A local first law for black hole thermodynamics

TL;DR

The paper introduces a quasilocal formulation of black hole thermodynamics by employing a family of near-horizon observers, establishing a local first law $δE = κ̄/(8π) δA$ with horizon energy $E = A/(8πℓ)$ and universal local surface gravity $κ̄ = 1/ℓ$ (up to $o(ℓ)$) for stationary black holes and extending this to isolated horizons. It demonstrates that the local surface gravity corresponds to a true local temperature $T = κ̄/(2π)$, with the local spectrum observed by these reporters being Planckian, $ obreak{⟨N⟩ = Γ/(e^{(2π/κ̄)ω}-1)}$, after relating local and asymptotic frequencies. The results unify the local thermodynamics of stationary BHs and IHs, providing a universal, locally defined energy and a Gibbs relation $E = T S$ with $S = A/(4ℓ_p^2)$ and $T = ℓ_p^2 κ̄ /(2π)$. This local framework supports semiclassical analyses and the quantum horizon description, notably within loop quantum gravity, and clarifies the horizon's energetic content via quasilocal quantities rather than global spacetime data.

Abstract

We first show that stationary black holes satisfy an extremely simple local form of the first law δE=κ(l) δA/(8 π) where the thermodynamical energy E=A/(8πl) and (local) surface gravity κ(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.