A local first law for black hole thermodynamics

TL;DR

This paper derives a simple local form of the first law of black hole thermodynamics applicable to stationary black holes and isolated horizons, with implications for quantum gravity and semiclassical physics.

Contribution

It introduces a local first law involving a universal local surface gravity and extends the framework to isolated horizons.

Findings

Local first law = ext{k}(l)  extdelta A/(8  extpi)

Universal local surface gravity  extk(l)=1/l

Implications for loop quantum gravity and semiclassical black hole physics

Abstract

We first show that stationary black holes satisfy an extremely simple local form of the first law \delta E=\kappa(l) \delta A/(8 \pi) where the thermodynamical energy E=A/(8\pi l) and (local) surface gravity \kappa(l)=1/l, where A is the horizon area and l is a proper length characterizing the distance to the horizon of a preferred family of local observers suitable for thermodynamical considerations. Our construction is extended to the more general framework of isolated horizons. The local surface gravity is universal. This has important implications for semiclassical considerations of black hole physics as well as for the fundamental quantum description arising in the context of loop quantum gravity.