Inner Mechanics of 3d Black Holes

TL;DR

The paper investigates inner-horizon thermodynamics in three-dimensional higher-derivative gravity (TMG) across BTZ, spacelike warped AdS, and warped dS black holes, confirming an inner-horizon first law and demonstrating that the product of inner and outer entropies, $S_+ S_-$, generally depends on the black hole mass due to the diffeomorphism anomaly ($c_L \neq c_R$). It provides explicit entropy formulas for each family of solutions and links mass dependence of the product to the central charge asymmetry. The analysis extends to New Massive Gravity (NMG), where inner-horizon relations generalize and the entropy product can be mass-independent, highlighting both the reach and limitations of inner-horizon thermodynamics in 3D gravity. These results deepen the understanding of holographic and thermodynamic aspects of inner horizons in higher-derivative theories.

Abstract

We investigate properties of the inner horizons of certain black holes in higher-derivative three-dimensional gravity theories. We focus on BTZ and Spacelike Warped Anti-de Sitter black holes, as well as on asymptotically Warped de-Sitter solutions exhibiting both a cosmological and a black hole horizon. We verify that a First Law is satisfied at the Inner horizon, in agreement with the proposal of \cite{Castro:2012av}. We then show that, in Topologically Massive Gravity, the product of the areas of the inner and outer horizons fails to be independent on the mass, and trace this to the diffeomorphism anomaly of the theory.