natural structures. For these reasons, it is useful to recognize the typical forms and meaning of the connection and curvature of Weyl geometry. Here we use the techniques of modern gauge theory [19, 20] to develop the properties of Weyl geometry, beginning in the next section with a review of the conformal properties of Riemannian spacetimes. Decomposition of the Riemann curvature into trace and traceless parts allows an easy proof that the Weyl curvature tensor is the conformally invariant part of the Riemann curvature, and shows the explicit change in the Ricci and Schouten tensors required to insure conformal invariance. We include a proof of the well-known condition for the existence of a conformal transformation to a Ricci-flat spacetime, and generalize this to a derivation of the condition for the existence of a conformal transformation to a spacetime satisfying the Einstein equation with matter sources. Then, in the final section, we enlarge the symmetry from Poincaré to Weyl to develop the Cartan structure equations of Weyl geometry, the form of the curvature tensor, and its relationship to the Riemann curvature of the corresponding Riemannian geometry. We conclude with a simple theory of Weyl-covariant gravity based on a curvature-linear action, and show that its vacuum solutions are conformal equivalence classes of Ricci-flat metrics in a trivial Weyl geometry. This theory is invariant under dilatations, but not the full conformal group.

2 2.1

Conformal transformations in Riemannian geometry Structure equations for Riemannian geometry

The Cartan structure equations of a Riemannian geometry are Rab

=

0 =

dαab − αcb αac

dea − eb αab

where the solder form, ea = eµ a dxµ , provides an orthonormal basis, αab is the spin connection 1-form, and the curvature 2-form is Rab = 12 Rabcd ec ed . Differential forms are written in boldface and the wedge product is always assumed between adjacent forms, e.g., eb αab = eb ∧ αab . 1 The second is currently defined as the duration of 9, 192, 631, 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom [physics.nist.gov]. The metre is defined as the 1 second [17th General Conference on Weights and Measures (1983), length of the path travelled by light in a vacuum in 299792458 Resolution 1].

2

The structure equations satisfy integrability conditions, the Bianchi identities, found by applying the Poincaré lemma, d2 ≡ 0: 0 ≡ d2 αab

0 ≡ d2 e a

= d (αcb αac + Rab )

= dαcb αac − αcb dαac + dRab = (αeb αce + Rcb ) αac − αcb (αec αae + Rac ) + dRab

= dRab + Rcb αac − αcb Rac ≡ DRab = d eb αab

= deb αab − eb dαab = (ec αcc ) αab − eb (αcb αac + Rab )

= −eb Rab In components, these take the familiar forms

Rab[cd;e]

= 0

Ra[bcd]

= 0

Here we use Greek and Latin indices to distinguish different vector bases. Use of the covariantly constant coefficient matrix of the solder form, eµ a , allows us to convert freely between orthonormal components (Latin indices) and coordinate components (Greek indices), Rαβµν = ea α eβ b eµ c eν d Rabcd .

2.2

Conformal transformation of the metric, solder form, and connection

A conformal transformation of the metric is the transformation gµν → g˜µν = e2φ gµν

(1)

This is not an invariance of Riemannian geometry, but it is an invariance of Weyl geometry. If we make a change of this type in a Riemannian geometry, the solder form changes by ˜ a = e φ ea ea → e

(2)

since the solder form and metric are related via the orthonormal metric, ηab = diag (−1, 1, 1, 1) by gµν = eµ a eν b ηab The corresponding structure equation then gives the altered form of the metric compatible connection, d˜ ea = d e φ ea =

eφ (dea + dφea ) = eb αab + dφea

=

˜b α e ˜ ab e φ eb α ˜ ab e φ eb α ˜ ab eb α ˜ ab

Since the spin connection is antisymmetric, α ˜ ab = −η ad ηbc α ˜ cd , this is solved by setting α ˜ ab = αab + 2∆ac db ec

where ec

µ

is inverse to eµ

a

and ∆ac db ≡

1 2

µ

∂µ φ ed

(δda δbc − η ac ηbd ) is the antisymmetric projection operator on

tensors. Checking, we have, eb α ˜ ab

=

eb αab + 2∆ac db ec

=

eb αab + ec ec

=

eb αab + dφea

µ

µ

∂µ φeb ed

∂µ φea − η ac ec

3

µ

∂µ φηbd eb ed

(3) 1 1

as required. Since the spin connection is uniquely determined (up to local Lorentz transformations) by the structure equation, this is the unique solution.

2.3

Transformation of the curvature

Now compute the new curvature tensor. For this longer calculation it is convenient to define φb ≡ ec d Then α ˜ ab = αab + 2∆ac db φc e and the conformal transformation changes the curvature to: 1 ã c d R e e 2 bcd

= =

µ

∂µ φ.

˜ ab − α ˜ cb α ˜ ac dα

dαab − αcb αac

a ag d c f ae d ce d ce d +d 2∆ac φ e − α 2∆ φ e 2∆ φ e − 2∆ φ e α − 2∆ φ e db c b dc e db e c db e fc g

=

=

=

=

= =

1 a c d R bcd e e + d φb ea − η ac ηbd φc ed − αcb (δda δce − η ae ηcd ) φe ed 2 − (δdc δbe − η ce ηbd ) φe ed αac − φb ec − η ce ηbd φe ed φc ea − η ag ηf c φg ef 1 a c d R e e + dφb ea − ηbd dφa ed + φb dea − ηbd φa ded − φc αcb ea + ηcd φa αcb ed 2 bcd −φb ec αac + ηbd φc ed αac − φb φc ec ea + φc φc ηbd ed ea − ηbd φa φe ed ee 1 a c d R e e + (dφb − φc αcb ) ea − (dφa + φc αac ) ηbd ed + φb (dea − ec αac ) 2 bcd 1 1 −φa ηbc dec − ed αcd − φb φc ec ea + φc φc ηbd ed ea − ηbd φa φe ed ee + φc φc ηbd ed ea 2 2 1 a c d a e ae c R e e + (δc δb − η ηbc ) Dφe e 2 bcd 1 1 +δca δbe φd φe − φf φf ηde ec ed − ηbc η ae φd φe − ηed φf φf ec ed 2 2 1 f 1 a c d c c d Dφ e + φ φ − e e R bcd e e + 2∆ae φ φ η e d e f de cb 2 2 1 a c d 1 f e ed R bcd e e + 2∆ac Dφ − φ dφ + φ φ η e c c f ce db 2 2

Writing D(α) φc = φc;d ed and d(x) φ = φd ed this becomes ˜ a = Ra + 2∆ac φc;e − φe φc + 1 φf φf ηce ee ed R b b db 2

(4)

In components: 1 f 1 f ae a a ae ˜ R bcd = R bcd + 2∆db φe;c − φc φe + φ φf ηec − 2∆cb φe;d − φd φe + φ φf ηed 2 2 The Ricci tensor and scalar are ˜ bd R

1 1 = Rbd + φb;d − ηbd φc;c − φb φd + ηbd φc φc − (n − 1) φf φf ηbd − (n − 1) φb;d − φd φb + φf φf ηbd 2 2 c c = Rbd − (n − 2) φb;d − ηbd φ ;c + (n − 2) φb φd + (n − 2) ηbd φ φc

and ˜ R

˜ ab = gãb R = e−2φ g ab (Rbd − (n − 2) φb;d − η ce φe;c ηbd + (n − 2) φd φb − (n − 2) φc φc ηbd ) = e−2φ R − 2 (n − 1) φc ;c + (n − 2) φc φc − n (n − 2) φc φc = e−2φ R − 2 (n − 1) φc ;c − (n − 1) (n − 2) φc φc 4

2.4

Invariance of the Weyl curvature tensor

In general, we may split the Riemann curvature Rabcd into its trace, the Ricci tensor, and its traceless part, called the Weyl curvature. This decomposition is most concisely expressed if we first define the Schouten tensor, 1 1 Rbd ≡ Rbd − ηdb R (5) (n − 2) 2 (n − 1) where Rab is the Ricci tensor,

Rab ≡ Rcacb

The Schouten tensor often arises as a 1-form, Ra = Rab eb . Except in 2-dimensions, it is equivalent to the Ricci tensor, since we may invert eq.5 to write Rbd

=

R

=

(n − 2) Rbd + ηdb R

2 (n − 1) R

(6) (7)

In terms of Rab , the Weyl curvature 2-form is defined as d Cab ≡ Rab + 2∆ae db Re e

(8)

Expanding to find the components, C abcd , of Cab , C abcd

= = = =

ae Rabcd + 2∆ae db Rec − 2∆cb Red

Rabcd + (δda δbe − η ae ηbd ) Rec − (δca δbe − η ae ηcb ) Red Rabcd + δda Rbc − Rac ηbd − δca Rbd + Rad ηbc R 1 (δca Rbd − δda Rbc − Rad ηbc + Rac ηbd ) + (δ a ηbd − δda ηbc ) Rabcd − (n − 2) (n − 1) (n − 2) c

we readily verify its tracelessness, C cbcd

= Rbd − = 0

R 1 (nRbd − Rbd − Rbd + ηbd R) + (n − 1) ηbd n−2 (n − 1) (n − 2)

with all other nontrivial traces equivalent to this one. By contrast, the second term is equivalent to knowing a ae ae the Ricci or Schouten tensor, since the components of the ∆ae cb term, D bcd ≡ ∆db Rec − ∆cb Red in eq.(8) give 2 1 Rbd = δac − Dabcd + η f g Daf cg ηbd n−2 (n − 1) (n − 2) Check: Rbd

= = = = = =

2 1 δac − η f g Daf cg ηbd Dabcd + n−2 (n − 1) (n − 2) 1 2 f g ae f g ae ae η η ∆ R − η ∆ R (∆ae R − ∆ R ) + δac − bd f g ec cf eg db ec cb ed n−2 (n − 1) (n − 2) 2 1 1 a e 1 fg a e 1 1 − δf δg − η ae ηf g Rea − (n − 1) R ηbd (δd δb − η ae ηdb ) Rea − (n − 1) Rbd + η n−2 2 2 (n − 1) (n − 2) 2 2 1 1 − (Rbd − ηbd R − (n − 1) Rbd ) + (1 − n − (n − 1)) Rηbd n−2 2 (n − 1) (n − 2) 1 [−Rbd + (n − 1) Rbd + ηbd R − Rηbd ] n−2 Rbd 5

We now have the decomposition of the Riemann curvature into traceless and trace parts, d Rab = Cab − 2∆ae db Re e

(9)

After a conformal transformation, the new Riemann curvature 2-form may also be decomposed in the same way, ã = C ˜ a − 2∆ae R ˜ ee ˜d R b

b

db

Combining this with eq.(4) we have ˜ a − 2∆ae R ˜ ee ˜d C b db

= =

1 2 φe;c − φe φc + φ ηce ec ed − + 2 1 2 c d Cab − 2∆ae db Rec − φe;c + φe φc − φ ηce e e 2

Cab

d 2∆ae db Re e

2∆ae db

Equality of the traceless and trace parts shows immediately that both ã C b

= Cab

˜e R

1 = e−φ Rec − φe;c + φe φc − φ2 ηce ˜ ec 2

so the Weyl curvature 2-form is conformally invariant. The components of each part transform as C˜ abcd ˜ ab R ˜ ηãb R

= e−2φ C ab 1 = e−2φ Rab − φa;b + φa φb − φ2 ηab 2 1 = ηab e−2φ R − φa;a − (n − 2) φa φa 2

(10) (11) (12)

ã = eφ ea . This proves that the Weyl curvature tensor is where the factor of e−2φ comes from replacing e covariant with weight −2 under a conformal transformation of the metric, and yields the expression for the change in the Schouten (and therefore, Ricci) tensor under conformal transformation.

2.5

Conditions for conformal Ricci flatness

Next, we find the condition required for the metric of a Riemannian geometry to be conformally related to the metric of a Ricci-flat spacetime. This follows as a pair of integrability conditions for φ when we set eq.(11) equal to zero. First, we rewrite eq.(11) as a 1-form equation, ˜ c = e−φ Rc − Dφc + φc dφ − 1 η ab φa φb ηce ee R 2 where Dφc = dφc −φe ω ec . Defining the vector field φc ≡ ec µ ∂µ φ, and the corresponding 1-form φ ≡ φc ec = ˜ c = 0 has a solution for φ. This may be written as a pair of dφ, we ask for the conditions under which R equations, 1 = Rc + φe ω ec + φc dφ − η ab φa φb ηce ee 2 dφ = 0

dφc

The integrability conditions follow from the Poincaré lemma, d2 ≡ 0, 0 = = d2 φ ≡

d2 φc

1 dRc + dφe ω ec + φe dω ec + dφc dφ − η ab φa dφb ηce ee − φ2 ηce dee 2 0 6

(13) (14)

The second condition is identically satisfied by the definition of φ. Substituting the original equation for dφc , eq.(13), into the first conditon, 1 2 1 2 d e e e d e 0 = dRc + Re + φd ω e + φe dφ − φ ηed e ω c + φe dω c + Rc + φe ω c + φc dφ − φ ηce e dφ 2 2 1 1 −η ab φa Rb + φd ω db + φb dφ − φ2 ηbd ed ηce ee − φ2 ηce dee 2 2 1 = (dRc + Re ω ec ) + Rc dφ − η ab φa Rb ηce ee + φe dωec + φd ωde ω ec − φ2 ηce dee + ηed ed ω ec 2 1 1 − φ2 ηce ee dφ − η ab φa φd ωdb ηce ee − φ2 dφηce ee + φ2 φa ea ηce ee 2 2 1 = DRc + φa δcb δea − η ab ηce Rb ee + φe Rec − φ2 ηec Dee − ed ω ed 2 1 1 e e e 2 ηce dφe − ηce dφe + ηce dφe − φa φd ω da ηce ee +φ 2 2 e R e = DRc + φa Rac + 2∆ab b ec which we see from eq.(9) becomes

0 = DRc + φa Cac Though this well-known condition still depends on the gradient of the conformal factor, φa , Szekeres has shown using spinor techniques that it can be broken down into two integrability conditions depending only on the curvature [21].

2.6

Conditions for conformal Einstein equation with matter

We may apply the same approach to the Einstein equation with conformal matter. Let the matter be of ˜ → ekφ Ψ for a generic field Ψ. Then the covariant form of the stress-energy definite conformal weight, Ψ tensor will be of conformal weight −2, Tãb = e−2φ Tab to have the correct weight for the Einstein equation. The Einstein tensor, of course, is not of definite conformal weight, but it acquires an overall factor of e−2φ . We assume that Tab is of definite weight. Then, writing the Einstein equation, Rab − 21 ηab R = Tab in terms of the Schouten tensor using eqs.(6) and (7), gives 1 Rab − ηab R = Tab n−2 Now define, for arbitrary curvatures, not necessarily solutions, Eab ≡ Rab − ηab R −

1 Tab n−2

ãb such that E ãb = 0. The We would like to know when there exists a conformal transformation, Eab → E calculation is simpler if we notice that Eab = 0 if and only if 1 1 1 Tab − Eηab = Rab − T ηab = 0 Eab − n−1 n−2 n−1 Defining Tab ≡

1 n−2

ãb − we ask for a conformal gauge in which E

Tab −

1 ˜ n−1 Eηab

7

1 T ηab n−1

= 0.

ãb and to E ãb − 1 Eη ˜ ab are different, we check Though it looks like the conformal transformations to E n−1 ãb , that they are equivalent. Substituting the conformally transformed fields to find E Eãb

= =

1 Rab − ηab R − Tab n−2 1 1 −2φ 1 2 c −2φ c −2φ R − φ ;c − (n − 2) φ φc − e Tab Rab − φa;b + φa φb − φ ηab − ηab e e 2 2 n−2

so we examine integrability of 1 1 1 Tab 0 = Rab − φa;b + φa φb − φ2 ηab − ηab R − φc;c − (n − 2) φc φc − 2 2 n−2 However, by solving the trace of this equation for φc;c , 0 = = φc;c

=

1 1 1 R − φa;a + φa φa − nφc φc − nR − nφc;c − n (n − 2) φc φc − T 2 2 n−2 1 1 − (n − 1) R + (n − 1) φc;c + (n − 1) (n − 2) φc φc − T 2 n−2 1 1 T R − (n − 2) φc φc + 2 (n − 1) (n − 2)

we may write the equivalent equation, 1 1 1 1 1 2 c c T − (n − 2) φ φc − Tab 0 = Rab − φa;b + φa φb − φ ηab − ηab R − R + (n − 2) φ φc − 2 2 (n − 1) (n − 2) 2 n−2 1 1 1 = Rab − φa;b + φa φb − φ2 ηab − ηab T Tab − 2 n−2 n−1 ˜ ab = 0. ãb − 1 Eη and this is just E n−1 b ãb − Setting Ta ≡ Tab e , and writing E 0

=

1 ˜ n−1 Eηab

= 0 as a 1-form equation,

1 Ra − dφa + φb ω ba + φa dφ − φ2 ηab eb 2

−Ta

We require dφa dφ

1 = Ra + φb ω ba + φa dφ − φ2 ηab eb − T a 2 = 0

dea Rab

= eb ω ab = dω ab − ω cb ω ac

dφa

1 = Ra − T a + φb ω ba + φa dφ − φ2 ηab eb 2

with the trace relation,

8

with the integrability condition, 0 ≡ = =

=

=

= = =

d2 φa 1 2 b b d Ra − T a + φb ω a + φa dφ − φ ηab e 2 1 d (Ra − T a ) + φb dω ba − φ2 ηab deb 2 1 2 c c + Rb − T b + φc ω b + φb dφ − φ ηbc e ωba 2 1 + Ra − T a + φb ω ba + φa dφ − φ2 ηab eb dφ 2 1 −φc Rc − T c + φb ω bc + φc dφ − φ2 ηcb eb ηad ed 2 1 d (Ra − T a ) + φb dω ba − φ2 ηab deb 2 1 b c b + (Rb − T b ) ω a + φc ω b ω a + φb dφω ba − φ2 ηbc ec ωba 2 1 2 b + (Ra − T a ) dφ + φb ω a dφ + φa dφdφ − φ ηab eb dφ 2 1 −φc (Rc − T c ) ηad ed − φc φb ω bc ηad ed − φc φc dφηad ed + φ2 dφηad ed 2 d (Ra − T a ) + (Rb − T b ) ω ba + (Ra − T a ) dφ − φc (Rc − T c ) ηad ed 1 +φb dω ba − ωca ωbc − φ2 ηab deb − ec ωbc 2 1 c 1 b b + φb dφω a + φb ω a dφ + φ φc − φc φc + φc φc dφηad ed 2 2

D (Ra − T a ) + φb Rba + φb δac δdb (Rc − T c ) ed − φb η bc ηad (Rc − T c ) ed d D (Ra − T a ) + φb Rba + 2φb ∆cb ad (Rc − T c ) e d DRa + φb Cba − DT a − 2φb ∆cb ad T c e

leaving us with DRa + φb Cba

=

d DT a + φb 2∆bc da T c e

This is the same condition as that for Ricci flatness, but with the Schouten tensor replaced by Ra − T a .

3

Weyl geometry

A simple extension of the Poincaré symmetry underlying Riemannian geometry leads to the Cartan structure equations for the Weyl group: Rab

=

Ta Ω

= =

dωab − ω cb ω ac

dea − eb ω ab − ωea dω

where the most general case includes both the torsion, Ta = 21 T abc eb ec , and the dilatational curvature, Ω = 12 Ωab eb ec . We will be interested in the torsion-free case, Ta = 0.

9

A conformal transformation of the metric, eq.1, now transforms both the solder form and the Weyl vector, according to ã e ˜ ω

=

e φ ea

=

ω + dφ

The final equation then remains unchanged, since ˜ = dω dω The basis equation transforms as ã T

= = = =

˜b ω ˜ ab − ω˜ ˜ ea d˜ ea − e ˜ ab − (ω + dφ) eφ ea eφ dφea + eφ dea − eφ eb ω

˜ ab − ωea − dφea eφ dφea + (Ta + ec ω ac + ωea ) − eb ω ˜ ab ) eφ Ta + eφ eb (ωab − ω

We conclude that it is sufficient to take the spin connection to be conformally invariant, and the torsion a weight-1 conformal tensor: ˜ ab ω ã T

= ω ab = eφ Ta

These results are correct, as may be shown directly from the gauge transformation properties of the Cartan ˜ a = Ra . connection. Since the spin connection is invariant, the Lorentz curvature 2-form is also invariant, R b b 2 We again use the Poincaré lemma, d ≡ 0 to find the integrability conditions: DRab DTa DΩ

= 0 = eb Rab − Ωea

= 0

where the covariant derivatives are given by DRab DTa DΩ

≡ dRab + Rcb ω ac − Rac ω cb

≡ dTa + Tb ω ab − ωTa ≡ dΩ

When the torsion vanishes, we have a pair of algebraic identities since the Weyl-Ricci tensor may have an antisymmetric part. From eb Rab

=

Ωea

Ra[bcd]

=

a δ[b Ωcd]

we find the symmetric and antisymmetric parts, Rabcd + Racdb + Radbc Rbd − Rdb

= δba Ωcd + δca Ωdb + δda Ωbc = − (n − 2) Ωbd

While the Lorentz curvature 2-form is conformally invariant, the components Rabcd , Rab and Ωab all have conformal weight −2.

10

3.1

The Weyl-Schouten tensor

˜ a = Ra , means that not only is the Weyl curvature of a Weyl The invariance of the full curvature, R b b geometry conformally covariant, but so is the Weyl-Schouten tensor, Ra . By the Weyl-Schouten tensor, we mean the conformally covariant Ricci tensor of a Weyl geometry. To compute it for a torsion-free Weyl geometry, we must expand Rab

=

dea

=

dω ab − ω cb ω ac eb ω ab + ωea

Notice that these have exactly the same form as the conformally transformed structure equations of a Riemannian geometry studied in Section 1. The solution for the spin connection is completely analogous, d ω ab = αab − 2∆ac db Wc e

This again solves the second structure equation: dea

=

eb ωab + ωea d + ωea eb αab − 2∆ac db Wc e

=

eb αab

= =

αab − Wb eb ea + η ac ηbd Wc eb ed + ωea

with αab still the metric-compatible spin connection. The difference is that now all of Rab will be conformally covariant. Substituting into the curvature, the algebra is identical to that leading up to eq.4, with φa replaced by −Wa . This results in 1 2 W η Rab = Rab − 2∆ac W − W W + ee ed ce c;e e c db 2 1 2 d ac ee ed W η W + W − W W + = Rab − 2∆ac R e − 2∆ ce e c c [c;e] (c;e) db db 2 1 2 e d R + W − W W + = Rab − 2∆ac ee ed − 2∆ac W η c e c ce (c;e) db db Ωec e e 2 This decomposes into three parts. With Ω

=

dω

Ωab

=

W[b;a]

we have Rab

1 2 = − − W[c;e] + W(c;e) − We Wc + W ηce ee ed 2 1 e d e 2 ed − 2∆ac = Cab − 2∆ac db Ωec e e db Rc + W(c;e) − We Wc + W ηce e 2 C ab

d 2∆ac db Rc e

2∆ac db

where Rab = dαab − αcb αac is the Riemannian part of the curvature. Carrying out the decomposition of the curvature into trace and trace-free parts, we find the relationship between the Weyl and Schouten tensors of the Weyl and Riemannian geometries. In addition, the asymmetry of the Ricci tensor gives rise to a third independent component, the dilatational curvature: Cab

=

Cab

Ra

=

1 Re + W(c;e) − We Wc + W 2 ηce ee 2

Ω

=

W[b;a] ea eb 11

or in components,

1 Rab = Rab + W(a;b) − Wa Wb + W 2 ηab 2 We define the Weyl-Schouten tensor Rab to be this symmetric part only. In a trivial Weyl geometry, defined as one in which the dilatational curvature, Ω, vanishes, there exists a conformal transformation which makes the Weyl vector vanish, Wa = 0. In this gauge, the Weyl-Schouten tensor reduces to the Schouten tensor. The gravitational field, Cab , is the same in all gauges.

3.2

The covariant derivative of Weyl geometry in a coordinate basis

Even when differentiating a scalar such as ϕ with conformal weight λ, the covariant derivative in Weyl geometry is not just the partial derivative, but includes the weight of the field times the field, times the Weyl vector, Dµ ϕ = ∂µ ϕ − λWµ ϕ This means that metric compatibility gives a different expression for the connection. 0 =

Dµ gαβ ˆ νβµ − 2Wµ gαβ ˜ ναµ − gαν Γ ∂µ gαβ − gνβ Γ ˆ βαµ − Γ ˆ αβµ − 2Wµ gαβ ∂µ gαβ − Γ

= = Cycling this expression,

ˆ βαµ + Γ ˆ αβµ = ∂µ gαβ − 2Wµ gαβ Γ ˆ αµβ + Γ ˆ µαβ = ∂β gµα − 2Wβ gµα Γ ˆ µβα + Γ ˆ βµα = ∂α gβµ − 2Wα gβµ Γ Each of these three expressions is a conformal tensor since ˜ µ g˜αβ ∂µ g˜αβ − 2W

= =

∂µ e2φ gαβ − 2 (Wµ + ∂µ φ) e2φ gαβ e2φ (∂µ gαβ − 2Wµ gαβ )

then adding the first two and subtracting the third, ˆ αβµ Γ

1 (gαβ,µ + gµα,β − gβµ,α ) − (Wµ gαβ + Wβ gµα − Wα gβµ ) 2 = Γαβµ − (Wµ gαβ + Wβ gµα − Wα gβµ ) =

Now, if we raise the first index, ˆ αβµ = 1 g αν (gνβ,µ + gµν,β − gβµ,ν ) − δβα Wµ + δµα Wβ − W α gβµ Γ 2 = Γαβµ − δβα Wµ + W α gβµ − Wβ gµα

ˆ α is not only scale covariant but also of weight zero. It is invariant under a conformal we see that Γ βµ transformation. The derivative of a contravariant vector of weight λ is then Dµ v α

ˆ α − λv α Wµ = ∂µ v α + v β Γ βµ

which transforms as ˜ µ v˜α D

˜ µ eλφ v α = D ˜ αβµ − λeλφ v α (Wµ + ∂µ φ) = ∂µ eλφ v α + eλφ v β Γ ˜ α − λv α Wµ = eλφ ∂µ v α + v β Γ βµ

= e

λφ

Dµ v

α

12

and is therefore properly covariant.

4

Scale invariant gravity

We now turn to the formulation of a scale invariant gravity theory, based in a Weyl geometry. For this we must construct a Lorentz- and dilatation-invariant action functional from the curvature and any other available tensors. As we have noted, the conformal weight of the curvature components in an orthonormal basis is −2. Since gµν = eµa eνb ηab , the Minkowski metric has conformal weight zero, making the conformal weight of the Weyl-Ricci scalar equal to −2 as expected. This introduces a difficulty in writing a scale invariant action in dimensions greater than two, since the volume element has weight +n in n-dimensions.

4.1

Curvature-quadratic actions

In 2n-dimensions, we may use n-products of the curvature: ˆ S = Rab Rcd · · · Ref Qabcd···ef where Qabcd···ef is a rank-n invariant tensor. In 4-dim the most general curvature-quadratic action is ˆ √ αRabcd Rabcd + βRab Rab + γR2 S= −gd4 x but the variation of the Gauss-Bonnet combination for the Euler character χ, ˆ δχ = δ Rab Rcd εabcd ˆ d δω ab − δωeb ω ae − (δω ae ) ωbe Rcd εabcd = 2 ˆ = 2 D δω ab Rcd εabcd ˆ D δω ab Rcd εabcd + δω ab DRcd εabcd = 2

vanishes identically when we use the second Bianchi identity, DRcd ≡ 0, and let the variation vanish on the boundary, ˆ δχ = 2 D δω ab Rcd εabcd V

= 2

ˆ V

d δω ab Rcd εabcd

= 2 δω ab Rcd εabcd

δV

= 0

The additon of any multiple of the Euler character to the action therefore makes no contribution to the field equations. Expanding the 4 -form, ˆ χ = Rab Rcd εabcd ˆ 1 = Rab ef Rcd gh ee ef eg eh εabcd 4 ˆ 1 = − Rab ef Rcd gh εef gh εabcd Φ 4 13

Define a convenient volume element as the dual of unity, Φ ≡ ∗ 1. Then: Φ

≡ =

∗

Φ

= = =

1 1 εabcd ea eb ec ed 4! 1 ∗ εabcd ea eb ec ed 4! 1 εabcd εabcd 4! −1 ∗

√ 1√ In a coordinate basis, Φ = 4! −gεµναβ dxµ dxν dxα dxβ , and if we ignore orientation this is simply −gd4 x. It follows from the definition that ea eb ec ed

=

−εabcd Φ

εabcd ea eb ec ed

= =

−εabcd εabcd Φ 4!Φ

since then

The pair of Levi-Civita tensors may be written as −εef gh εabcd = δae δbf δcg δdh − δch δdg + δbg δch δdf − δcf δdh + δbh δcf δdg − δcg δdf −δaf δbe δcg δdh − δch δdg + δbg δch δde − δce δdh + δbh (δce δdg − δcg δde ) −δag δbf δce δdh − δch δde + δbe δch δdf − δcf δdh + δbh δcf δde − δce δdf −δah δbf (δcg δde − δce δdg ) + δbg δce δdf − δcf δde + δbe δcf δdg − δcg δdf (Check2 ) This enables us to explicitly write out the integrand of the Euler character in the Gauss-Bonnet 2 We

check the normalization by contracting all pairs of indices, (ae) , (bf ) , (cg) , (dh): 4! = 4 δbb δcc δdd − δcd δdc + δbc δcd δdb − δcb δdd + δbd δcb δdc − δcc δdb − δab δba δcc δdd − δcd δdc + δbc δcd δda − δce δdd + δbd (δca δdc − δcc δda ) − δac δbb δca δdd − δcd δda + δba δcd δdb − δcf δdd + δbd δcb δda − δca δdb − δad δbb (δcc δda − δca δdc ) + δbc δca δdb − δcb δda + δba δcb δdc − δcc δdb = 4 (4 (16 − 4) + 4 − 16 + 4 − 16) − (64 − 16 + 4 − 16 + 4 − 16) − (64 − 16 + 4 − 16 + 4 − 16) − (64 − 16 + 4 − 16 + 4 − 16) = 64 − 16 + 4 − 16 + 4 − 16 = 24

14

form, (details: 3 ) Rab ef Rcd gh ee ef eg eh εabcd

= =

−Rab ef Rcd gh εef gh εabcd Φ

4 R2 − 4Rbd Rdb + Rabcd Rabcd

so that χ

ˆ 1 = − Rab ef Rcd gh εef gh εabcd Φ 4 ˆ R2 − 4Rbd Rdb + Rabcd Rabcd Φ =

The invariance of χ allows us to replace ˆ ˆ abcd R Rabcd Φ = χ −

R2 − 4Rbd Rdb Φ

leaving the most general curvature-quadratic action in the form ˆ √ −gd4 x aR2 + bRab Rab S=

(15)

for constants a, b. Quadratic gravity theories, especially the b = 0 case, have often been studied because the scale invariance allows the theory to be renormalizable. However, fourth order field equations such as those resulting from eq.(15) are sometimes found to introduce ghosts in the quantum theory. The Einstein-Hilbert term maybe included as well, but while this has desirable effects, it breaks the scale invariance we examine here. For references, see the bibliography of arXiv:1505.07657v2 [hep-th]. Quadratic gravity applies only in four dimensions, with dimension 2n theories having correspondingly higher order field equations. Instead, we turn to a variant of Dirac’s theory. By including an additional field, these theories hold in any dimension. The Palatini style variation we employ makes this additional field purely auxiliary. 3

−Rab ef Rcd gh εef gh εabcd

=

=

=

=

Rab ef Rcd gh δae δbf δcg δdh − δch δdg + δbg δch δdf − δcf δdh + δbh δcf δdg − δcg δdf −Rab ef Rcd gh δaf δbe δcg δdh − δch δdg + δbg δch δde − δce δdh + δbh δce δdg − δcg δde −Rab ef Rcd gh δag δbf δce δdh − δch δde + δbe δch δdf − δcf δdh + δbh δcf δde − δce δdf −Rab ef Rcd gh δah δbf δcg δde − δce δdg + δbg δce δdf − δcf δde + δbe δcf δdg − δcg δdf 2Rab af Rcd gh δbf δcg δdh + δbg δch δdf + δbg δcf δdh −2Rab ea Rcd gh δbe δcg δdh + δbg δch δde + δbh δce δdg −2Rab ef Rcd ah δbf δce δdh + δbe δch δdf + δbh δcf δde −2Rab ef Rcd ga δcg δbf δde + δbe δcf δdg + δbg δce δdf 2 Rab ab Rcd cd + Rab ad Rcd bc + Rab ac Rcd bd −2 Rab ba Rcd cd + Rab da Rcd bc + Rab ca Rcd db −2 Rab cb Rcd ad + Rab bd Rcd ac + Rab dc Rcd ab −2 Rab db Rcd ca + Rab bc Rcd da + Rab cd Rcd ba 4 R2 − 4Rbd Rdb + Rabcd Rabcd

15

4.2

The Dirac theory

An alternative approach to scale invariant gravity was developed by Dirac in an attempt to give rigor to his Large Numbers Hypothesis, the idea that extremely large dimensionless numbers in the description of nature should be related to one another. In [23], Dirac presents a scale invariant gravity theory in which the gravitational constant varies with time in such a way that the large dimensionless magnitude constructed from the fundamental charge e and G is related to the age of the universe. The result follows from a single, simple solution to the scale invariant theory. Here we examine the scale invariant theory without further discussion of the Large Numbers Hypothesis. In the Dirac theory, scale invariance is achieved by including a gravitationally coupled scalar field in addition to the curvature. We take a similar but slightly different approach, and find a locally scale invariant theory that exactly reproduces general relativity as soon as a suitable definition of the unit of length is made. 4.2.1

A curvature linear, scale invariant gravitational action in any dimension

Beginning with the action for a Klein-Gordon scalar field, ϕ, in curved, n-dimensional Weyl geometry, ˆ √ m2 c2 g µν Dµ ϕDν ϕ + 2 ϕ2 Sϕ = −gdn x ~ where the covariant derivative of ϕ is Dµ ϕ = ∂µ ϕ − λWµ , we include a gravitational term of the form ˆ ˆ 1 ϕk Rab ef εef c...d εabc···d Φ ϕk Rab ec · · · ed εabc···d = − 2 ˆ = 2 ϕk Rab ef δae δbf − δaf δbe Φ ˆ = 4 ϕk RΦ ˆ √ → αϕk R −gdn x 1 εa···b ea · · · eb in n-dim and the power k will be chosen to make the full action scale where Φ ≡ ∗ 1 = n! invariant. There is no scalar we can form which is linear in the dilatational curvature. Thus, we arrive at ˆ √ m2 c2 αϕk R + g µν Dµ ϕDν ϕ + 2 ϕ2 SD = −gdn x ~

If the scalar field, metric, curvature and geometric mass scale as ϕ gµν g R m2 c2 ~2 then SD scales as S˜D =

→ eλφ ϕ → e2φ gµν

→ e2nφ g → e−2φ R m2 c2 → e−2φ 2 ~

ˆ m2 c2 2 p k˜ µν −˜ gdn x αϕ˜ R + g˜ Dµ ϕD ˜ ν ϕ˜ + 2 ϕ˜ ~

16

S˜D

= = = =

ˆ p m2 c2 ˜ + g˜µν Dµ ϕD αϕ˜k R ˜ ν ϕ˜ + 2 ϕ˜2 −˜ gdn x ~ ˆ 2 2 ˜ + e−2φ e2λφ g µν Dµ ϕDν ϕ + e−2φ e2λφ m c ϕ2 enφ √−gdn x αe(λk−2)φ ϕ˜k R ~2 ˆ 2 2 ˜ + e−2φ e2λφ g µν Dµ ϕDν ϕ + e−2φ e2λφ m c ϕ2 enφ √−gdn x αe(λk−2)φ ϕ˜k R ~2 ˆ √ m2 c2 e(λk−2+n)φ αϕk R + e(n−2+2λ)φ g µν Dµ ϕDν ϕ + 2 ϕ2 −gdn x ~

and is therefore locally scale invariant if λ = k

=

n−2 2 n−2 =2 − λ

−

We may make these assignments in any dimension greater than two, resulting in ˆ m2 c2 2 √ 2 µν αϕ R + g Dµ ϕDν ϕ + 2 ϕ −gdn x SD = ~ We consider the Palatini variation of SD . 4.2.2

The action in differential forms

This may be written as well in differential forms. We note the following equivalences: ϕ2 Rab ec · · · ed εabc···d

= ϕ2 Rab ec · · · ed εabc···d

= ϕ2 Rab ef ee ef ec · · · ed εabc···d

= ϕ2 Rab ef εef c···d εabc···d = −ϕ2 Rab ef (n − 2)! δae δbf − δaf δbe = − (n − 2)!ϕ2 R

Dϕ∗ Dϕ

1 (∂ν ϕ + λWν ϕ) εναβρ dxα dxβ dxρ 3! √ 1 1 (∂µ ϕ + λWµ ϕ) (∂ν ϕ + λWν ϕ) −gεναβρ √ εµαβρ Φ = 3! −g 1 = (∂µ ϕ + λWµ ϕ) (∂ν ϕ + λWν ϕ) (−3!) g νµ Φ 3! = − (∂µ ϕ + λWµ ϕ) (∂ν ϕ + λWν ϕ) g νµ Φ = (∂µ ϕ + λWµ ϕ) dxµ

= −Dµ ϕDµ ϕΦ 1 √ ϕ∗ ϕ = ϕ ϕ −gεµναβ dxµ dxν dxα dxβ 4! = −ϕ2 Φ and also, quite generally, f ∗g

= = =

1 √ g −gεµναβ dxµ dxν dxα dxβ 4! −f gΦ g∗f f

17

(16)

With these, the action may be written as ˆ α ϕ2 Rab ec · · · ed εabc···d + Dϕ∗ Dϕ + m2 ϕ∗ ϕ SD = − (n − 2)!

(17)

This form is convenient for variation.

5

Field equations

5.1

Variation

5.1.1

Spin connection

Only the curvature tensor depends on the spin connection, Rab = dω ab − ωcb ωac so varying ω ab in eq.(17) leads to ˆ α ϕ2 dδω ab − δω cb ωac − ωcb δω ac ec · · · ed εabc···d δωab S = − (n − 2)! ˆ α ϕ2 Dδω ab ec · · · ed εabc···d 0 = − (n − 2)! ˆ α = − D ϕ2 δωab ec · · · ed εabc···d − 2ϕ (Dϕ) δω ab ec · · · ed εabc···d + ϕ2 δω ab D ec · · · ed εabc···d (n − 2)! The first term is a total divergence, ˆ ˆ 2 ab c d D ϕ δω e · · · e εabc···d = d ϕ2 δωab ec · · · ed εabc···d which integrates to the boundary where the variation vanishes. For the derivative of the solder forms, we have = (n − 2) Tc ee · · · ed εabce···d D ec · · · ed εabc···d

Therefore, 0

= = = =

0 0

ˆ α 2ϕ (Dϕ) δω ab ec · · · ed εabc···d − ϕ2 δω ab D ec · · · ed εabc···d (n − 2)! ˆ α − δω ab f ef 2ϕ (Dϕ) ec · · · ed εabc···d + (n − 2) ϕ2 Tc ee · · · ed εabce···d (n − 2)! ˆ 1 α δω ab f 2ϕ (De ϕ) ef ee ec · · · ed εabc···d + (n − 2) ϕ2 T cgh ef eg eh ee · · · ed εabce···d − (n − 2)! 2 ˆ α 1 − δω ab f 2ϕ (De ϕ) εf ec···d εabc···d + (n − 2) ϕ2 T cf g εf ghe···d εabce···d Φ (n − 2)! 2

= δω ab f ef 2ϕ (Dϕ) ec · · · ed εabc···d + ϕ2 (n − 2) Tc ee · · · ed εabce···d 1 = 2ϕ (De ϕ) ef ee ec · · · ed εabc···d + ϕ2 (n − 2) ef T cgh eg eh ee · · · ed εabce···d 2 1 2 c f ghe···d f ec···d εabce···d Φ = 2ϕ (De ϕ) ε εabc···d + ϕ (n − 2) T gh ε 2 18

so that

1 (n − 2) ϕ2 T cgh εf ghe···d εabce···d 2 Now define ϕa ≡ Da ϕ and use the expansion of the pairs of Levi-Civita tensors, 0 = 2ϕ (De ϕ) εf ec···d εabc···d +

0 = =

=

=

1 2ϕϕe εf ec···d εabc···d + (n − 2) ϕ2 T cgh εf ghe···d εabce···d 2 −2ϕϕe (n − 2)! δaf δbe − δae δbf 1 − (n − 2) ϕ2 T cgh (n − 3)! δaf δbg δch − δbh δcg + δag δbh δcf − δbf δch + δah δbf δcg − δbg δcf 2 −2ϕ (n − 2)! δaf ϕb − δbf ϕa 1 − (n − 2)!ϕ2 2δaf T cbc + T fab − δbf T cac + δbf T cca − T fba 2 −2ϕ (n − 2)! δaf ϕb − δbf ϕa − (n − 2)!ϕ2 T fab + δaf T cbc − δbf T cac

Solving for the torsion in terms of its trace, 2 f T fab = − δa ϕb − δbf ϕa + δbf T cac − δaf T cbc ϕ and solving this for the trace, T bab

=

T cac

=

2 (ϕa − nϕa ) + nT cac − T cac ϕ 2 n−1 − ϕa ϕn−2

−

we substitute to find the full torsion in terms of the gradient of the scalar, T cab = 5.1.2

2 1 c (δ ϕb − δbc ϕa ) n−2ϕ a

(18)

Solder form

Next, expand the Hodge duals to see the full dependence on the solder form: ˆ α ϕ2 Rab ec · · · ed εabc···d + Dϕ∗ Dϕ + m2 ϕ∗ ϕ SD = − (n − 2)! ˆ α 1 1 = − ϕ2 Rab ec · · · ed εabc···d + Dϕ Da ϕεab···d eb · · · ed + m2 ϕ ϕεab···d ea eb · · · ed (n − 2)! (n − 1)! n!

and vary, 0

= δea S ˆ (n − 1) n α (n − 2) ϕ2 Rab δec ed · · · ee εabcd···e + DϕDa ϕεabc···d δeb ec · · · ed + m2 ϕ ϕεcb···d δec eb · · · ed = − (n − 2)! (n − 1)! n! ˆ α 1 1 = − δec ϕ2 Rab ed · · · ee εabcd···e − DϕDa ϕεacb···d eb · · · ed + m2 ϕ ϕεcb···d eb · · · ed (n − 3)! (n − 2)! (n − 1)!

Now expand the variation of the solder form in terms of the solder form, δec = Acf ef , and use ea · · · ed = −εa···d Φ , ˆ α 1 2 f ab 1 1 c 0 = − Af ϕ e R gh eg eh ed · · · ee εabcd···e − De ϕDa ϕεacb···d ef ee eb · · · ed + m2 ϕ ϕεcb···d ef eb · (n − 3)! 2 (n − 2)! (n − 1)! ˆ 1 2 ab 1 1 α f ghd···e a f eb···d 2 f b···d c Φ ϕ R gh ε εabcd···e − De ϕDa ϕε cb···d ε +m ϕ ϕεcb···d ε = Af (n − 3)! 2 (n − 2)! (n − 1)! 19

Therefore, again setting Da ϕ = ϕa , 0

α 1 2 ab 1 1 ϕ R gh εf ghd···e εabcd···e − ϕe ϕa εacb···d εf eb···d + m2 ϕ ϕεcb···d εf b···d (n − 3)! 2 (n − 2)! (n − 1)! 1 2 ab α ϕ R gh (n − 3)! δaf δbg δch − δbh δcg + δag δbh δcf − δbf δch + δah δbf δcg − δbg δcf = (n − 3)! 2 1 1 − ϕe ϕa (n − 2)! η af δce − η ae δcf + m2 ϕ ϕ (n − 1)!δcf (n − 2)! (n − 1)! α 2 fb = ϕ 2R bc + Rδcf − Raf ac + Raf ca − δcf Rab ba 2 −ϕf ϕc + δcf ϕa ϕa + m2 ϕ2 δcf =

so we have

2αϕ2 Rf

1 f c − Rδc 2

= −ϕf ϕc + δcf ϕa ϕa + m2 ϕ2

so we have the Einstein equation in the form 1 1 Rab − Rηab = − ϕa ϕb − ηab ϕc ϕc + m2 ϕ2 2 2αϕ2

(19)

The trace gives a useful relation,

1−

n R 2 R

5.1.3

1 1 1 Da ϕDa ϕ + nDc ϕDc ϕ + nm2 ϕ2 2αϕ2 2αϕ2 2αϕ2 1 n−1 a n = − 2 D ϕDa ϕ − m2 αϕ n − 2 α

= −

Weyl vector

Now we must expand the covariant derivatives, ˆ α ϕ2 Rab ec · · · ed εabc···d + Dϕ∗ Dϕ + m2 ϕ∗ ϕ SD = − (n − 2)! ˆ α ϕ2 Rab ec · · · ed εabc···d + (d − λω) ϕ∗ (d − λω) ϕ + m2 ϕ∗ ϕ = − (n − 2)! Only the kinetic term contains the Weyl vector, so we have simply 0 = =

δSD ˆ − 2λδωϕ∗ (d − λω) ϕ

which immediately gives 2λϕ∗ Dϕ = 0 Therefore, either ϕ = 0 (which is nonviable since it would make the full action vanish) or ∗ Dϕ = 0 and taking the dual in the latter case we have Dϕ = 0 (20) This may immediately be solved for the Weyl vector dϕ − λωϕ

= 0

ω

= d

20

1 ln ϕ λ

which implies a trivial Weyl geometry and the existence of a gauge in which the Weyl vector vanishes. We easily find the gauge transformation φ required to remove the Weyl vector. ˜ ω 0 φ

= ω + dφ 1 = d ln ϕ + dφ λ 1 1 ln a − ln ϕ = λ λ

˜ = 0. This remains the case for arbitrary global scale transformations. With this transformation, we have ω The same transformation changes the scalar field according to ϕ˜ = ϕeλφ 1 1 = ϕeλ( λ ln a− λ ln ϕ) = ϕeln a e− ln ϕ = a so the transformation that removes the Weyl vector makes the scalar field constant. If we were to allow curvature squared terms in the action, we could include a term Ω∗ Ω. Such a kinetic term for the dilatation would lead, in general, to a nontrivial Weyl geometry. However, the physical constraints against such a geometry are extremely strong – we do not experience changes of relative physical size. 5.1.4

Scalar field

Varying the scalar field, ˆ

α ϕ2 Rab ec · · · ed εabc···d + Dϕ∗ Dϕ + m2 ϕ∗ ϕ (n − 2)!

SD

=

−

0

=

δϕ SD ˆ α 2ϕδϕ ϕRab ec · · · ed εabc···d + 2Dδϕ ϕ∗ Dϕ + 2m2 δϕ ϕ∗ ϕ − (n − 2)!

=

Integrating the middle term by parts and discarding the surface term, ˆ α 2ϕRab ec · · · ed εabc···d − 2D∗ Dϕ + 2m2∗ ϕ 0 = − δϕ ϕ (n − 2)! ˆ α 2 1 = − δϕ ϕ ϕRab ef ee ef ec · · · ed εabc···d − Da Db ϕεbc···d ea ec · · · ed + 2m2 ϕεc···d ec · · · ed (n − 2)! (n − 1)! n! ˆ 2 1 α ϕRab ef εef c···d εabc···d − Da Db ϕεbc···d εac···d + 2m2 ϕεc···d εc···d Φ = δϕ ϕ (n − 2)! (n − 1)! n! so that 0

= =

α 2 ϕRab ef (n − 2)! δae δbf − δaf δbe − Da Db ϕ (n − 1)!η ba + 2m2 ϕ (n − 2)! (n − 1)!

2αϕR − 2Da Da ϕ + 2m2 ϕ

and we have a wave equation for ϕ Da Da ϕ − αRϕ − m2 ϕ = 0

21

(21)

5.2

Collected field equations

Collecting the variational field equations, eqs.(18)-(21), 1 Rab − Rηab 2

1 ϕa ϕb − ηab ϕc ϕc + m2 ϕ2 2 2αϕ 2 1 c = (δ ϕb − δbc ϕa ) n−2ϕ a = 0 = 0 = −

T cab Dϕ Da Da ϕ − αRϕ − m2 ϕ

Using the third equation to set ϕa = 0 in the others, these reduce to 1 Rab − Rηab 2 T cab

= =

Dϕ = R

=

1 ηab m2 2α 0 0 1 − m2 α

The trace of the first equation is inconsistent with the fourth unless m2 = 0, leaving us with 1 Rab − Rηab = 2 T cab = Dϕ =

0 0 0

Finally, Dϕ = 0 requires that there exist a local gauge in which the Weyl vector vanishes and ϕ is constant. In this gauge, the Weyl-Riemann tensor reduces to the usual Riemann curvature, 1 2 ee ed = Rab W η Rab = Rab − 2∆ac W − W W + ce c;e e c db 2 and the Einstein tensor takes the usual Riemannian form. The system has reduced to exactly the vacuum Einstein equation in a Riemannian geometry with the usual global scale invariance still present.

References [1] Weyl, Hermann, Sitz. Königlich Preußischen Akademie Wiss. (1918) 465; H. Weyl, Ann. d. Physik (4) 59, (1919) 101; H. Weyl, Gött. Nachr. (1921) 99; H. Weyl, Raum, Zeit, Materie, Springer, Berlin, (1919-1923) . [2] Hermann Weyl, Math. Zeitschr., 2 (1918b) 384. [3] A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. 26, 478 (1918), including Weyl’s reply. [4] F. London, Z. Phys. 42, 375 (1927). [5] L. O’Raifeartaigh, The Dawning of Gauge Theory, Princeton Series in Physics, Princeton U. Press, Princeton (1997). [6] J. Ehlers, A. E. Pirani, and A. Schild, in General Relativity, edited by L. 0 Raifeartaigh (Oxford University, Oxford, 1972). [7] Vladimir S. Matveev and Andrzej Trautman, Commun. Math. Phys. 329, (2014) pp 821-825. 22

[8] Rudolf Bach, Mathematische Zeitschrift 9 (1-2) 110. [9] Jorge Crispim Romao, Alan Ferber, and Peter G. O. Freund, Nucl.Phys. B126 (1977) 429. [10] M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, Phys. Lett. B 69 (1977) 304. [11] M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Phys. Rev. D17 (1978) 3179. [12] Jorge Crispim-Romao, Nuc.Phys.B145 (1978) 535. [13] Conformal supergravity E.S. Fradkin, A.A. Tseytlin, Physics Reports Volume 119, Issues 4–5, March 1985, pp 233–362. [14] James T. Wheeler, Phys. Rev. D 44 6 (1991) 1769. [15] E. A. Ivanov and J. Niederle, Phys.Rev.D, Vol. 25, 4, (1982) 976. [16] E. A. Ivanov and J. Niederle, Phys.Rev.D, 25, 4 (1982) 988. [17] Wheeler, James T., J. Math. Phys. 39 (1998) 299. [18] André Wehner and James T. Wheeler, Nuc. Phys. B 557 (1999) 380. [19] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, John Wiley and Sons (1963). [20] Y. Ne’eman and T. Regge, Phys.Lett.B 74 1–2 (1978) 54. [21] P. Szekeres, Proc Roy Soc London. Series A, Mathematical and Physical Sciences, 274, 1357 (1963) 206. [22] Thomas, T.Y., On the projective and equi-projective geometries of paths, Proc. Natl. Acad. Sci. USA 11, 199–203 (1925) [23] Dirac, P. A. M.: Proc. R. Soc. Lond. 333, 403 (1973)

6

Appendix: Bianchi identities

The Cartan structure equations, dω ab dea

= =

ωcb ω ac + Rab eb ωab + ωea + Ta

dω

=

µ

Wµ,ν dx dx

=

Wµ,ν − Wν,µ

=

Ω 1 Ωµν dxµ dxν 2 Ωµν

ν

have integrability conditions, which for gravity theories are called Bianchi identities. These follow from the Poincaré lemma, d2 ≡ 0: dRab

=

0

= =

DRab d2 e a

= =

0

=

0 DTa

= =

dΩ

=

d2 ω ab − dω cb ω ac + ω cb dω ac

dRab + (Rcb + ωeb ωce ) ω ac − ωcb (Rac + ω ec ωae ) dRab + Rcb ω ac − Rac ωcb

0 deb ωab − eb dω ab + dωea − ωdea + dTa ec ω bc + ωeb + Tb ω ab − eb (ωcb ω ac + Rab ) + Ωea − ω eb ωab + ωea + Ta + dTa −eb Rab + Ωea + dTa + Tb ω ab − ωTa eb Rab − Ωea 0

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Summary: DRab DTa

≡ =

DΩ

=

dRab + Rcb ω ac − Rac ω cb ≡ 0 eb Rab − Ωea 0

where DRab DTa DΩ

≡ dRab + Rcb ω ac − Rac ω cb

≡ dTa + Tb ω ab − ωTa ≡ dΩ

When the torsion vanishes, we have an algebraic identity, eb Rab Ra[bcd] Rabcd + Racdb + Radbc Rbd − Rbd

= Ωea a Ωcd] = δ[b a = δb Ωcd + δca Ωdb + δda Ωbc

= − (n − 2) Ωbd

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