Mass, Angular Momentum and Thermodynamics in Four-Dimensional Kerr-AdS Black Holes

TL;DR

The paper develops a Lorentz‑covariant boundary term framework for four‑dimensional AdS gravity by coupling the Einstein–Hilbert action to a topological Euler term under a boundary condition of constant negative asymptotic curvature, leading to a finite action and background‑independent Noether charges without a Gibbons–Hawking term. A tensorial expression for the conserved charges is obtained via a carefully chosen adapted frame and a second fundamental form, with the boundary term $B_{3}$ explicitly derived. The formalism yields finite Kerr–AdS charges and a consistent thermodynamics: the physical energy and angular momentum emerge from the appropriate timelike Killing vector, and the Euclidean action produces the correct entropy $S=\frac{1}{4}\text{Area}$, validating the approach. The results link the regularization to topological invariants and suggest straightforward generalization to higher even dimensions using higher Euler terms or corresponding boundary forms, providing an alternative to the standard counterterm program.

Abstract

In this paper, the connection between the Lorentz-covariant counterterms that regularize the four-dimensional AdS gravity action and topological invariants is explored. It is shown that demanding the spacetime to have a negative constant curvature in the asymptotic region permits the explicit construction of such series of boundary terms. The orthonormal frame is adapted to appropriately describe the boundary geometry and, as a result, the boundary term can be expressed as a functional of the boundary metric, extrinsic curvature and intrinsic curvature. This choice also allows to write down the background-independent Noether charges associated to asymptotic symmetries in standard tensorial formalism. The absence of the Gibbons-Hawking term is a consequence of an action principle based on a boundary condition different than Dirichlet on the metric. This argument makes plausible the idea of regarding this approach as an alternative regularization scheme for AdS gravity in all even dimensions, different than the standard counterterms prescription. As an illustration of the finiteness of the charges and the Euclidean action in this framework, the conserved quantities and black hole entropy for four-dimensional Kerr-AdS are computed.