Effective Lagrangian for Quantum Black Holes

TL;DR

This paper develops a general effective Lagrangian framework for quantizing black hole horizons, capturing quantum fluctuations and computing black hole entropy within a covariant semiclassical formalism.

Contribution

It introduces a novel effective Lagrangian approach based on horizon degrees of freedom, incorporating quantum fluctuations and providing a method to calculate black hole entropy.

Findings

Derived a general effective Lagrangian depending on horizon geometry.

Formulated a covariant semiclassical expansion for quantum fluctuations.

Computed black hole entropy using the new formalism.

Abstract

We discuss the most general effective Lagrangian obtained from the assumption that the degrees of freedom to be quantized, in a black hole, are on the horizon. The effective Lagrangian depends only on the induced metric and the extrinsic curvature of the (fluctuating) horizon, and the possible operators can be arranged in an expansion in powers of $\mpl/M$, where $\mpl$ is the Planck mass and $M$ the black hole mass. We perform a semiclassical expansion of the action with a formalism which preserves general covariance explicitly. Quantum fluctuations over the classical solutions are described by a single scalar field living in the 2+1 dimensional world-volume swept by the horizon, with a given coupling to the background geometry. We discuss the resulting field theory and we compute the black hole entropy with our formalism.