Asymptotically Anti-de Sitter spacetimes and their stress energy tensor

TL;DR

The paper analyzes asymptotically AdS spacetimes and the infrared divergences of the gravitational action. It develops a holographic renormalization framework using boundary counterterms, showing that in odd boundary dimensions the renormalized action is not invariant under bulk diffeomorphisms that implement boundary Weyl transformations, which manifests as a holographic Weyl anomaly. It provides explicit expressions for the holographic stress-energy tensor in terms of Fefferman–Graham coefficients, including dimension-dependent corrections, and discusses how these tensors transform under diffeomorphisms and relate to the anomaly. An AdS5 example demonstrates finiteness, conservation, and, in certain backgrounds, tracelessness of the stress tensor, linking the results to the AdS/CFT dictionary and boundary Casimir energy.

Abstract

We consider asymtotically anti-de Sitter spacetimes in general dimensions. We review the origin of infrared divergences in the on-shell gravitational action, and the construction of the renormalized on-shell action by the addition of boundary counterterms. In odd dimensions, the renormalized on-shell action is not invariant under bulk diffeomorphisms that yield conformal transformations in the boundary (holographic Weyl anomaly). We obtain formulae for the gravitational stress energy tensor, defined as the metric variation of the renormalized on-shell action, in terms of coefficients in the asymptotic expansion of the metric near infinity. The stress energy tensor transforms anomalously under bulk diffeomorphisms broken by infrared divergences.