Conformal Invariance of Black Hole Temperature

TL;DR

This work analyzes how black hole thermodynamics behaves under conformal rescalings $g_{ab}\rightarrow \Omega^2 g_{ab}$ that are unity at infinity. It introduces a conformal-invariant surface gravity $\kappa_1$ for conformal Killing horizons, showing it matches standard definitions on Killing horizons and yields $T_H=\kappa_1/2\pi$, thus preserving the Hawking temperature under such transformations. The authors examine relations among $\kappa_1$, $\kappa_2$, and $\kappa_3$, proving invariance of $\kappa_1$ under static conformal maps and discussing the (in)variance of other definitions ($\kappa_4$, $\kappa_5$) under more general transformations. They also demonstrate that Hawking radiation remains conformally invariant and provide a reconciliation with the trace anomaly, showing the asymptotic flux is unchanged by regular conformal rescalings, thereby maintaining consistency between conformal transformations and black hole thermodynamics.

Abstract

It is shown that the surface gravity and temperature of a stationary black hole are invariant under conformal transformations of the metric that are the identity at infinity. More precisely, we find a conformal invariant definition of the surface gravity of a conformal Killing horizon that agrees with the usual definition(s) for a true Killing horizon and is proportional to the temperature as defined by Hawking radiation. This result is reconciled with the intimate relation between the trace anomaly and the Hawking effect, despite the {\it non}invariance of the trace anomaly under conformal transformations.