Conformal boundaries near extremal black holes

We examine four dimensional, near-extremal black hole solutions in the presence of a finite boundary obeying conformal boundary conditions, where the conformal class of the induced metric and the trace of the extrinsic curvature are fixed. Working in Euclidean signature and at fixed charge, we find the near-extremal regime is dominated by a double-scaling limit which reveals new scaling laws for the quasi-local conformal entropy at low temperatures. Upon spherical dimensional reduction, we obtain the effective two-dimensional dilaton-gravity theory that describes the near-extremal regime. In contrast to Dirichlet boundaries, for conformal boundaries a linear dilaton potential is not sufficient to capture the leading correction away from extremality and higher orders are needed. We also examine near-Nariai solutions and the circular reduction of pure three-dimensional gravity in (Anti-) de Sitter space. In the latter, provided the boundary is placed near the conformal boundary of three-dimensional Anti-de Sitter space, the dynamics of the spherically symmetric boundary mode is governed by a Liouville equation that descends from a (minus) Schwarzian effective action.