Quasilocal first law for black hole thermodynamics

We first show that stationary black holes satisfy an extremely simple quasilocal form of the first law, $\delta E=\frac{\overline{\kappa }}{8\pi }\delta A$, where the (quasilocal) energy $E=A/(8\pi \ell )$ and (local) surface gravity $\overline{\kappa }=1/\ell$, with $A$ the horizon area and $\ell$ is a proper length characterizing the distance to the horizon of a preferred family of quasilocal 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.

Figure 1

Conformal diagram representing the perturbation of an initially stationary black hole with a bifurcate horizon. The dashed line represents the true BH horizon, the stationary observers world sheet is denoted by $\mathcal{W}$. The quantity $A_{\mathrm{in}}$ is the area of the initially stationary background while $A_{\mathrm{out}}$ is the final area of the BH horizon.