A Stress Tensor for Anti-de Sitter Gravity

TL;DR

The paper develops a finite, covariant boundary stress tensor for asymptotically anti-de Sitter spacetimes by adding intrinsic boundary counterterms, avoiding ambiguities in reference-spacetime subtraction. This holographic renormalization yields conserved charges that reproduce masses and angular momenta, and its AdS/CFT interpretation ties the bulk stress tensor to the boundary CFT expectation value, including correct conformal anomalies in d=2 and d=4 and a Casimir-energy-consistent mass for global AdS$_5$. Explicit counterterms are given for AdS$_3$, AdS$_4$, and AdS$_5$, and the work clarifies how conformal data (central charges, anomalies) arise from bulk gravity through asymptotic symmetries. The results provide a robust framework for defining gravitational energy in AdS, with implications for holography and potential extensions to flat spacetimes via appropriate renormalization concepts.

Abstract

We propose a procedure for computing the boundary stress tensor associated with a gravitating system in asymptotically anti-de Sitter space. Our definition is free of ambiguities encountered by previous attempts, and correctly reproduces the masses and angular momenta of various spacetimes. Via the AdS/CFT correspondence, our classical result is interpretable as the expectation value of the stress tensor in a quantum conformal field theory. We demonstrate that the conformal anomalies in two and four dimensions are recovered. The two dimensional stress tensor transforms with a Schwarzian derivative and the expected central charge. We also find a nonzero ground state energy for global AdS_5, and show that it exactly matches the Casimir energy of the dual N=4 super Yang-Mills theory on S^3 x R.