Black holes are usually introduced in asymptotically flat spacetimes [237, 238, 240, 534], and hence, it is natural to derive the formal laws of black hole mechanics/thermodynamics in the asymptotically flat context (see, e.g., [49, 67, 68], and for a comprehensive review, [539]). The discovery of Hawking radiation [239] showed that the laws of black hole thermodynamics are not only analogous to the laws of thermodynamics, but black holes are genuine thermodynamic objects: black hole temperature is a physical temperature, that is times the surface gravity, and its entropy is a physical entropy, times the area of the horizon (in the traditional units with the Boltzmann constant , speed of light , Newton’s gravitational constant , and Planck’s constant ) (see, also, [537]). Apparently, the detailed microscopic (quantum) theory of gravity is not needed to derive black hole entropy, and it can be derived even from the general principles of a conformal field theory on the horizon of black holes [124, 125, 126, 409, 127, 128].

However, black holes are localized objects, thus, one must be able to describe their properties and dynamics even at the quasi-local level. Nevertheless, beyond this rather theoretical claim, there are pragmatic reasons that force us to quasi-localize the laws of black hole dynamics. In particular, it is well known that the Schwarzschild black hole, fixing its temperature at infinity, has negative heat capacity. Similarly, in an asymptotically anti-de Sitter spacetime, fixing black hole temperature via the normalization of the timelike Killing vector at infinity is not justified because there is no such physically-distinguished Killing field (see [116]). These difficulties lead to the need of a quasi-local formulation of black hole thermodynamics. In [116], Brown, Creighton, and Mann investigated the thermal properties of the Schwarzschild–anti-de Sitter black hole. They used the quasi-local approach of Brown and York to define the energy of the black hole on a spherical two-surface outside the horizon. Identifying the Brown–York energy with the internal (thermodynamic) energy and (in the units) times the area of the event horizon with the entropy, they calculated the temperature, surface pressure, and heat capacity. They found that these quantities do depend on the location of the surface . In particular, there is a critical value such that for temperatures greater than there are two black hole solutions, one with positive and one with negative heat capacity, but there are no Schwarzschild–anti-de Sitter black holes with temperature less than . In [157] the Brown–York analysis is extended to include dilaton and Yang–Mills fields, and the results are applied to stationary black holes to derive the first law of black hole thermodynamics. The Noether charge formalism of Wald [536], and Iyer and Wald [287] can be interpreted as a generalization of the Brown–York approach from general relativity to any diffeomorphism invariant theory to derive quasi-local quantities [288]. However, this formalism gave a general expression for the black hole entropy, as well. That is the Noether charge derived from the Hilbert Lagrangian corresponding to the null normal of the horizon, and explicitly this is still times the area of the horizon. (For related work see, e.g., [205, 253]). A comparison of the various proposals for the surface gravity of dynamic black holes in spherically-symmetric black hole spacetimes is given by Nielsen and Yoon [396].

There is extensive literature on the quasi-local formulation of the black hole dynamics and relativistic thermodynamics in the spherically-symmetric context (see, e.g., [250, 252, 251, 256] and for non–spherically-symmetric cases [372, 254, 96]). These investigations are based on the quasi-locally defined notion of trapping horizons [246]. A trapping horizon is a smooth hypersurface that can be foliated by (e.g., future) marginally-trapped surfaces such that the expansion of the outgoing null normals is decreasing along the incoming null normals. (On the other hand, the investigations of [248, 246, 249] are based on gauge-dependent energy and angular momentum definitions; see also Sections 4.1.8 and 6.3.) For reviews of the quasi-local formulations and the various aspects of black hole dynamics based on the notion of trapping horizons, see [41, 294, 395, 255], and, for a recent one with an extended bibliography, see [292].

The idea of isolated horizons (more precisely, the gradually more restrictive notion of nonexpanding, weakly isolated and isolated horizons, and the special weakly isolated horizon called rigidly rotating) generalizes the notion of Killing horizons by keeping their basic properties without the existence of any Killing vector in general. Thus, while the black hole is thought to be settled down to its final state, the spacetime outside the black hole may still be dynamic. (For a review see [32, 41] and references therein, especially [34, 31].) The phase space for asymptotically flat spacetimes containing an isolated horizon is based on a three-manifold with an asymptotic end (or finitely many such ends) and an inner boundary. The boundary conditions on the inner boundary are determined by the precise definition of the isolated horizon. Then the Hamiltonian is the sum of the constraints and boundary terms, corresponding both to the ends and the horizon. Thus, the appearance of the boundary term on the inner boundary makes the Hamiltonian partly quasi-local. It is shown that the condition of the Hamiltonian evolution of the states on the inner boundary along the evolution vector field is precisely the first law of black hole mechanics [34, 31].

Booth [93] applied the general idea of Brown and York to a domain whose boundary consists not only of two spacelike submanifolds and and a timelike one , but a further, internal boundary as well, which is null. Thus, he made the investigations of the isolated horizons fully quasi-local. Therefore, the topology of and is , and the inner (null) boundary is interpreted as (a part of) a nonexpanding horizon. Then, to have a well-defined variational principle on , the Hilbert action had to be modified by appropriate boundary terms. However, by requiring to be a rigidly-rotating horizon, the boundary term corresponding to and the allowed variations are considerably restricted. This made it possible to derive the ‘first law of rigidly rotating horizon mechanics’ quasi-locally, an analog of the first law of black hole mechanics. The first law for rigidly-rotating horizons was also derived by Allemandi, Francaviglia, and Raiteri in the Einstein–Maxwell theory [9] using their Regge–Teitelboim-like approach [191]. The first law for ‘slowly evolving horizons’ was derived in [96].

Another concept is the notion of a dynamic horizon [39, 40]. This is a smooth spacelike hypersurface that can be foliated by a geometrically distinguished family of (e.g., future) marginally-trapped surfaces, i.e., it is a generalization of the trapping horizon above. The isolated horizons are thought to be the asymptotic state of dynamic horizons. The local existence of such horizons was proven by Andersson, Mars and Simon [19]: If is a (strictly stably outermost) marginally trapped surface lying in a leaf, e.g., , of a foliation of the spacetime, then there exists a hypersurface (the ‘horizon’) such that lies in , and which is foliated by marginally outer-trapped surfaces. (For the related uniqueness properties of the structure of the dynamic horizons see [35]). This structure of the dynamic horizons makes it possible to derive balance equations for the areal radius of the surfaces and the angular momentum given by Eq. (11.3) [32, 40] (see also [41]). In particular, the difference of the areal radius of two marginally-trapped surfaces of the foliation, e.g., and , is just the flux integral on the portion of between and of a positive definite expression: This is the flux of the energy current of the matter fields and terms that can be interpreted as the energy flux carried by the gravitational waves. Interestingly enough, the generator vector field in this flux expression is proportional to the geometrically distinguished outward null normal of the surfaces , just as in the derivation of black hole entropy as a Noether charge by Wald [536] and Iyer and Wald [287] above. Thus, the second law of black hole mechanics is proven for dynamic horizons. Moreover, this supports the view that the energy that we should associate with marginally-trapped surfaces is the irreducible mass. For further discussion (and generalizations) of the basic flux expressions see [227, 228]. For a different calculation of the energy flux in the Vaidya spacetime, see [531].

In [97, 98] Booth and Fairhurst extended their previous investigations [93, 95] (see above and Section 10.1.5). In [97] a canonical analysis, based on the extended phase space, is given such that the underlying three-manifold has an inner boundary, which can be any of the horizon types above. Though the formalism does not give any explicit expression for the energy on the horizons, an argument is given that supports the expectation that this must be the irreducible mass of the horizon. The variations of marginally trapped surfaces, generated by vector fields orthogonal to the surfaces, are investigated and the corresponding variations of various geometric objects (intrinsic metric, expansions, connection one-form on the normal bundle, etc.) on the surfaces are calculated in [98]. In terms of these, several basic properties of marginally trapped or future outer trapped surfaces (and hence, of the horizons themselves) are derived in a straightforward way.

Living Rev. Relativity 12, (2009), 4
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