A Cardy-like expression for charged rotating solitons and black holes

TL;DR

The paper develops a generalized Cardy-like entropy formula ${\cal S}_{C}$ for charged and/or rotating black holes and their double Wick-rotated solitons, incorporating the dynamical exponent $z$ and the effective spatial dimensionality $d_{\text{eff}}$ to ensure proper scaling with the first law and Smarr relations. It derives spinning configurations from static ones via a Lorentz boost and analyzes several concrete examples, including a new charged rotating black hole in four-dimensional Critical Gravity, spinning hyperscaling-violation black holes, and a three-dimensional rotating black hole with scalar hair, validating ${\cal S}_{C}$ against the Wald entropy ${\cal S}_{W}$ across these cases. A key finding is that reproducing the semiclassical entropy in some models requires nontrivial soliton thermodynamics (e.g., magnetic charge or rotation) encoded in ${\cal M}_{\text{sol}}$, ${\cal J}_{\text{sol}}$, ${\cal Q}_{m}$, and $\Phi_{m}$, which enter the Cardy-like expression. Overall, the work extends the Cardy-like entropy framework to a broad class of AdS/CFT-relevant geometries, including hyperscaling violation and anisotropic scaling, with potential implications for holographic microstructure and phase structure of dual field theories.

Abstract

This paper aims to propose a Cardy-like formula characterized by the mass, charge, and angular components of the black hole, along with their corresponding solitonic configuration, obtained through a double Wick rotation. The expression also incorporates the dynamical exponent and effective spatial dimensionality as key elements. To validate the proposal, we first present a new concrete example in which recovering the semiclassical entropy requires the soliton to possess thermodynamic quantities beyond its mass. Additionally, we show more examples derived from static black hole solutions, employing a Lorentz boost to calculate their thermodynamic parameters. Finally, we include a case of a rotating configuration where the Lorentz boost is not required.