Abbott-Deser-Tekin Research Articles

Conserved charges of GMMG in arbitrary backgrounds

We extend the Abbott–Deser–Tekin (ADT) method to the computation of the global charges associated with Killing vectors of the background for solutions of general minimal massive gravity (GMMG) linearized about an arbitrary background. Then, we implement our result in various solutions to evaluate the mass and angular momentum with arbitrary asymptotics.

Off-shell ADT conserved quantities in Palatini gravity

In this paper we generalize the off-shell Abbott–Deser–Tekin (ADT) conserved charge formalism to Palatini theory of gravity with torsion and non-metricity. Our construction is based on the coordinate formalism and the independent dynamic fields are the metric and the affine connection. For a general Palatini theory of gravity, which is diffeomorphism invariant up to a boundary term, we obtain the most general expression for off-shell ADT potential. As explicit examples, we derive the off-shell ADT potentials for Einstein–Hilbert action, the most general theories and the teleparallel Palatini gravity.

Generalized off-shell ADT conserved charges in the presence of matter Chern–Simons term

In this paper we elevate the formalism for off-shell Abbott–Deser–Tekin (ADT) conserved charges in the presence of arbitrary matter fields to including the internal gauge transformation, when the gauge fields are present. For this purpose, we resort to exact symmetry, and the symmetry generator is combined by diffeomorphism and internal gauge transformation. For universality, we consider an apparently non-covariant Lagrangian, which contains a matter Chern–Simons term. We show that the elevated off-shell ADT formalism is equivalent to the covariant phase space method and the Barnich–Brandt–Compère (BBC) formalism. To check the validity of our method, we explicitly compute the conserved charges of general non-extremal rotating charged Gödel black holes in minimal five-dimensional gauged supergravity, reproducing the previously known results.

Mass and angular momentum of charged rotating Gödel black holes in five-dimensional minimal supergravity

In this paper, for the sake of providing a concrete comparison between the usual Abbott–Deser–Tekin (ADT) formalism and its off-shell extension, as well as comparing the latter with the Barnich–Brandt–Compere (BBC) approach, we carry out these methods to compute the mass and angular momentum of the rotating charged Gödel black holes in five-dimensional minimal supergravity. We first present the off-shell ADT potential of the supergravity theories in arbitrary odd dimensions, which is consistent with the superpotential via the BBC approach. Then the off-shell generalized ADT method is applied to evaluate the mass and angular momentum of the Gödel-type black holes by including the contribution from the gauge field. Finally, we strictly obey the rules of the original ADT formalism to incorporate the contribution from the gauge field within the potential. With the help of the modified potential, we try to seek for appropriate reference backgrounds to produce the mass and angular momentum. It is observed that the ADT formalism has to incorporate the contribution from the matter fields to yield physical charges for the Gödel-type black holes.

Field-theoretical construction of currents and superpotentials in Lovelock gravity

Conserved currents and related superpotentials for perturbations on arbitrary backgrounds in the Lovelock theory are constructed. We use the Lagrangian based field-theoretical method where perturbations are considered as dynamical fields propagating on a given background. Such a formulation is exact (not approximate) and equivalent to the theory in the original metric form. From the very start, using Noether theorem, we derive the Noether–Klein identities and adopt them for the purposes of the current work. Applying these identities in the framework of Lovelock theory, we construct conserved currents, energy-momentum tensors out of them, and related superpotentials with arbitrary displacement vectors, not restricting to Killing vectors. A comparison with the well known Abbott–Deser–Tekin approach is given. The developed general formalism is applied to give conserved quantities for perturbations on anti-de Sitter (AdS) backgrounds. As a test we calculate mass of the Schwarzschild–AdS black hole in the Lovelock theory in arbitrary D dimensions. Proposals for future applications are presented.

Abbott–Deser–Tekin like conserved quantities in Lanczos–Lovelock gravity: beyond Killing diffeomorphisms

We obtain the conserved Abbott–Deser–Tekin (ADT)- like current for the Lanczos–Lovelock gravity for a diffeomorphism vector, which defines the horizon of a spacetime and, importantly, is not necessarily a Killing vector. As the original ADT current is defined only for the presence of a background Killing vector, one cannot use it extensively for the thermodynamic description of the wide classes of non-Killing horizons which appear in gravity. On the other hand, this general approach can be utilized for those horizons. Here, the conserved current can be written as the derivative of the two-rank anti-symmetric potential, the connection of which is apparent with the conserved Noether potential from our analysis. If one assumes the diffeomorphism vector as the Killing one, the results match to the ADT case, whereas non-trivial result comes for the conformal Killing vectors and other horizon defining vectors.

Noether and Abbott-Deser-Tekin conserved quantities in scalar-tensor theory of gravity both in Jordan and Einstein frames

We revisit the thermodynamic aspects of the scalar-tensor theory of gravity in the Jordan and in the Einstein frame. Examining the {\it missing links} of this theory carefully, we establish the thermodynamic descriptions from the conserved currents and potentials by following both the Noether and the Abbott-Deser-Tekin (ADT) formalism. With the help of conserved Noether current and potential, we define the thermodynamic quantities, which we show to be {\it conformally invariant}. Moreover, the defined quantities are shown to fit nicely in the laws of (the first and the second) black hole thermodynamics formulated by the Wald's method. We stretch the study of the conformal equivalence of the physical quantities in these two frames by following the ADT formalism. Our further study reveals that there is a connection between the ADT and the Noether conserved quantities, which signifies that the ADT approach provide the equivalent thermodynamic description in the two frames as obtained in Noether prescription. Our whole analysis is very general as the conserved Noether and ADT currents and potentials are formulated {\it off-shell} and the analysis is exempted from any prior assumption or boundary condition.

Revisiting the ADT mass of the five-dimensional rotating black holes with squashed horizons

We evaluate the Abbott–Deser–Tekin (ADT) mass of the five-dimensional rotating black holes with squashed horizons on two different on-shell reference backgrounds, which are the flat background and the boundary matched Kaluza–Klein (KK) monopole. The mass on the former, identified with the one on the background of the asymptotic geometry, differs from the mass on the latter by that of the KK monopole. However, each mass satisfies the first law of black hole thermodynamics. To test the results in five dimensions, we compute the mass in the context of the dimensionally reduced theory. Finally, in contrast with the original ADT formulation, its off-shell generalisation is applied to calculate the mass as well.

A note on the mass of Kerr–AdS black holes in the off-shell generalized ADT formalism☆☆Project supported by the National Natural Science Foundation of China (Grant Nos. 11505036 and 11275157) and partially supported by the Science Fund from the Technology Department of Guizhou Province, China (Grant No. (2016)1104).

In this note, the off-shell generalized Abbott–Deser–Tekin (ADT) formalism is applied to explore the mass of Kerr–anti-de Sitter (Kerr–AdS) black holes in various dimensions within asymptotically rotating frames. The cases in four and five dimensions are explicitly investigated. It is demonstrated that the asymptotically rotating effect may make the charge non-integrable or unphysical when the asymptotic non-rotating timelike Killing vector associated with the charge is allowed to vary and the fluctuation of the metric is determined by the variation of all the mass and rotation parameters. To obtain a physically meaningful mass, it is proposed that one can let the non-rotating timelike Killing vector be fixed or perform calculations in the asymptotically static frame. The results further support that the ADT formalism is background-dependent.

Conserved charge of a gravity theory with p -form gauge fields and its property under Kaluza-Klein reduction

In this paper, we investigate the conserved charges of generally diffeomorphism invariant gravity theories with a wide variety of matter fields, particularly of the theories with multiple scalar fields and $p$-form potentials, in the context of the off-shell generalized Abbott-Deser-Tekin (ADT) formalism. We first construct a new off-shell ADT current that consists of the terms for the variation of a Killing vector and expressions of the field equations as well as the Lie derivative of a surface term with respect to the Killing vector within the framework of generally diffeomorphism invariant gravity theories involving various matter fields. After deriving the off-shell ADT potential corresponding to this current, we propose a formula of conserved charges for these theories. Next, we derive the off-shell ADT potential associated with the generic Lagrangian that describes a large range of gravity theories with a number of scalar fields and $p$-form potentials. Finally, the properties of the off-shell generalized ADT charges for the theory of Einstein gravity and the gravity theories with a single $p$-form potential are investigated by performing Kaluza-Klein dimensional reduction along a compactified direction. The results indicate that the charge contributed by all the fields in the lower-dimensional theory is equal to that of the higher-dimensional one at mathematical level with the hypothesis that the higher-dimensional spacetime allows for the existence of the compactified dimension. In order to illustrate our calculations, the mass and angular momentum for the five-dimensional rotating Kaluza-Klein black holes are explicitly evaluated as an example.

Abbott–Deser–Tekin Charge of Dilaton Black Holes with Squashed Horizons**Supported by the National Natural Science Foundation of China under Grant Nos 11275157 and 11505036, the Doctoral Research Fund of Guizhou Normal University in 2014, the Technology Department of Guizhou Province Fund under Grant No [2015]2114, and the Science and Technology Innovation Talent Team of Guizhou Province under Grant No (2015)4015.

We consider the conserved charge of static black holes with squashed horizons in the Einstein-Maxwell-dilaton theory via both the Abbott–Deser–Tekin (ADT) method and its off-shell generalization. We first make use of the original ADT method to compute the mass of the dilaton squashed black holes in terms of three different reference spacetimes, which are the asymptotic geometry, the flat background and the spacetime of the Kaluza-Klein monopole with boundary matched to the original metric, respectively. Each mass satisfies the first law of black hole thermodynamics, although the mass computed on the basis of the boundary matching the Kaluza-Klein monopole is different from that of the other two reference spacetimes. Then the mass of the black holes is evaluated through the off-shell generalized ADT method.

Mass and angular momentum of black holes in low-energy heterotic string theory

We investigate conserved charges in the low-energy effective field theory describing heterotic string theory. Starting with a general Lagrangian that consists of a metric, a scalar field, a vector gauge field, together with a two-form potential, we derive off-shell Noether potentials of the Lagrangian and generalize the Abbott–Deser–Tekin (ADT) formalism to the off-shell level by establishing one-to-one correspondence between the ADT potential and the off-shell Noether potential. It is proved that the off-shell generalized ADT formalism is conformally invariant. Then, we apply the formulation to compute mass and angular momentum of the four-dimensional Kerr–Sen black hole and the five-dimensional rotating charged black string in the string frame without a necessity to transform the metrics into the Einstein frame.

Conserved charges of black holes in Weyl and Einstein–Gauss–Bonnet gravities

An off-shell generalization of the Abbott-Deser-Tekin (ADT) conserved charge was recently proposed by Kim et al. They achieved this by introducing off-shell Noether currents and potentials. In this paper, we construct the crucial off-shell Noether current by the variation of the Bianchi identity for the expression of motion equation, with the help of the property of Killing vector. Our Noether current, which contains an additional term that is just one half of the Lie derivative of a surface term with respect to the Killing vector, takes a different form in comparison with the one in their work. Then we employ the generalized formulation to calculate the quasi-local conserved charges for the most general charged spherically symmetric and the dyonic rotating black holes with AdS asymptotics in four-dimensional conformal Weyl gravity, as well as the charged spherically symmetric black holes in arbitrary dimensional Einstein-Gauss-Bonnet gravity coupled to Maxwell or nonlinear electrodynamics in AdS spacetime. Our results confirm those through other methods in the literature.

Covariant symplectic structure and conserved charges of new massive gravity

We show that the symplectic current obtained from the boundary term, which arises in the first variation of a local diffeomorphism invariant action, is covariantly conserved for any gravity theory described by that action. Therefore, a Poincare invariant 2-form can be constructed on the phase space, which is shown to be closed without reference to a specific theory. Finally, we show that one can obtain a charge expression for gravity theories in various dimensions, which plays the role of the Abbott-Deser-Tekin (ADT) charge for spacetimes with non-constant curvature backgrounds, by using the diffeomorphism invariance of the symplectic 2-form. As an example, we calculate the conserved charges of some solutions of New Massive Gravity (NMG) and compare the results with the previous works.

Witten–Nester energy in topologically massive gravity

We formulate topologically massive supergravity with a cosmological constant in the first-order formalism and construct the Noether supercurrent and superpotential associated with its local supersymmetry. Using these results, we construct in ordinary topologically massive gravity the Witten–Nester integral for conserved charges containing spinors which satisfy a generalized version of the Witten equation on the initial value surface. We show that the Witten–Nester charge, represented as an integral over the boundary of the initial value surface, produces the Abbott–Deser–Tekin energy for asymptotically anti-de Sitter spacetimes. We consider all values of the Chern–Simons coupling constant, including the critical value known as the chiral point, and study the cases of standard Brown–Henneaux boundary conditions, as well as their weaker version that allows a slower fall-off. Studying the Witten–Nester energy as a bulk integral over the initial value surface instead, we find a bound on the energy, and through it the sufficient condition for the positivity of the energy. In particular, we find that spacetimes of Petrov type N that admit globally well-defined solutions of the generalized Witten equation have positive energy.

Black hole mass and angular momentum in topologically massive gravity

We extend the Abbott–Deser–Tekin approach to the computation of the Killing charge for a solution of topologically massive gravity (TMG) linearized around an arbitrary background. This is then applied to evaluate the mass and angular momentum of black hole solutions of TMG with non-constant curvature asymptotics. The resulting values, together with the appropriate black hole entropy, fit nicely into the first law of black hole thermodynamics.