Dynamical boundary for anti-de Sitter space

We argue that a natural boundary condition for gravity in asymptotically anti-de Sitter (AdS) spaces is to hold the renormalized boundary stress tensor density fixed, instead of the boundary metric. This leads to a well-defined variational problem, as well as new counterterms and a finite on-shell action. We elaborate this in various (even and odd) dimensions in the language of holographic renormalization. Even though the form of the new renormalized action is distinct from the standard one, once the cutoff is taken to infinity, their values on classical solutions coincide when the trace anomaly vanishes. For AdS 4, we compute the Arnowitt-Deser-Misner 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.