Laws of Black Hole Mechanics from Holst Action

TL;DR

This paper derives the laws of black hole mechanics for Weak Isolated Horizons using the Holst action, establishing the first law and revealing a U(1) Chern-Simons theory on spherical horizons.

Contribution

It constructs a covariant phase space for WIH with Holst action, deriving the first law and identifying the boundary theory as U(1) Chern-Simons.

Findings

First law depends on the chosen action.

A foliation-independent symplectic structure is achieved.

Boundary theory on spherical horizons is U(1) Chern-Simons.

Abstract

The formulation of Weak Isolated Horizons (WIH) based on the Isolated Horizon formulation of black hole horizons is reconsidered. The first part of the paper deals with the derivation of laws of mechanics of a WIH. While the zeroth law follows from the WIH boundary conditions, first law depends on the action chosen. We construct the covariant phase space for a spacetime having an WIH as inner boundary for the Holst action. This requires the introduction of new potential functions so that the symplectic structure is foliation independent. We show that a precise cancellation among various terms leads to the usual first law for WIH. Subsequently, we show from the same covariant phase space that for spherical horizons, the topological theory on the inner boundary is a U(1) Chern-Simons theory.