Mass in anti-de Sitter spaces

TL;DR

The paper develops a holographic renormalization framework for five-dimensional AdS gravity with matter, showing that a finite boundary counterterm proportional to $\phi^2$ is essential to obtain linear mass–charge relations for $R$-charged black holes and to obtain consistent masses for rotating Gutowski–Reall solutions. By combining standard gravitational counterterms with a carefully chosen $\phi^2$ term, nonlinear charge contributions cancel in the boundary stress tensor, yielding linear expressions for the mass in terms of charges, e.g. $M \sim \tfrac{3}{2}\mu + q$ for single charge and $M \sim \tfrac{3}{2}\mu + q_1+q_2+q_3$ for STU. The results align with Behrndt–Cveti–Youm thermodynamics and Ashtekar–Das masses up to Casimir energy shifts, and highlight the scheme dependence of holographic energies controlled by finite counterterms. The work also connects the finite counterterm approach to Hamilton–Jacobi methods and supersymmetry considerations, outlining implications for AdS/CFT in both AdS$_5$ and AdS$_4$ with scalar fields.

Abstract

The boundary stress tensor approach has proven extremely useful in defining mass and angular momentum in asymptotically anti-de Sitter spaces with CFT duals. An integral part of this method is the use of boundary counterterms to regulate the gravitational action and stress tensor. In addition to the standard gravitational counterterms, in the presence of matter we advocate the use of a finite counterterm proportional to phi^2 (in five dimensions). We demonstrate that this finite shift is necessary to properly reproduce the expected mass/charge relation for R-charged black holes in AdS_5.