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

Abstract

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 pform 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 lowerdimensional 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. *

Generalized Komar currents for vector fields

“…( 3.3) can be applied to understand the Noether current corresponding to the diffeomorphism symmetry of a spacetime manifold, generated by an arbitrary vector field V µ , in the context of the Einstein gravity equipped with the Lagrangian (2.17). According to the standard Noether method, the conserved current associated with the diffeomorphism symmetry is given by J N C = − δdV [24,25]. On the other hand, as what has been mentioned above, supposed that the current depends solely on the terms proportional to at most the second-order derivatives of a vector, the satisfactory conserved current for V µ has to be JV .…”

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

“…As usual, the off-shell Noether current corresponding to K µν is expressible as J µ = ∇ ν K µν . Furthermore, with the help of the surface term Θ µ and the potential K µν , it is feasible to define the off-shell ADT potential Q µν involved in the Lagrangian (2.1) [26,27,28,47,48].…”

“…In practice, such a modification leads to the advantage that it becomes more operable to derive the potential in terms of the corresponding conserved current. What is more, by contrast with the original ADT formalism that involves no matter fields in the perturbation of fields, it is of great convenience to incorporate the contributions from the matter fields within the off-shell formulation [27,28] along the lines of the well-known covariant phase space approach, put forward by Wald and his collaborators [29,30,31].…”

“…In this section, taking into consideration of the contribution from the matter field, we shall derive the off-shell Noether and ADT potentials associated with (2n + 1)-dimensional bosonic supergravity theories along the lines of the work [28], which recently developed the off-shell generalized ADT formalism in [26,27]. Then a general formula of conserved charges for these theories will be presented on basis of the potentials.…”

Thermodynamics of inhomogeneously mass-deformed ABJM model and pressure anisotropy

“…We use the off-shell Abbott-Deser-Tekin(ADT) formalism invented and developed in [19][20][21][22][23][24] to find the first law. In addition the reduced action formalism [25][26][27] is considered to obtain the Smarr relation.…”