The thermodynamics of a black hole in equilibrium implies the breakdown of Einstein equations on a macroscopic near-horizon shell

TL;DR

This paper investigates the thermodynamics of a black hole in equilibrium with Hawking radiation within a box, revealing a macroscopic near-horizon shell where Einstein equations break down, based on classical and thermodynamic principles.

Contribution

It demonstrates that a large, entropy-rich shell forms near the horizon, where classical Einstein equations are violated, derived solely from thermodynamics and classical gravity without new physics assumptions.

Findings

The shell width is of order √M, much larger than Planck length.

The shell entropy is proportional to M.

The shell temperature is of order 1/√M.

Abstract

We study a black hole of mass $M$, enclosed within a spherical box, in equilibrium with its Hawking radiation. We show that the spacetime geometry inside the box is described by the Oppenheimer-Volkoff equations for radiation, except for a {\em thin shell} around the horizon. We use the maximum entropy principle to show that the invariant width of the shell is of order $\sqrt{M}$, its entropy is of order $M$ and its temperature of order $1/\sqrt{M}$ (in Planck units). Thus, the width of the shell is {\em much larger} than the Planck length. Our approach is to insist on thermodynamic consistency when classical general relativity coexists with the Hawking temperature in the description of a gravitating system. No assumptions about an underlying theory are made and no restrictions are placed on the origins of the new physics near the horizon. We only employ classical general relativity and the principles of thermodynamics. Our result is strengthened by an analysis of the trace anomaly associated to the geometry inside the box, i.e., the regime where quantum field effects become significant corresponds to the shells of maximum entropy around the horizon.