Black holes as quantum membranes: path integral approach

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 MPl. When M becomes as low as (10 − 100)MPl a phase transition takes place and the specific heat of the black hole becomes positive. In a recent paper [1] we have put forward a proposal for a quantum description of black holes. Following the work of ’t Hooft [2] and of Susskind and coworkers [3], we have considered the black hole horizon, as seen by an external static observer, as a dynamical surface endowed with physical degrees of freedom, and we have suggested that the dynamics of this surface is governed by an action principle. The simplest action describing a relativistic bosonic membrane in 3+1 dimensions is [4]