The First Law of Black Hole Mechanics

A simple proof of a strengthened form of the first law of black hole mechanics is presented. The proof is based directly upon the Hamiltonian formulation of general relativity, and it shows that the the first law variational formula holds for arbitrary nonsingular, asymptotically flat perturbations of a stationary, axisymmetric black hole, not merely for perturbations to other stationary, axisymmetric black holes. As an application of this strengthened form of the first law, we prove that there cannot exist Einstein-Maxwell black holes whose ergoregion is disjoint from the horizon. This closes a gap in the black hole uniqueness theorems. 1. Derivation of the First Law It was noted by Hilbert at the inception of general relativity that the Einstein field equations are derivable from an action principle, S = 1 16π ∫ R √−g dx (1) Thus, general relativity has a Lagrangian formulation. The corresponding Hamiltonian formulation was given many years later in a collaboration between Charles Misner, Richard Arnowitt, and Stanley Deser. The main results of this collaboration are summarized in [1]. The Hamiltonian formulation of general relativity is employed as a starting point in all attempts to formulate a quantum theory of gravity via the canonical approach. It plays a less essential role within the context of purely classical general relativity. However, even in that context, the Hamiltonian formulation of general relativity provides some penetrating insights into the structure of the theory. In this paper, I shall illustrate this point by showing how a strengthened form of the first law ∗Enrico Fermi Institute and Department of Physics, University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637, USA. This research was supported in part by the National Science Foundation under Grant No. PHY89-18388.