Mechanics of Isolated Horizons

TL;DR

Addresses how to formulate the laws of black hole mechanics for isolated horizons, i.e., horizons whose intrinsic geometry is time independent even when the exterior spacetime contains radiation. It employs a quasi-local, connection-variable framework to define horizon quantities such as the horizon mass $M_Δ$ and the surface gravity $κ$ without reference to infinity, and establishes the zeroth and first laws for isolated horizons. By introducing a horizon boundary term in the Hamiltonian and relating $M_Δ$ to the ADM/Bondi masses in radiating spacetimes, the approach reconciles equilibrium thermodynamics with dynamical exterior fields. The results generalize black hole thermodynamics to non-stationary settings and provide a foundation for entropy calculations in quantum gravity, with applicability to cosmological horizons as well.

Abstract

A set of boundary conditions defining an undistorted, non-rotating isolated horizon are specified in general relativity. A space-time representing a black hole which is itself in equilibrium but whose exterior contains radiation admits such a horizon. However, the definition is applicable in a more general context, such as cosmological horizons. Physically motivated, (quasi-)local definitions of the mass and surface gravity of an isolated horizon are introduced and their properties analyzed. Although their definitions do not refer to infinity, these quantities assume their standard values in the static black hole solutions. Finally, using these definitions, the zeroth and first laws of black hole mechanics are established for isolated horizons.

Figures (2)

• Figure 1: (a) A typical gravitational collapse. The portion $\Delta$ of the horizon at late times is isolated. The space-time $\mathcal{M}$ of interest is the triangular region bounded by $\Delta$, $\mathcal{I}^+$ and a partial Cauchy slice $M$. (b) Space-time diagram of a black hole which is initially in equilibrium, absorbs a small amount of radiation, and again settles down to equilibrium. Portions $\Delta_1$ and $\Delta_2$ of the horizon are isolated.
• Figure 2: A spherical star of mass $M$ undergoes collapse. Later, a spherical shell of mass $\delta{M}$ falls into the resulting black hole. While $\Delta_1$ and $\Delta_2$ are both isolated horizons, only $\Delta_2$ is part of the event horizon.