Extrinsic Renormalization of New Massive Gravity

Abstract

The renormalization of asymptotically AdS spacetimes in New Massive Gravity is studied using boundary terms constructed with the extrinsic curvature. For generic values of the theory couplings, a single extrinsic boundary term produces a consistent holographic description, as the variation of the total action is finite and expressible in terms of the holographic source. This prescription applies to constant-curvature geometries and non-Einstein solutions with Brown-Henneaux asymptotic conditions. A suitable asymptotic expansion shows the agreement of the resulting holographic stress tensor with the one obtained from the auxiliary-field formulation. At the degenerate point, where the two maximally symmetric vacua coalesce, the Fefferman-Graham expansion admits an additional mode associated with relaxed AdS boundary conditions. A quadratic extrinsic counterterm removes the new divergences and leads naturally to a holographic picture with two independent sources. The corresponding responses obey a modified holographic Ward identity and determine finite conserved charges. Finally, the proper use of the Noether-Wald formalism reproduces the correct charges of the BTZ black hole, non-Einstein AdS waves, and the rotating hairy black hole.