Black holes as quantum membranes: path integral approach

TL;DR

This paper models the black hole horizon as a quantum membrane using a path integral approach, deriving thermodynamic properties and phase transition behavior near the Planck mass.

Contribution

It introduces a novel formalism describing the black hole horizon as a dynamical quantum membrane governed by a relativistic bosonic membrane action.

Findings

Derives black hole entropy and temperature consistent with standard results.

Predicts a phase transition with positive specific heat for black holes near the Planck mass.

Discusses corrections to Hawking temperature for small-mass black holes.

Abstract

We describe the horizon of a quantum black hole in terms of a dynamical surface which defines the boundary of space-time as seen by external static observers, and we define a path integral in the presence of this dynamical boundary. Using renormalization group arguments, we find that the dynamics of the horizon is governed by the action of the relativistic bosonic membrane. {}From the thermodynamical properties of this bosonic membrane we derive the entropy and the temperature of black holes, and we find agreement with the standard results. With this formalism we can also discuss the corrections to the Hawking temperature when the mass $M$ of the black hole approaches the Planck mass $M_{\rm Pl}$. When $M$ becomes as low as $(10-100) M_{\rm Pl}$ a phase transition takes place and the specific heat of the black hole becomes positive.