Elsevier

Physics Reports

Volumes 834–835, 20 November 2019, Pages 1-85
Physics Reports

Conserved charges in extended theories of gravity

https://doi.org/10.1016/j.physrep.2019.08.003Get rights and content

Abstract

We give a detailed review of construction of conserved quantities in extended theories of gravity for asymptotically maximally symmetric spacetimes and carry out explicit computations for various solutions. Our construction is based on the Killing charge method, and a proper discussion of the conserved charges of extended gravity theories with this method requires studying the corresponding charges in Einstein’s theory with or without a cosmological constant. Hence we study the ADM charges (in the asymptotically flat case but in generic viable coordinates), the AD charges (in generic Einstein spaces, including the anti-de Sitter spacetimes) and the ADT charges in anti-de Sitter spacetimes. We also discuss the conformal properties and the behavior of these charges under large gauge transformations as well as the linearization instability issue which explains the vanishing charge problem for some particular extended theories. We devote a long discussion to the quasi-local and off-shell generalization of conserved charges in the 2+1 dimensional Chern–Simons like theories and suggest their possible relevance to the entropy of black holes.

Section snippets

Reading guide and conventions

This review is naturally divided into two parts: In Part I, we discuss global conserved charges for generic modified or extended gravity theories. Global here refers to the fact that the integrals defining the charges are on a spacelike surface on the boundary of the space. In Part II, we discuss quasi-local and off-shell conserved charges; particularly, for 2 + 1 dimensional gravity theories with Chern–Simons like actions. In what follows, the meaning of these concepts will be elaborated.

ADT energy

Let us consider a generic gravity theory defined by the field equations Eμνg,R,R,R2,=κτμν,where Eμν is a divergence-free, symmetric two-tensor possibly coming from an action. Here, R denotes the Riemann tensor or its contractions, κ is the n-dimensional Newton’s constant and τμν represents the nongravitational matter source. It is clear that μEμν=0 does not yield a globally conserved charge without further structure. A moment of reflection shows that given an exact Killing vector ξμ, one can

Charges of quadratic curvature gravity

We will work out the construction of conserved charges for generic gravity theories with an action of the form I=dDx|g|fRρσμν,where F is a smooth function of the Riemann tensor and its contractions given in the form RρσμνRμνρσμν, and hence contractions do not require the metric. For example, Rνσν=Rμσμν and R=Rμνμν. Here, we start with the quadratic theory I=dnxg1κR2Λ0+αR2+βRμνRμν+γRμνρσRμνρσ4RμνRμν+R2,whose results can be easily extended to the above more general action as we shall lay

Charges of fRρσμν gravity

In principle, using the above construction for quadratic gravity, we can find the conserved charges for asymptotically (A)dS spacetimes of the more generic theory defined by the action I=dnxgfRρσμν,whose field equation can be found from the following variation δgI=dnxδgfRαβμν+gfRρσμνδRρσμν.For generic variations of the metric, the variation of the Riemann tensor reads δRρσμν=12gαρσνgασρνδgμα12gαρσμgασρμδgαν12Rρσναρσνδgμα+12Rρσμαρσμδgαν, which follows from usual easy to

Conserved charges of topologically massive gravity

In the above examples, we studied explicitly diffeomorphism invariant theories, in this section we will study the celebrated topologically massive gravity (TMG) [87], [88] in three dimensions whose action is diffeomorphism invariant only up to a boundary term. The conserved charges of this theory for flat backgrounds was constructed in [88] and its extension for AdS backgrounds was given in [89] following the AD construction laid out above. The theory also admits non-Einsteinian nonsingular

Charges in Scalar–Tensor Gravities

Let us consider a generic scalar–tensor modification of Einstein’s theory given by the action I=12κdnxgUϕRg2Λ0WϕμϕμϕVϕ+HϕLmatterψ,where Lmatterψ represents all the matter besides the scalar. One can add higher order curvature terms, but for the sake of simplicity we will consider the above theory and study the conformal properties of the charges we defined above. With the following conformal transformation to the so-called Einstein frame gμνEUϕ2ngμν,the action can be recast as I=12κdnx

Conserved charges in the first order formulation

It is well-known that when fermions are introduced to gravity, the metric formulation is not sufficient and one has to introduce the vierbein and spin connection. As this is the case in the supergravity theories, we will introduce the construction of conserved charges in the first order formalism in this section. This will also be a useful background material for the rest of this review where we shall define the three-dimensional Chern–Simons like theories in this formalism. The following

Vanishing conserved charges and linearization instability

The astute reader might have realized that in the above construction of global conserved charges for Einstein’s theory or for generic gravity theories, two crucial ingredients are the Stokes’ theorem and the existence of asymptotic rigid symmetries (or Killing vectors). Once Stokes’ theorem is invoked, one necessarily resorts to perturbative methods: namely, a background spacetime M,ḡ is assigned zero charges and subsequently the conserved charges of a perturbed spacetime M,g that has the same

Chern–Simons-like theories of gravity

There is a class of gravitational theories in (2 + 1)-dimensions that are naturally expressed in terms of first order formalism. Some examples of such theories are topological massive gravity (TMG) [88], new massive gravity (NMG) [134], minimal massive gravity (MMG) [135], zwei-dreibein gravity (ZDG) [136], generalized minimal massive gravity (GMMG) [137], etc. We shall refer to all such theories as Chern–Simons-like theories of gravity [138].

Some examples of Chern–Simons-like theories of gravity

Here, we will discuss briefly some examples of the above construction. Further examples can be found in [144].

Extended off-shell ADT current

Historically, Regge and Teitelboim were the pioneers who showed that it is possible to write the charges as boundary terms in gravity theories [9], [155]. Arnowitt, Deser, and Misner (ADM) [3] then introduced a way to associate a conserved mass for an asymptotically flat spacetime in general relativity. Their construction was generalized to the asymptotically AdS spacetimes for the cosmological Einstein’s gravity [4], [156]. In a further development Deser and Tekin generalized the ADM’s

Off-shell extension of the covariant phase space method

In this section, we define the Lee–Wald symplectic current [86], [124], [127], [128], [129], [158] in the first order formalism. For diffeomorphism covariant theories, the symplectic current was introduced in [127] while the generalization to noncovariant theories was carried out in [129]. As we discussed above, since the total variation is covariant, the covariant phase space method can be used to obtain conserved charges for the gravity theories formulated in the first order formalism.

To

Quasi-local conserved charge

In this section, we define quasi-local conserved charges with the basic assumption that the spacetime is globally hyperbolic. The quasi-local charge perturbation for a diffeomorphism generator ξ can be defined as δQ(ξ)=18πGVJADT(a,δa;ξ),where VC for a Cauchy surface C. Using (361) and Stokes’ theorem, one can have δQ(ξ)=18πGΣQADT(a,δa;ξ),or, using the definition of QADT, one explicitly has δQ(ξ)=18πGΣδKξKδξiξΘ(a,δa),where Σ is the boundary of V. Note that the first term is the charge

Black hole entropy in Chern–Simons-like theories of gravity

Wald prescription will be utilized to find an expression for black hole entropy in the Chern–Simons like theories. Wald’s suggestion is to identify the black hole entropy as the conserved charge associated with the horizon-generating Killing vector field ζ which vanishes on the bifurcation surface B. Now, take Σ in (377) to be the bifurcation surface B then one has Q(ζ)=18πGg̃ωrBχζar.Up to now, λξab and so χξa have been considered to arbitrary functions of the spacetime coordinates and of the

Asymptotically AdS3 spacetimes

The Brown–Henneaux boundary conditions [26] are appropriate for both cosmological Einstein’s gravity and general massive gravity models.

Asymptotically spacelike warped anti-de Sitter spacetimes in general minimal massive gravity

In this section, we consider asymptotically spacelike warped anti-de Sitter spacetimes in the context of GMMG model. We find conserved charges and algebra among them in the given model.

Conserved charges of the rotating Oliva–Tempo–Troncoso black hole

The rotating OTT black hole spacetime (301), is a solution of the NMG in the special choice of the parameters that leads to a unique vacuum as discussed. To find the conserved charges of this black hole, we take the AdS3 spacetime (415) as the background, i.e. the AdS3 spacetime corresponds to s=0 [176]. The non-zero components of the flavor metric are given in (294), with (300), one can show that the extended off-shell ADT charge (360) QADT(ξ)=g̃rsiξarg̃ωsχξδas,reduces to QADT(ξ)=2iξēδΩϕ+

Explicit examples of the black hole entropy

In this section we shall use the black hole entropy formula (408) to compute entropy of some black holes for various models. In fact, based on papers [161], [170], [176], [177], we simplify entropy formula in the context of MMG, GMG, GMMG and NMG.

Near horizon symmetries of the non-extremal black hole solutions of the general minimal massive gravity

An arbitrary variation of the Lagrangian the GMMG (307) is given by δLGMMG=δeEe+δωEω+δfEf+δhEh+dΘ̃(a,δa),where Ee=Eω=Ef=Eh=0 are the field equations (308)–(311) and Θ̃(a,δa) is a surface term which reads explicitly as Θ̃(a,δa)=σδωe+12μδωω1m2δωf+δeh.Now, assume that the variation is due to a diffeomorphism which is generated by the vector field ξ, then the variation of the Lagrangian (307) is δξL=δξeEe+δξωEω+δξfEf+δξhEh+dΘ̃(a,δξa).The Lorentz Chern–Simons term in the Lagrangian

Extended near horizon geometry

In [184], the authors proposed the following metric with new fall-off conditions for the near horizon of a non-extremal black hole ds2=lρf+ζ++fζ+l24ζ+ζ2dv2+2ldvdρ+lJ+ζ+Jζdρdϕ+lρJ+ζ+Jζf+ζ++fζdvdϕ+l24J++J2lρζ+ζf+ζ++fζJ+Jdϕ2,where ζ± are constant parameters, J±=J±(ϕ) are arbitrary functions of ϕ and f±=f±(ρ) are given as f±(ρ)=1ρ2lζ±.This metric is written in the ingoing Eddington–Finkelstein coordinates: v, ρ and ϕ are the advanced time, the radial coordinate and the angular

Horizon fluffs in generalized minimal massive gravity

Banados geometries have the metric [191] ds2=l2dr2r2rdx+l2Lrdxrdxl2L+rdx+,where x±=tl±ϕ. r, t and ϕϕ+2π are respectively radial, time and angular coordinates and L±=L±(x±) are two arbitrary periodic functions. This metric solves cosmological Einstein’s gravity, then we can use the expression (558). The metric under transformations generated by ξ transforms as δξgμν=£ξgμν. The variation generated by the following Killing vector field preserves the form of the metric [192] ξr=r2+T++T,

Asymptotically flat spacetimes in general minimal massive gravity

We apply the fall of conditions presented in [198] for the asymptotically flat spacetime solutions of GMMG model.

Minimal massive gravity coupled to a scalar field

In [202], it was demonstrated that from the Lorentz–Chern–Simons action, a topologically massive gravity (TMG) non-minimally coupled to a scalar field can be constructed. Given LCS(ω)=ωbadωab+23ωbaωcbωac,where ωab are the components of spin-connection 1-form, we can decompose the spin-connection in two independent parts (227), where Ωab is the torsion-free part which is known as Riemannian spin-connection and Cab is contorsion 1-form. As discussed before, we denote contorsion 1-form by κab

Conclusions

In the first part of this review, we gave detailed account of Killing charge construction of global conserved quantities in generic gravity theories following the works of Deser–Tekin which also relied on the work of Abbott–Deser that was carried out for cosmological Einstein’s theory. All of these constructions of course extend the well-known ADM mass for asymptotically flat spacetimes. We have discussed subtle issues about the decay conditions, large gauge transformations, and linearization

Acknowledgments

Bayram Tekin would like to thank Stanley Deser for many useful discussions since 2001 to this day on conserved charges of gravity theories, particularly generic gravity theories. The work of Mohammad Reza Setare and Hamed Adami has been financially supported by Research Institute for Astronomy Astrophysics of Maragha (RIAAM) .

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