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.
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Living Rev. Relativity 12, (2009), 4
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