A Dynamical Boundary for Anti-de Sitter Space

TL;DR

This paper proposes fixing the renormalized boundary stress tensor in asymptotically AdS spaces, leading to a well-defined variational principle, new counter-terms, and consistent black hole thermodynamics, with implications for holographic renormalization.

Contribution

It introduces a boundary condition fixing the renormalized stress tensor instead of the boundary metric, providing a new variational framework and insights into AdS black hole thermodynamics.

Findings

Derived a well-defined variational principle for AdS gravity.

Showed the equivalence of the new and standard actions on classical solutions.

Demonstrated correct thermodynamics for Kerr-AdS black holes.

Abstract

We argue that a natural boundary condition for gravity in asymptotically AdS spaces is to hold the {\em renormalized} boundary stress tensor density fixed, instead of the boundary metric. This leads to a well-defined variational problem, as well as new counter-terms and a finite on-shell action. We elaborate this in various (even and odd) dimensions in the language of holographic renormalization. Even though the {\em form} of the new renormalized action is distinct from the standard one, once the cut-off is taken to infinity, their {\em values} on classical solutions coincide when the trace anomaly vanishes. For AdS$_4$, we compute the ADM form of this renormalized action and show in detail how the correct thermodynamics of Kerr-AdS black holes emerge. We comment on the possibility of a consistent quantization with our boundary conditions when the boundary is dynamical, and make a connection to the results of Compere and Marolf. The difference between our approach and microcanonical-like ensembles in standard AdS/CFT is emphasized.