Infrared modification of gravity from conformal symmetry
aa r X i v : . [ g r- q c ] A p r Infrared modification of gravity from conformal symmetry
Jack Gegenberg, ∗ Shohreh Rahmati, † and Sanjeev S. Seahra ‡ Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada E3B 5A3 (Dated: September 9, 2018)We reconsider a gauge theory of gravity in which the gauge group is the conformal group SO(4,2)and the action is of the Yang-Mills form, quadratic in the curvature. The resulting gravitationaltheory exhibits local conformal symmetry and reduces to Weyl-squared gravity under certain condi-tions. When the theory is linearized about flat spacetime, we find that matter which couples to thegenerators of special conformal transformations reproduces Newton’s inverse square law. Conversely,matter which couples to generators of translations induces a constant and possibly repulsive forcefar from the source, which may be relevant for explaining the late time acceleration of the universe.The coupling constant of theory is dimensionless, which means that it is potentially renormalizable.
I. INTRODUCTION
General relativity (GR) is a well-tested theory of grav-ity on scales ranging from microns to the size of the solarsystem. But on larger scales, cracks appear in the edificeof observational support for the theory. The gravitationalinteraction over galactic distances requires dark matterto account for rotation curves and dynamics of clusters,while over cosmological distances we require dark energyto drive accelerated expansion. While it is quite possi-ble, and some may argue probable, that one of both ofthese effects have an explanation within particle physics,it seems worthwhile to explore the possibility that GRitself is not the correct theory of gravity on large scales.An intriguing line of attack involves replacing GR witha theory respecting local conformal (Weyl) symmetry.There are diverse motivations for such an approach (see[1] for a recent discussion): One of the earliest involvesnoting that Maxwell’s equations are not only invariantunder the Poincare group of transformations ISO(3,1),they are also invariant under the larger SO(4,2) groupof conformal transformations of Minkowski spacetime [2](as are other important massless field theories [3], includ-ing the standard model with zero Higgs mass). It seemslogical that the theory of gravity should have the samesymmetries as electromagnetism, at least locally, yet GRretains local ISO(3,1), not SO(4,2), symmetry.Shortly after the appearance of GR, Weyl [4] and Bach[5] tried to rectify this by writing down a locally confor-mally invariant theory of gravity and electromagnetism.This approach failed as a unified theory, giving unac-ceptable gravity-EM couplings, and also failed to repro-duce solar system dynamics as linear gravity was gov-erned by a fourth order Poisson equation. To recoverthe inverse square law in conformal gravity, Mannheimhas recently suggested that point sources are describedby highly singular distributions involving derivatives of δ -functions [5, 6]. ∗ [email protected] † [email protected] ‡ [email protected] Another approach to alternative theories of gravitywith a long history is based on Yang-Mills (YM) gaugetheories. The YM action (here written assuming a com-pact gauge group), S = − g Z d x √− gg µν g αβ Tr( F µα F νβ ) , (1)is quadratic in the curvature F αβ of a Lie-algebra val-ued connection A α on a principle bundle over the space-time manifold ( M, g ), where g αβ is a given non-dynamicalLorentzian metric on M . The coupling constant g YM isdimensionless, which is what makes the theory perturba-tively renormalizable. By contrast, the Einstein-Hilbertaction of GR is linear in the curvature of the Christoffelconnection, there is no background (the metric is dynam-ical), and the coupling constant has dimension (mass) .The latter presents one of the main obstructions to quan-tizing GR.To construct a YM gauge theory of gravity one firstchooses a gauge group, and then identifies componentsof the gauge potential A α with a vierbein e aα (or sol-dering form) and spin-connection ω abα on ( M, g ). Thisintroduces an explicit algebraic dependence of the metricon the gauge potential, implying that YM gravity modelsgenerally exhibit less gauge symmetry than their purelyYM cousins. Also, like the Einstein-Cartan action forGR, the YM gravity action is in first order form, butunlike the Einstein-Cartan action, the torsion of the con-nection is not constrained to be zero [7–9].If one selects one of the Poincare, de Sitter, or anti-de Sitter groups as the gauge group, the identificationof e aα and spin-connection ω abα exhausts all the compo-nents of A α . (The de Sitter case has been investigatedin [9].) Here, we follow Wheeler et al [8] and considerthe gauging of the SO(4,2) group. In this case, there areextra components of the gauge potential associated withspecial conformal transformations and dilatations. Thisallows for more general matter gravity coupling than inGR. The resulting equations exhibit local conformal sym-metry.By analyzing the linearization of this model about flatspace, we find that matter that couples directly to thetetrad e aα sources a long range gravitational potentialthat may be repulsive. Conversely, matter which cou-ples to the special conformal transformations sources aNewton-like gravitational potential that gives rise to thefamiliar inverse square law. Hence, in this model it isin principle possible to realize infrared modifications ofgravity while still maintaining Newtonian gravity in thesolar system. II. THE MODEL
Our model is based on the SO(4,2) conformal group ofMinkowski spacetime, which is the largest group of trans-formations that leaves null geodesics invariant. With a, b = 0 . . .
A, B = 1 . . .
15, the fifteen generators J A of this group can be subdivided into: four transla-tions P a , four special conformal transformations K a , sixLorentz rotations J ab = − J ba , and one dilatation D . Thenon-zero commutators of generators are:[ J ab , J cd ] = η ad J bc + η bc J ad + η ac J db + η bd J ca , [ P a , J bc ] = η ba P c − η ca P b , [ D , K a ] = K a , [ K a , J bc ] = η ba K c − η ca K b , [ P a , D ] = P a , [ P a , K b ] = 2( η ab D − J ab ) , (2)where η ab is the Minkowski metric.We define an so(4,2)-Lie algebra-valued vector poten-tial by A α = A Aα J A = e aα P a + l aα K a + ω abα J ab + q α D , (3)where α = 0 . . . F αβ = F Aαβ J A are given by F Aαβ = ∂ α A Aβ − ∂ β A Aα + f ABC A Bα A Cβ , (4)with the structure constants defined by [ J A , J B ] = f CAB J C .We identify various components of A α in the J A basiswith geometric quantities in a 4-dimensional Lorentzianmanifold M with metric g αβ and affine connection Γ αβδ .In particular, we take e aα as the components of an or-thonormal frame fields on M , with ω abα as the associatedconnection one-forms. Hence, the metric and connectionare given by: g αβ = η ab e aα e bβ , Γ γαβ = e γa ( ∂ α e aβ + ω acα e cβ ) . (5)In these expressions, lowercase Greek and Latin indicesare raised and lowered with g αβ and η ab , respectively.The curvature one-forms are anti-symmetric in theirframe indices ω ( ab ) α = 0, from which it follows that theaffine connection is metric compatible [10]:0 = ∇ α g βγ , (6)where ∇ α is the derivative operator defined by Γ αβδ . TheRiemann curvature and torsion tensors of M are given by: R µναβ = e µa e νb ( dω ab + ω ac ∧ ω cb ) αβ , (7) T αβγ = e αa ( de a + ω ac ∧ e c ) βγ . (8) Note that we do not assume T αβγ = 2Γ α [ βγ ] = 0.To write down the action of our model, we start withthe curved spacetime Yang-Mills action assuming a com-pact gauge group: S = − g Z d x √− gg αµ g βν Tr ( F αβ F µν ) + S m . (9)Here, S m = S m [ A Aα , ψ ] is the action for matter ψ . To getthis into a form suitable for a non compact gauge groupsuch as SO(4,2), we merely substitute in F αβ = F Aαβ J A : S = − g Z d x √− gg αµ g βν h AB F Aαβ F Bµν + S m , (10)where h AB = Tr( J A J B ) is the Cartan-Killing metric onso(4,2). This can be related to the structure constantsvia the formula: h AB = f M AN f N BM . (11)The non-trivial components of h AB are: h a ¯ b = h ¯ ab = − η ab , h , = 2 .h [ ab ][ cd ] = h [ cd ][ ab ] = − η a [ c η d ] b . (12)The notation here is that a, ¯ a = 0 , , , P a and specialconformal transformations K a , respectively. The six in-dices [ ab ] consist of [12] , [23] , [31] , [01] , [02] , [03] and de-note directions along the distinct non-zero generators J ab of Lorentz transformations. Finally, the index 14 denotesthe component in the direction of the generator D of di-latations.We view (10) and (11) as the defining relationships forour model . Notice that the generators J A do not explic-itly appear in either of these formulae, yet the particularchoice of basis for so(4,2) does influence h AB and F Aαβ .That is, if we change basis according to˜ J A = C AB J B , (13)where C AB is a matrix such that C AD C DB = δ AB , thenthe field strength components and Killing metric trans-form as˜ h AB = C AC C BD h CD , ˜ F Aαβ = C BA F Bαβ . (14)It follows that (10) itself is invariant under such a trans-formation; i.e., the action for our model is invariant un-der a change of so(4,2) basis. Furthermore, the action ismanifestly diffeomorphism invariant.Making use of the definitions above, we can rewrite theaction (10) exclusively in terms of spacetime tensors: S = 2 g Z d x √− g h ( R αβγδ − φ αβγδ ) + 2( ∇ α f µβ + f µσ T σαβ − f µα q β ) × ( T µαβ + 2 g µ [ α q β ] )+( ∂ [ α q β ] + F αβ ) i + S m . (15)Here, we have defined: f αβ := η ab e aα l bβ , F αβ := f [ αβ ] ,φ αβγδ := g γ [ α f β ] δ − g δ [ α f β ] γ . (16)We demonstrate in Appendix A that under certain re-strictive circumstances, (15) reduces to the action ofWeyl-squared gravity.Variation of the action (10) with respect to the gaugepotential yields the equation of motion: D µ F Bµν = k Bν + j Bν , (17)where D µ is the gauge covariant derivative: D µ F Bµν := ˆ ∇ µ F Bµν + f BCD A Cµ F Dµν , (18)and ˆ ∇ µ is the derivative operator defined from the Levi-Civita connection:ˆΓ αβγ = g αρ ( ∂ β g ργ + ∂ γ g ρβ − ∂ ρ g βγ ) . (19)Note that ∇ α = ˆ ∇ α if and only if T αβγ = 0. The currentsin (17) are given by: k Aν J A = τ µν e bµ K b , j νC = − g √− g δ ( √− g L m ) δA Cν , (20)where L m is the Lagrangian density of the matter fields, j νC = h BC j Bν and τ ρσ = h AB (cid:0) F Aρµ F Bσµ − g ρσ F Aµν F Bµν (cid:1) . (21)We parametrize the matter current as j Bν J B = a aν P a + b aν K a + c abν J ab + d ν D , (22)and define a αν := e αa a aν , etc. The structure of h AB im-plies that a αβ is proportional to the functional deriva-tive of the matter action with respect to l aα ; that is, a αβ characterizes matter which couples to the generators ofspecial conformal transformations. Similarly, b αβ , c αβγ ,and d α describe matter with coupling to e aα , ω abα , or q α ,respectively. III. GAUGE SYMMETRIES
The action (10) is not invariant under the usual in-finitesimal YM gauge transformations, A Aα A Aα + ∂ α ǫ A + f ABC A Bα ǫ C , (23)with arbitrary gauge parameters ǫ A . The reason is thatalthough (10) closely resembles the familiar YM action, itdiffers in one key respect: The metric carries an explicitdependence on the gauge potential via the identification g αβ = η ab e aα e bβ , which breaks the full YM gauge symme-try. To see this explicitly, let us parametrize an arbitrarygauge transformation as ǫ A J A = χ a P a + λ a K a + Λ ab J ab + Ω D . (24)We note following gauge transformations: δe dα = ∂ α χ d − e cα Λ dc + e dα Ω + ω dcα χ c − q α χ d , (25a) δl dα = ∂ α λ d − l cα Λ dc − l dα Ω + ω dcα λ c + q α λ d , (25b) δω abα = ∂ α Λ ab + ω αac Λ cb + ω αbc Λ ac + ( λ [ a e b ] α − χ [ a l b ] α ) , (25c) δq α = ∂ α Ω + ( λ α − χ β f βα ) , (25d) δg αβ = 2 ∇ ( α χ β ) + Ω g αβ − ( q α χ β + q β χ α ) , (25e) δf αβ = ( ∇ β + q β ) λ α + f µβ ( ∇ α − q α ) χ µ , (25f)where we have written χ σ = e σa χ a , etc. Temporarilyswitching off the matter fields, we find the variation ofthe action under this gauge transformation is δS = − g Z d x √− gχ σ ( ∇ ρ τ ρσ + q ρ τ ρσ ) . (26)Hence, the action is invariant only if we take χ σ = 0.In other words, this theory is invariant under an elevenparameter subgroup of SO(4,2) gauge transformationsparametrized by ǫ A J A = λ a K a + Λ ab J ab + Ω D . (27)The absence of the P a generators in (27) is not surprising:as in GR, the Poincar´e translational symmetry of theMinkowski metric is supplanted by the diffeomorphisminvariance of curved space in our model. Under the transformation (27), the component of A α parallel to D transforms as δq α = ∂ α Ω + λ α . (28)It is obvious that via a simple series of gauge transforma-tions of the form ǫ A J A = λ a K a we can impose the gaugecondition q α = 0. For the remainder of this paper, wewill work exclusively in such a gauge. Note that there isstill some residual gauge freedom: the condition q α = 0is preserved under transformations parametrized by: ǫ A J A = − e aα ∂ α Ω K a + Λ ab J ab + Ω D . (29)Under this, the frame fields and metric transform as δe aα = ( − Λ ab + Ω δ ab ) e bα , δg αβ = Ω g αβ . (30) When thinking about gauge transformations in this model, onemay be tempted to interpret P a as the generator or diffeomor-phisms rather than translations . As is clear from (10) and (26),the action is invariant under diffeomorphisms but not invariantunder gauge transformations generated by P a . Therefore, P a isnot the generator of diffeomorphims. We see that Λ ab generates infinitesimal Lorentz rotationsof the frame fields and Ω generates infinitesimal localconformal transformations. Hence, the invariance of theaction under (30) means that our model is conformallyinvariant.The following gauge transformation generates a con-formal transformation of the metric and preserves thegauge condition q α = 0: ǫ A J A = − e aα ∂ α Ω K a + Ω D . (31)Under this, the torsion tensor transforms as δT aαβ = T aαβ Ω . (32)So, under transformations that preserve the q α = 0gauge, the torsion tensor is gauge invariant, and thetorsion-free sector is preserved under this restricted groupof gauge transformation. IV. EXPLICIT EQUATIONS OF MOTIONA. With torsion
In a gauge where q α = 0, we can re-write the compo-nents of the equation of motion (17) explicitly in termsof spacetime tensors: a αν = ˆ ∇ µ T αµν + R αν − ( f ( αν ) + f g αν ) ,b αν = ˆ ∇ µ θ αµν − f λµ Φ αλµν − f αµ F µν − τ αν ,c αβν = ˆ ∇ µ Φ αβµν + θ [ αβ ] ν − f [ α | µ | T β ] νµ ,d ν = (2 ˆ ∇ µ + ∇ µ ) F µν + ∇ µ ( f ( µν ) − g µν f )+2 F µσ T [ σµ ] ν . (33)Here, R αβ is the Ricci tensor and we have defined θ µαβ := ∇ α f µβ − ∇ β f µα + f µσ T σαβ , (34)Φ αβγδ := R αβγδ − ( g γ [ α f β ] δ − g δ [ α f β ] γ ) . (35) B. Without torsion
In general, solutions of the equations of motion (33)will have non-zero torsion. However, in the reminder ofthe paper we will concentrate on the torsion-free sectorof the solution space. In future work, we will explore themore general case in detail.If there is no torsion, equation (8) implies that: ∂ [ β e aγ ] + ω ab [ β e γ ] b = 0 . (36)Making use of the fact that ω ( ab ) α = 0, this can be solvedto give the connection one-forms in terms of the tetradand its derivatives: ω abβ = e αa e γb ( ξ βγα − ξ αβγ − ξ γαβ ) , (37) where ξ αβγ = e αa ∂ β e aγ − e αa ∂ γ e aβ . (38)Making note of g αβ = e αa e aβ , this can be re-written as ω abβ = e αa e γb ( ∂ β g γα + ∂ γ g αβ − ∂ α g βγ − e αc ∂ β e cγ ) . (39)Substituting this into the second member of (5), we getΓ αβγ = g αρ ( ∂ β g γρ + ∂ γ g ρβ − ∂ ρ g βγ ) . (40)Hence, if the torsion is zero we find that Γ αβγ reducesto the Levi-Civita connection and ∇ α is the familiar co-variant derivative used in general relativity. Of course wecould have come to this conclusion without any calcula-tions: We have a priori assumed that our connection ismetric compatible, so the additional restriction that thetorsion vanishes means that we must necessarily recover(40).When we assume that T αβγ = 0 in the equations of mo-tion (33), several simplifications occur. The first memberof (33) can be algebraically solved for f ( αβ ) : f ( αβ ) = 4 S αβ − a αβ . (41)where ¯ a αβ := a αβ − g αβ a , a = a µµ , and S αβ := ( R αβ − Rg αβ ) (42)is known as the Schouten tensor. The third and fourthmembers of (33) yield a consistency condition satisfiedby matter fields to ensure T αβγ = 0,0 = ∇ α a αβ + c βαα − d β . (43)as well as relations resembling Maxwell’s equations in thepresence of electric and magnetic charges: ∇ [ α F βν ] = g νβ c αµµ − g να c βµµ − c αβν , (44) ∇ α F αβ = d β − c βαα − ∇ β a. (45)Finally, the second member of (33) gives B αν = − b αν + ∇ µ ( ∇ [ ν ¯ a µ ] α + ∇ [ ν F µ ] α ) + Q αν , (46)where B µν = −∇ α ∇ α S µν + ∇ α ∇ µ S αν + C µανβ S αβ , (47)is the Bach tensor, C αβγδ is the Weyl tensor, and Q αν isa tensor quadratic in F αβ , a αβ and the curvature: Q αν = a λµ C αλµν − τ αν − (2 S λµ − ¯ a λµ + F λµ ) × ( g λ [ µ ¯ a ν ] α − g α [ µ ¯ a ν ] λ + g α [ µ F ν ] λ − g λ [ µ F ν ] α + g αλ F µν ) . (48) V. WEAK FIELDS
We now consider the linearization of the torsion-freesector of the model about a Minkowski background inwhich R αβγδ , F αβ and all matter fields vanish. For sim-plicity, we will consider perturbative matter sources with c αβγ = 0 = d α ⇒ ∂ α a αβ = 0 . (49)As in [5, 6], we define a traceless metric perturbation H αβ : g αβ = η αβ + h αβ , H αβ = h αβ − η αβ h. (50)Under these assumptions and expanding to linear order,we find that ∇ [ α F βν ] = 0 , ∇ α F αβ = − ∇ β a, (cid:3) a = 0 , (51)and − ( ∂ α ∂ β ∂ µ ∂ ν H µν − (cid:3) ∂ ν ∂ ( α H β ) ν + η αβ ∂ µ ∂ ν (cid:3) H µν + (cid:3) H αβ ) = (cid:3) a TF αβ + b αβ + ∂ α ∂ β a. (52)Here, a TF αβ indicates the trace-free part of a αβ . We seethat the trace of a αβ acts as a source for both F αβ and H αβ . We can hence consider H αβ to be the sum of con-tributions sourced by a and a TF αβ , respectively. Concen-trating on the latter, we can impose the transverse gaugecondition ∂ α H αβ = 0 to simplify the equations of mo-tion. For static sources and fields, the equations can besolved explicitly via Green’s functions in this gauge: H αβ ( r ) = Z d r ′ a TF αβ ( r ′ )2 π | r − r ′ | + Z d r ′ | r − r ′ | π b αβ ( r ′ ) . (53)We see that the metric perturbations sourced by a TF αβ fall-off as inverse distance, just as in the static weak field limitof general relativity. However, the perturbations sourcedby b αβ increase proportionally with distance, implyinga modification of the gravitational interaction on longwavelengths. The existence of distinct short and longrange gravitational forces is directly related to the ap-pearance of (cid:3) a TF αβ in (52) and not a TF αβ , which in turnfollows from (46).If we do not impose the transverse condition ( ∂ α H αβ =0) and assume that a αβ and b αβ both have δ -functionsupport at the origin, we can solve (52) explicitly: ds = − (1 + 2 φ ) dt + (1 − ψ )( dx + dy + dz ) , (54) φ + ψ − r a r − rr b . (55)Here, r a and r b are constants proportional to the am-plitude of the δ -functions in a αβ and b αβ , respectively.Note that r a and r b are not necessarily positive; theirsigns cannot be determined in the absence of a specificmatter model. Also note that the equation of motion (52) does not fix the difference φ − ψ of the metric po-tentials. This is because the gauge transformations (29)gives δh αβ = ǫη αβ , which implies: δ ( φ + ψ ) = 0 , δ ( φ − ψ ) = ǫ ; (56)i.e., φ + ψ is gauge invariant while φ − ψ is a purely gaugedegree of freedom. We can use this freedom to imposea parametrized-post-Newtonian (PPN) [11] gauge condi-tion φ = − r a /r from which it follows that the standardPPN parameter γ is γ = ψφ = 1 + 2 r | r a r b | . (57)Hence, in order to satisfy the Cassini constraint | γ − | . − , we need | r a r b | ≫ r in the solar system.One may be concerned that when we assume that a αβ ∝ δ ( r ), the source term on the righthand side of(52) contains derivatives of a δ -function; i.e., our mech-anism for recovering Newton’s law involves the samehighly singular sources as envisioned by Mannheim et al[5, 6]. However, there is a subtle but important distinc-tion: Mannheim assumes the matter density for pointlikesources (and hence stress-energy tensors) involves factorsof the form → ∇ δ ( r ), which are hard to obtain from thevariation of a non-singular classical action. In our calcu-lations, matter fields obtained from variation of the ac-tion (i.e., a αβ , b αβ , etc.) have no worse than δ -functionsingularities, and hence imply the action is finite. (Fur-ther discussion of the viability of the Newtonian limit inMannheim’s theory, including the possible need for ex-tended sources, can be found in Refs. [12, 13]. ) VI. DISCUSSION
We have reconsidered a gauge theory of gravity inwhich the gauge group is the conformal group SO(4,2)and the action is of the Yang-Mills form, quadratic in thecurvature. By identifying the “background metric” withthe fields gauged by the translation generator, the fullSO(4,2) gauge invariance is broken to that generated bythe Lorentz rotations, special conformal transformationsand dilatations. Under certain restrictions, the vacuumequations of motion of our model are solved by the solu-tions of the vacuum equations of Weyl squared gravity.We then considered the linearization about torsion-free Minkowski spacetime. We found that matter whichcouples to the generators of special conformal transfor-mations induces gravitational forces which fall off as r − , as in the weak field limit of GR. Conversely, mat-ter that couples directly to the vierbein induces forces It should be noted that the conclusions of [12] have been rebuttedin [5]. which are asymptotically constant in the far field. Un-like Weyl’s original conformal gravity (in the absence ofthe highly singular sources proposed in [5, 6]), the the-ory is potentially consistent with solar systems tests ofgravity (with judicious parameter choices). Furthermore,the long range behaviour of the gravitational interactioncould provide an explanation of the late time accelerationof the universe. The fact that matter with nonstandardcoupling is responsible for Newtonian gravity is an ex-otic feature of our model that deserves further study. Fi-nally, the theory may be perturbatively renormalizable,since the gauge coupling constant (in four dimensions) isdimensionless.However, we must state an important caveat: As (52)shows, the linearized sector of the theory is governed bya fourth-order wave equation. According to “standardfolklore” such a theory should be prone to ghost insta-bilities upon quantization [14]; however we note that thisis a controversial assertion and there are known counterexamples [15]. In future work, we plan to study this issuein detail by examining the canonical form of the theoryand its quantization. Also of interest for further workis the role of torsion and F αβ in astrophysics at boththe linear and non-linear level, as well as cosmologicalsolutions.We conclude by noting that local conformal symmetryis clearly not an observed property of low-energy physics.Therefore, our model will require a mechanism to breakthis symmetry at low energy in order to be consistentwith familiar phenomena. There are several ways thiscan be accomplished, one of which involves coupling themodel to non-conformally invariant classical matter. Or,as advocated by Mannheim [16] and t’Hooft [1], one canassume conformal symmetry is broken spontaneously inanalogy to the breaking of electroweak symmetry in thestandard model via the Brout–Englert–Higgs mechanism.That is, the vacuum state of some quantum field picksout a preferred conformal frame. We hope to report onsuch issues in the future. ACKNOWLEDGMENTS
This work was supported by NSERC of Canada.
Appendix A: Reduction to Weyl squared gravity
In this appendix, we examine our model under the fol-lowing (somewhat restrictive) assumptions: ( i ) Torsion-free: T αβγ = 0;( ii ) q α = 0;( iii ) F µν = 0; and( iv ) S m only depends on g αβ and matter fields, and noneof the other components of A α ; i.e, a αβ = c αβγ = d α = 0.We will see that these lead us to Weyl-squared gravity.Under these circumstances, the action (15) reduces to S = 2 g Z d x √− g ( R αβγδ − φ αβγδ ) + S m . (A1)Referring to the definition (16) of φ αβγδ , we see thatthis action contains no derivatives of f αβ . Hence, f αβ isessentially a Lagrange multiplier. Variation of the actionwith respect to f αβ yields: δSδf αβ δf αβ = 8 g Z d x √− g × [ R αβ − (cid:0) f αβ + f g αβ (cid:1) ] δf αβ . (A2)Setting this equation to zero yields a constraint, whichafter some algebra gives: f αβ = 2 (cid:0) R αβ − Rg αβ (cid:1) . (A3)Substituting this constraint back into the action (A1)gives S = 2 g Z d x √− g C αβγδ C αβγδ + S m , (A4)where C αβγδ is the ordinary Weyl-tensor. Now, variationwith respect to the metric yields δS = 4 g Z d x √− g (cid:0) − B αβ − g T αβ (cid:1) δg αβ , (A5)where T αβ is the ordinary stress energy tensor, and theBach tensor is defined in (47). We then obtain the fieldequation B µν = − g T µν . (A6)This is the equation originally studied by Bach [5]. Noticethat the Bach tensor is traceless, which places restrictionson the form of material sources of the form g αβ T αβ = 0. [1] G. T. Hooft, (2014), arXiv:1410.6675 [gr-qc].[2] E. Cunningham, Proc. London Math. Soc. (Ser. 2) , 77(1910); T. Bateman, ibid . , 223 (1910).[3] P. Dirac, Ann. Math. , 429 (1936); M. Drew andJ. Gegenberg, Il Nuovo Cimento A , 41 (1980). [4] H. Weyl, Space-Time-Matter (Methuen, 1918).[5] R. Bach, Math. Zeitschr. , 110 (1921); P. D.Mannheim, Phys. Rev. D75 , 124006 (2007),arXiv:gr-qc/0703037 [gr-qc]. [6] P. D. Mannheim and D. Kazanas,Gen.Rel.Grav. , 337 (1994); P. D.Mannheim, Found.Phys. , 709 (2000),arXiv:gr-qc/0001011 [gr-qc];Found.Phys. , 388 (2012), arXiv:1101.2186 [hep-th].[7] R. Utiyama, Phys.Rev. , 1597 (1956); T. W. B.Kibble, J. Math. Phys. , 212 (1961); P. Townsend,Phys. Rev. D 15 , 2795 (1977); S. W. MacDowell andF. Mansouri, Phys. Rev. Lett. , 739 (1977); F. Man-souri, Phys. Rev. Lett. , 1021 (1979); K. Hayashi andT. Shirafuji, Prog. Theor. Phys. , 866882 (1980); E. A.Ivanov and J. Niederle, Phys. Rev. D , 976 (1982).[8] J. T. Wheeler, Phys.Rev. D44 , 1769 (1991);J. S. Hazboun and J. T. Wheeler,Class.Quant.Grav. , 215001 (2014),arXiv:1305.6972 [gr-qc]; J. T. Wheeler,Phys.Rev. D90 , 025027 (2014), arXiv:1310.0526 [gr-qc].[9] C.-G. Huang, Y. Tian, X. Wu, and H.-Y. Guo, Front.Phys.China , 191 (2008),arXiv:0804.4339 [gr-qc]; C.-G. Huang andM.-S. Ma, Front.Phys.China , 525 (2009),arXiv:0906.2622 [gr-qc]; Phys.Rev. D80 , 084033 (2009), arXiv:0906.2837 [gr-qc]; J.-A. Lu and C.-G.Huang, Class.Quant.Grav. , 145004 (2013),arXiv:1306.3028 [gr-qc]; Gen.Rel.Grav. , 691 (2013),arXiv:1301.5796 [gr-qc].[10] S. M. Carroll, Spacetime and geometry: An introduction to general relativity (Addison-Wesley, San Francisco, USA, 2004).[11] C. M. Will, Living Rev.Rel. , 4 (2014),arXiv:1403.7377 [gr-qc].[12] E. E. Flanagan, Phys. Rev. D74 , 023002 (2006),arXiv:astro-ph/0605504 [astro-ph].[13] Y. Yoon, Phys. Rev.
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