A new formula for conserved charges of Lovelock gravity in AdS space–times and its generalization

Abstract

<jats:p> Within the framework of the Lovelock gravity theory, we propose a new rank-four divergenceless tensor consisting of the Riemann curvature tensor and inheriting its algebraic symmetry characters. Such a tensor can be adopted to define conserved charges of the Lovelock gravity theory in asymptotically anti-de Sitter (AdS) space–times. Besides, inspired with the case of the Lovelock gravity, we put forward another general fourth-rank tensor in the context of an arbitrary diffeomorphism invariant theory of gravity described by the Lagrangian constructed out of the curvature tensor. On basis of the newly-constructed tensor, we further suggest a Komar-like formula for the conserved charges of this generic gravity theory. </jats:p>

A Komar-like integral for mass and angular momentum of asymptotically AdS black holes in Einstein gravity

“…Second, the potential K µν is equivalent with the KBL superpotential in [12,13] and the potential via the topological regularization method in even dimensions [24,25,26,43,44,45]. Third, as what has been shown in [37],…”

“…First, in comparison with the superpotential K µν R given by Eq. ( 35) in [37], here represented by the 2-form K µν gr in the framework of the Einstein gravity armed with the Lagrangian (2.1), we follow the work [37] (see [38,39] as well) to compute K µν gr and find that…”

A note on the definition of gravitational energy for quadratic curvature gravity via topological regularization*

“…where the 2-form K µν R is one half of the Noether potential obtained via the covariant phase space method. [3][4][5] It should be pointed out that it has been illustrated [10] that the perturbation of the potential 𝒦 µν on the background AdS spacetime coincides with the result via the covariant phase space method, as well as that through the (off-shell) ADT approach. [6][7][8][9]11] Now, we concentrate on the most general fourdimensional quadratic curvature gravity with the Lagrangian (1).…”

“…As a warmup, we follow the work [10] to present the concrete expression of the 2-form potential 𝒦 µν in the formula (6). It is read off as…”

“…[6][7][8][9] On the other hand, quite recently, in Ref. [10], within the framework of a general D-dimensional gravity theory described by the Lagrangian…”