Stress-Energy Tensors for Higher Dimensional Gravity

TL;DR

This work develops and applies the boundary counterterm method to higher-dimensional AdS gravity to define a finite boundary stress-energy tensor and Weyl anomaly. The authors derive $T^{ab}_{(finite)}$ by combining the unrenormalized tensor with curvature-based counterterms, enabling computation of conserved charges without a reference background. They compute black hole masses in $d=5,7,9$, uncover a Casimir energy in the vacuum state, and identify a Weyl anomaly in $d=8$ with an explicit $a_{(8)}$ structure, providing new results for the $d=8$ case. The approach reinforces the AdS/CFT correspondence by linking gravitational boundary data to dual field theory vacuum energies and anomalies, and offers a practical, background-free renormalization scheme for higher-dimensional spacetimes.

Abstract

We calculate, in the context of higher dimensional gravity, the stress-energy tensor and Weyl anomaly associated with anti-de Sitter and anti-de Sitter black hole solutions. The boundary counter-term method is used to regularize the action and the resulting stress-energy tensor yields both the correct black hole energies as well as a vacuum energy contribution which is interpreted as a Casimir energy. This calculation is done up to d = 8 (d being the boundary dimension). We confirm some results for d < 8 as well as comment on some new results (some of which are relevant to (2,0) theory). All results for d=8 are new.