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13.3 Quasi-local laws of black hole dynamics

13.3.1 Quasi-local thermodynamics of black holes

Black holes are usually introduced in asymptotically flat spacetimes [172173175387], and hence it was natural to derive the formal laws of black hole mechanics/thermodynamics in the asymptotically flat context (see for example [345051], and for a recent review see [392Jump To The Next Citation Point]). The discovery of the Hawking radiation [174] showed that the laws of black hole thermodynamics are not only analogous to the laws of thermodynamics, but black holes are genuine thermodynamical objects: The black hole temperature is a physical temperature, that is hc/(2pk) times the surface gravity, and the entropy is a physical entropy, 3 kc /(4Gh) times the area of the horizon (in the traditional units with the Boltzmann constant k, speed of light c, Newton’s gravitational constant G, and Planck’s constant h) (see also [390Jump To The Next Citation Point]). Apparently, the detailed microscopic (quantum) theory of gravity is not needed to derive the black hole entropy, and it can be derived even from the general principles of a conformal field theory on the horizon of the black holes [100101102296103].

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 theoretic 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 the 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 [92Jump To The Next Citation Point]). These difficulties lead to the need of a quasi-local formulation of black hole thermodynamics. In [92] 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 2-surface S outside the horizon. Identifying the Brown-York energy with the internal (thermodynamical) energy and (in the k = h = c = 1 units) 1/(4G) 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 S. In particular, there is a critical value T0 such that for temperatures T greater than T0 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 T less than T0. In [121] 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 so-called Noether charge formalism of Wald [389], and Iyer and Wald [215] can be interpreted as a generalization of the Brown-York approach from general relativity to any diffeomorphism invariant theory to derive quasi-local quantities [216]. 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 1/(4G) times the area of the horizon. (For some recent related works see for example [149188]).

There is an extensive literature of the quasi-local formulation of the black hole dynamics and relativistic thermodynamics in the spherically symmetric context (see for example [185187186190] and for non-spherically symmetric cases [27518974]). However, one should see clearly that while the laws of black hole thermodynamics above refer to the event horizon, which is a global concept in the spacetime, the subject of the recent quasi-local formulations is to describe the properties and the evolution of the so-called trapping horizon, which is a quasi-locally defined notion. (On the other hand, the investigations of [183181184] are based on energy and angular momentum definitions that are gauge dependent; see also Sections 4.1.8 and 6.3.)

13.3.2 On the isolated horizons

The idea of the isolated horizons (more precisely, the gradually more restrictive notions of the non-expanding, the weakly isolated and isolated horizons, and the special weakly isolated horizon called the rigidly rotating horizons) is to generalize the notion of Killing horizons by keeping their basic properties without the existence of any Killing vector in general. (For a recent review see [19] and references therein, especially [21Jump To The Next Citation Point18Jump To The Next Citation Point].) The phase space for asymptotically flat spacetimes containing an isolated horizon is based on a 3-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, obviously, the Hamiltonian will be the sum of the constraints and boundary terms, corresponding both to the ends and the horizon. Thus, by 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 [2118].

Booth [75] applied the general idea of Brown and York to a domain D whose boundary consists not only of two spacelike submanifolds S1 and S2 and a timelike one 3B, but a further, internal boundary D as well, which is null. Thus he made the investigations of the isolated horizons fully quasi-local. Therefore, the topology of S1 and S2 is S2 × [a,b], and the inner (null) boundary is interpreted as (a part of) a non-expanding horizon. Then to have a well-defined variational principle on D, the Hilbert action had to be modified by appropriate boundary terms. However, requiring D to be a so-called rigidly rotating horizon, the boundary term corresponding to D 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 [4] using their Regge-Teitelboim-like approach [141].

Another concept is the notion of a dynamical horizon [2526]. This is a smooth spacelike hypersurface that can be foliated by a preferred family of marginally trapped 2-spheres. By an appropriate definition of the energy and angular momentum balance equations for these quantities, carried by gravitational waves, are derived. Isolated horizons are the asymptotic state of dynamical horizons.

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