Isolated Horizons: the Classical Phase Space

Abstract

A Hamiltonian framework is introduced to encompass non-rotating (but possibly charged) black holes that are ``isolated'' near future time-like infinity or for a finite time interval. The underlying space-times need not admit a stationary Killing field even in a neighborhood of the horizon; rather, the physical assumption is that neither matter fields nor gravitational radiation fall across the portion of the horizon under consideration. A precise notion of non-rotating isolated horizons is formulated to capture these ideas. With these boundary conditions, the gravitational action fails to be differentiable unless a boundary term is added at the horizon. The required term turns out to be precisely the Chern-Simons action for the self-dual connection. The resulting symplectic structure also acquires, in addition to the usual volume piece, a surface term which is the Chern-Simons symplectic structure. We show that these modifications affect in subtle but important ways the standard discussion of constraints, gauge and dynamics. In companion papers, this framework serves as the point of departure for quantization, a statistical mechanical calculation of black hole entropy and a derivation of laws of black hole mechanics, generalized to isolated horizons. It may also have applications in classical general relativity, particularly in the investigation of of analytic issues that arise in the numerical studies of black hole collisions.