Boundary Terms and Noether Current of Sphereical Black Holes

TL;DR

This paper compares two methods for defining entropy of spherical black holes with trapping horizons, showing both yield entropy proportional to the horizon area, and discusses their relation to a generalized first law.

Contribution

It generalizes Wald's Noether current method to dynamic black holes and compares it with boundary term approaches in spherical symmetry.

Findings

Both methods produce entropy proportional to the horizon area.

The generalized first law relates these entropy definitions to black hole dynamics.

The results unify different entropy proposals for dynamical black holes.

Abstract

We consider two proposals for defining black hole entropy in spherical symmetry, where the horizon is defined locally as a trapping horizon. The first case, boundary terms in a dual-null form of the reduced action in two dimensions, gives a result that is proportional to the area. The second case, Wald's Noether current method, is generalized to dynamic black holes, giving an entropy that is just the area of the trapping horizon. These results are compared with a generalized first law of thermodynamics.