Black holes are usually introduced in asymptotically flat spacetimes [172, 173, 175, 387], and hence it was
natural to derive the formal laws of black hole mechanics/thermodynamics in the asymptotically flat
context (see for example [34, 50, 51], and for a recent review see [392]). 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
times the surface gravity, and the 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 [390
]). 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 [100, 101, 102, 296, 103].
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 [92]). 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
outside the horizon.
Identifying the Brown-York energy with the internal (thermodynamical) 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 [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
times the area of the horizon. (For some recent related works see for
example [149, 188]).
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 [185, 187, 186, 190] and for non-spherically symmetric cases [275, 189, 74]). 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 [183, 181, 184] are based on energy and angular momentum definitions that are gauge dependent; see also Sections 4.1.8 and 6.3.)
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 [21, 18
].) 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 [21, 18].
Booth [75] 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 non-expanding horizon. Then to have a well-defined variational
principle on
, the Hilbert action had to be modified by appropriate boundary terms. However,
requiring
to be a so-called 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 [4] using their Regge-Teitelboim-like
approach [141].
Another concept is the notion of a dynamical horizon [25, 26]. 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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