Conserved Charges for Even Dimensional Asymptotically AdS Gravity Theories

TL;DR

This work develops a Noether‑theoretic framework for defining conserved charges in even‑dimensional gravity with locally Anti‑de Sitter asymptotics (ALAdS). By augmenting the Einstein–Hilbert action with a boundary term proportional to the Euler density, the authors obtain finite, background‑independent Noether charges that reproduce standard results (e.g., mass and angular momentum) without background subtraction, and demonstrate that locally AdS spacetimes have vanishing charges. The construction extends to Born–Infeld gravity in even dimensions and yields explicit charge expressions for diffeomorphisms and Lorentz transformations, with clear examples including Schwarzschild‑AdS, Kerr‑AdS, and brane geometries. The approach highlights the regulator role of the AdS scale $l$ and provides a unified, Lagrangian‑based method to compute charges in higher‑dimensional ALAdS spacetimes, while outlining directions for extending to other Lovelock theories and odd dimensions.

Abstract

Mass and other conserved Noether charges are discussed for solutions of gravity theories with locally Anti-de Sitter asymptotics in 2n dimensions. The action is supplemented with a boundary term whose purpose is to guarantee that it reaches an extremum on the classical solutions, provided the spacetime is locally AdS at the boundary. It is also shown that if spacetime is locally AdS at spatial infinity, the conserved charges are finite and properly normalized without requiring subtraction of a reference background. In this approach, Noether charges associated to Lorentz and diffeomorphism invariance vanish identically for constant curvature spacetimes. The case of zero cosmological constant is obtained as a limit of AdS, where $Λ$ plays the role of a regulator.