Black hole mechanics was recently extended by replacing the more commonly used event horizons in stationary space-times with isolated horizons in more general space-times ͑which may admit radiation arbitrarily close to black holes͒. However, so far the detailed analysis has been restricted to nonrotating black holes ͑although it incorporated arbitrary distortion, as well as electromagnetic, Yang-Mills, and dilatonic charges͒. We now fill this gap by first introducing the notion of isolated horizon angular momentum and then extending the first law to the rotating case. 64 044016-1 F∧૽F,͑2.13͒where the traces are taken over the internal ͑tetrad͒ indices, F abI J denotes the curvature of the connection A aI J , and the two-forms ⌺ abI J are defined in terms of the ͑co-͒tetrad field by ⌺ abI J ª⑀ I When the background fields (⌺ abI J ,A aI J ,A a ) satisfy the field equations of Einstein-Maxwell theory, one can easily verify that ␦ W satisfies the linearized equations of motion. It therefore represents a tangent vector field on covariant phase space. This vector field generates a canonical transformation if it preserves the symplectic structure, i.e., if L ␦ W ⍀ϭ0. Equivalently, ␦ W is a canonical transformation if and only if there exists a Hamiltonian function H W on phase space such that ␦H W ϭ⍀͑␦,␦ W ͒ ͑4.1͒ 5 As we saw in Sec. III, if (l) ϭ0, B W need not be constant on ⌬ for W a to define a symmetry of that horizon. However, our purpose here is to consider a smooth assignment of symmetry vectors to many different horizons. By continuity in phase space, therefore, B W should be constant in this case as well.

Geometrical structures intrinsic to non-expanding, weakly isolated and isolated horizons are analyzed and compared with structures which arise in other contexts within general relativity, e.g., at null infinity. In particular, we address in detail the issue of singling out the preferred normals to these horizons required in various applications. This work provides powerful tools to extract invariant, physical information from numerical simulations of the near horizon, strong field geometry. While it complements the previous analysis of laws governing the mechanics of weakly isolated horizons, prior knowledge of those results is not assumed.

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.

Boundary conditions defining a generic isolated horizon are introduced. They generalize the notion available in the existing literature by allowing the horizon to have distortion and angular momentum. Space-times containing a black hole, itself in equilibrium but possibly surrounded by radiation, satisfy these conditions. In spite of this generality, the conditions have rich consequences. They lead to a framework, somewhat analogous to null infinity, for extracting physical information, but now in the strong field regions. The framework also generalizes the zeroth and first laws of black hole mechanics to more realistic situations and sheds new light on the 'origin' of the first law. Finally, it provides a point of departure for black hole entropy calculations in non-perturbative quantum gravity.Pacs: 04070-m, 0425Dm, 0460-m A great deal of analytical work on black holes in general relativity centers around event horizons in globally stationary space-times (see, e.g., [1,2]). While it is a natural starting point, this idealization seems overly restrictive from a physical point of view. In a realistic gravitational collapse, or a black hole merger, the final black hole is expected to rapidly reach equilibrium. However, the exterior space-time region will not be stationary. Indeed, a primary goal of many numerical simulations is to study radiation emitted in the process. Similarly, since event horizons can only be determined retroactively after knowing the entire space-time evolution, they are not directly useful in many situations. For example, when one speaks of black holes in centers of galaxies, one does not refer to event horizons. The idealization seems unsuitable also for black hole mechanics and statistical mechanical calculations of entropy. Firstly, in ordinary equilibrium statistical mechanics, one only assumes that the system under consideration is stationary, not the whole universe. Secondly, from quantum field theory in curved space-times, thermodynamic considerations are known to apply also to cosmological horizons [3]. Thus, it seems desirable to replace event horizons by a quasi-local notion and develop a detailed framework tailored to diverse applications, from numerical relativity to quantum gravity, without the assumption of global stationarity. The purpose of this letter is to present such a framework.Specifically, we will provide a set of quasi-local boundary conditions which define an isolated horizon ∆ representing, for example, the last stages of a collapse or a merger, and focus on space-time regions admitting such horizons as an inner boundary. Although the boundary conditions are motivated purely by geometric considerations, they lead to a well-defined action principle and Hamiltonian framework. This, in turn, leads to a definition of the horizon mass M ∆ and angular momentum J ∆ . These quantities refer only to structures intrinsically available on ∆, without any reference to infinity, and yet lead to a generalization of the familiar laws of black hole mechanics. We will also introduce...