Conserved charges in extended theories of gravity
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 where is a divergence-free, symmetric two-tensor possibly coming from an action. Here, denotes the Riemann tensor or its contractions, is the -dimensional Newton’s constant and represents the nongravitational matter source. It is clear that does not yield a globally conserved charge without further structure. A moment of reflection shows that given an exact Killing vector , 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 where is a smooth function of the Riemann tensor and its contractions given in the form , and hence contractions do not require the metric. For example, and . Here, we start with the quadratic theory whose results can be easily extended to the above more general action as we shall lay
Charges of 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 whose field equation can be found from the following variation For generic variations of the metric, the variation of the Riemann tensor reads 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 where 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 the action can be recast as
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 is assigned zero charges and subsequently the conserved charges of a perturbed spacetime 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 can be defined as where for a Cauchy surface . Using (361) and Stokes’ theorem, one can have or, using the definition of , one explicitly has where is the boundary of . 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 which vanishes on the bifurcation surface . Now, take in (377) to be the bifurcation surface then one has Up to now, and so 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 [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) reduces to
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 where are the field equations (308)–(311) and is a surface term which reads explicitly as Now, assume that the variation is due to a diffeomorphism which is generated by the vector field , then the variation of the Lagrangian (307) is 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 where are constant parameters, are arbitrary functions of and are given as This metric is written in the ingoing Eddington–Finkelstein coordinates: , and are the advanced time, the radial coordinate and the angular
Horizon fluffs in generalized minimal massive gravity
Banados geometries have the metric [191] where . , and are respectively radial, time and angular coordinates and 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 transforms as . The variation generated by the following Killing vector field preserves the form of the metric [192]
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 where are the components of spin-connection 1-form, we can decompose the spin-connection in two independent parts (227), where is the torsion-free part which is known as Riemannian spin-connection and is contorsion 1-form. As discussed before, we denote contorsion 1-form by
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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