Quasi-local first law of black-hole dynamics

A property well known as the first law of black hole is a relation among infinitesimal variations of parameters of stationary black holes. We consider a dynamical version of the first law, which may be called the first law of black hole dynamics. The first law of black hole dynamics is derived without assuming any symmetry or any asymptotic conditions. In the derivation, a definition of dynamical surface gravity is proposed. In spherical symmetry it reduces to that defined recently by one of the authors (SAH). PACS number(s): 04.70.Dy Typeset using REVTEX 1 Black hole thermodynamics, analogies between the theory of black holes and thermodynamics, has been one of the hottest fields in black hole physics since Bekenstein’s introduction of black hole entropy [1]. The black hole entropy was introduced as a quantity proportional to horizon area. The proportionality coefficient was fixed by Hawking’s discovery that a black hole with surface gravity κ emits radiation with temperature TH = κ/2π [2]: by identifying TH with the temperature of the black hole, black hole entropy is determined to be one quarter of the horizon area. The expression of black hole entropy is called the Bekenstein-Hawking formula. To determine the coefficient in the Bekenstein-Hawking formula from the expression of the Hawking temperature TH , the first law of black holes [3] is used. The first law is a relation among infinitesimal variations of parameters of stationary black holes: horizon area, mass, angular momentum, etc. Strictly speaking, it does not relate dynamical evolutions of these quantities. Thus, it might be physically non-trivial to connect the temperature of dynamically emitted radiation with black hole entropy by using the first law. Nonetheless, the Bekenstein-Hawking formula obtained by using the first law is acceptable. For example, in Euclidean gravity the Bekenstein-Hawking formula for a Schwarzschild black hole is correctly obtained by requiring a regularity of the corresponding Euclidean section [7]. Moreover, the first law is used in (quasi-stationary but) dynamical situations to prove the generalized second law [4,5], which is a natural generalization of both the second law (or area law [6]) of black holes and the second law of usual thermodynamics. In the proof, by assuming quasi-stationarity, the use of the first law is extended to relate small changes of physical quantities from an initial near-stationary black hole to a final near-stationary one. However, this idea of quasi-stationarity is an approximation; if we intend to prove the generalized second law for finite changes between initial and final near-stationary black holes or to a purely dynamical situation, the stationary first law can not be used. Therefore, to make black hole thermodynamics self-consistent it must be shown that a dynamical version of the first law of black holes exists. In Ref. [8], it was derived assuming spherical symmetry and may be called the first law of black hole dynamics. The purpose of 2 this paper is to derive the first law of black hole dynamics without assuming any symmetry or any asymptotic conditions. In this paper we treat a dynamical and not necessarily asymptotically flat spacetime. Even for such a general situation, there is a definition of a black hole as a certain type of trapping horizon [9]. A trapping horizon is a three-surface foliated by marginal surfaces, where a marginal surface is a spatial two-surface on which one null normal expansion defined below vanishes. Geometrically this is where a light wave would have instantaneously parallel rays. The physical idea is that gravity can trap an expanding light wave and make it contract. We mention here that different types of trapping horizon can be regarded as defining a black hole, a white hole or a wormhole [10]. However, the distinctions are irrelevant for the purpose of this paper. The first law we shall obtain holds for any trapping horizon. To investigate the behavior of the trapping horizon, the so-called double-null formalism or (2+2) decomposition of general relativity is useful. Among several (2+2)-formalisms [11,12], we adopt one based on Lie derivatives w.r.t null vectors developed by one of the authors [12]. Let us review basic ingredients of the formalism. Suppose that a four-dimensional spacetime manifold (M, g) is foliated (at least locally) by null hypersurfaces Σ, each of which is parameterized by a scalar ξ, respectively. The null character is described by g(n, n) = 0, where n = −dξ are normal 1-forms to Σ. The relative normalization of the null normals defines a function f as g(n, n) = −e . The intersections of Σ(ξ) and Σ(ξ) define a two-parameter family of two-dimensional spacelike surfaces S(ξ, ξ). Hence, by introducing an intrinsic coordinate system (θ,θ) of the 2-surfaces, the foliation is described by the imbedding x = x(ξ, ξ; θ, θ). For the imbedding, the intrinsic metric on the 2-surfaces is found to be h = g+e(n⊗ n + n ⊗ n). Correspondingly, the vectors u± = ∂/∂ξ have ’shift vectors’ s± =⊥ u±, where ⊥ indicates projection by h. The 4-dimensional metric is written in terms of (h,f ,s±) as