In physical terms, a black hole is a region where gravity is
so strong that nothing can escape. In order to make this notion
precise, one must have in mind a region of spacetime to which one
can contemplate escaping. For an asymptotically flat spacetime
(representing an isolated system), the asymptotic portion of the
spacetime ``near infinity'' is such a region. The
black hole
region,
, of an asymptotically flat spacetime,
, is defined as
where
denotes future null infinity and
denotes the chronological past. Similar definitions of a black
hole can be given in other contexts (such as asymptotically
anti-deSitter spacetimes) where there is a well defined
asymptotic region.
The
event horizon,
, of a black hole is defined to be the boundary of
. Thus,
is the boundary of the past of
. Consequently,
automatically satisfies all of the properties possessed by past
boundaries (see, e.g., [55
] or [99
] for further discussion). In particular,
is a null hypersurface which is composed of future inextendible
null geodesics without caustics, i.e., the expansion,
, of the null geodesics comprising the horizon cannot become
negatively infinite. Note that the entire future history of the
spacetime must be known before the location of
can be determined, i.e.,
possesses no distinguished local significance.
If Einstein's equation holds with matter satisfying the null
energy condition (i.e., if
for all null
), then it follows immediately from the Raychauduri equation
(see, e.g., [99
]) that if the expansion,
, of any null geodesic congruence ever became negative, then
would become infinite within a finite affine parameter,
provided, of course, that the geodesic can be extended that far.
If the black hole is
strongly asymptotically predictable
- i.e., if there is a globally hyperbolic region containing
- it can be shown that this implies that
everywhere on
(see, e.g., [55
,
99
]). It then follows that the surface area,
A, of the event horizon of a black hole can never decrease with
time, as discovered by Hawking [53].
It is worth remarking that since
is a past boundary, it automatically must be a
embedded submanifold (see, e.g., [99
]), but it need not be
. However, essentially all discussions and analyses of black hole
event horizons implicitly assume
or higher order differentiability of
. Recently, this higher order differentiability assumption has
been eliminated for the proof of the area theorem [36].
The area increase law bears a resemblance to the second law of
thermodynamics in that both laws assert that a certain quantity
has the property of never decreasing with time. It might seem
that this resemblance is a very superficial one, since the area
law is a theorem in differential geometry whereas the second law
of thermodynamics is understood to have a statistical origin.
Nevertheless, this resemblance together with the idea that
information is irretrievably lost when a body falls into a black
hole led Bekenstein to propose [14,
15
] that a suitable multiple of the area of the event horizon of a
black hole should be interpreted as its entropy, and that a
generalized second law
(GSL) should hold: The sum of the ordinary entropy of matter
outside of a black hole plus a suitable multiple of the area of a
black hole never decreases. We will discuss this law in detail in
section
4
.
The remaining laws of thermodynamics deal with equilibrium and
quasi-equilibrium processes. At nearly the same time as
Bekenstein proposed a relationship between the area theorem and
the second law of thermodynamics, Bardeen, Carter, and
Hawking [12] provided a general proof of certain laws of ``black hole
mechanics'' which are direct mathematical analogs of the zeroth
and first laws of thermodynamics. These laws of black hole
mechanics apply to stationary black holes (although a formulation
of these laws in terms of isolated horizons will be briefly
described at the end of this section).
In order to discuss the zeroth and first laws of black hole
mechanics, we must introduce the notions of stationary, static,
and axisymmetric black holes as well as the notion of a Killing
horizon. If an asymptotically flat spacetime
contains a black hole,
, then
is said to be
stationary
if there exists a one-parameter group of isometries on
generated by a Killing field
which is unit timelike at infinity. The black hole is said to be
static
if it is stationary and if, in addition,
is hypersurface orthogonal. The black hole is said to be
axisymmetric
if there exists a one parameter group of isometries which
correspond to rotations at infinity. A stationary, axisymmetric
black hole is said to possess the ``
t
-
orthogonality property'' if the 2-planes spanned by
and the rotational Killing field
are orthogonal to a family of 2-dimensional surfaces. The
t
-
orthogonality property holds for all stationary-axisymmetric
black hole solutions to the vacuum Einstein or Einstein-Maxwell
equations (see, e.g., [56
]).
A null surface,
, whose null generators coincide with the orbits of a
one-parameter group of isometries (so that there is a Killing
field
normal to
) is called a
Killing horizon
. There are two independent results (usually referred to as
``rigidity theorems'') that show that in a wide variety of cases
of interest, the event horizon,
, of a stationary black hole must be a Killing horizon. The
first, due to Carter [35
], states that for a static black hole, the static Killing field
must be normal to the horizon, whereas for a
stationary-axisymmetric black hole with the
t
-
orthogonality property there exists a Killing field
of the form
which is normal to the event horizon. The constant
defined by Eq. (2
) is called the
angular velocity of the horizon
. Carter's result does not rely on any field equations, but
leaves open the possibility that there could exist stationary
black holes without the above symmetries whose event horizons are
not Killing horizons. The second result, due to Hawking [55] (see also [45]), directly proves that in vacuum or electrovac general
relativity, the event horizon of any stationary black hole must
be a Killing horizon. Consequently, if
fails to be normal to the horizon, then there must exist an
additional Killing field,
, which is normal to the horizon, i.e., a stationary black hole
must be nonrotating (from which staticity follows [84
,
85,
37]) or axisymmetric (though not necessarily with the
t
-
orthogonality property). Note that Hawking's theorem makes no
assumptions of symmetries beyond stationarity, but it does rely
on the properties of the field equations of general
relativity.
Now, let
be any Killing horizon (not necessarily required to be the event
horizon,
, of a black hole), with normal Killing field
. Since
also is normal to
, these vectors must be proportional at every point on
. Hence, there exists a function,
, on
, known as the
surface gravity
of
, which is defined by the equation
It follows immediately that
must be constant along each null geodesic generator of
, but, in general,
can vary from generator to generator. It is not difficult to
show (see, e.g., [99]) that
where
a
is the magnitude of the acceleration of the orbits of
in the region off of
where they are timelike,
is the ``redshift factor'' of
, and the limit as one approaches
is taken. Equation (4
) motivates the terminology ``surface gravity''. Note that the
surface gravity of a black hole is defined only when it is ``in
equilibrium'', i.e., stationary, so that its event horizon is a
Killing horizon. There is no notion of the surface gravity of a
general, non-stationary black hole, although the definition of
surface gravity can be extended to isolated horizons (see
below).
In parallel with the two independent ``rigidity theorems''
mentioned above, there are two independent versions of the zeroth
law of black hole mechanics. The first, due to Carter [35] (see also [78]), states that for any black hole which is static or is
stationary-axisymmetric with the
t
-
orthogonality property, the surface gravity
, must be constant over its event horizon
. This result is purely geometrical, i.e., it involves no use of
any field equations. The second, due to Bardeen, Carter, and
Hawking [12
] states that if Einstein's equation holds with the matter
stress-energy tensor satisfying the dominant energy condition,
then
must be constant on any Killing horizon. Thus, in the second
version of the zeroth law, the hypothesis that the
t
-
orthogonality property holds is eliminated, but use is made of
the field equations of general relativity.
A
bifurcate Killing horizon
is a pair of null surfaces,
and
, which intersect on a spacelike 2-surface,
(called the ``bifurcation surface''), such that
and
are each Killing horizons with respect to the same Killing field
. It follows that
must vanish on
; conversely, if a Killing field,
, vanishes on a two-dimensional spacelike surface,
, then
will be the bifurcation surface of a bifurcate Killing horizon
associated with
(see [101
] for further discussion). An important consequence of the zeroth
law is that if
, then in the ``maximally extended'' spacetime representing a
stationary black hole, the event horizon,
, comprises a branch of a bifurcate Killing horizon [78
]. This result is purely geometrical - involving no use of any
field equations. As a consequence, the study of stationary black
holes which satisfy the zeroth law divides into two cases:
``extremal'' black holes (for which, by definition,
), and black holes with bifurcate horizons.
The first law of black hole mechanics is simply an identity relating the changes in mass, M, angular momentum, J, and horizon area, A, of a stationary black hole when it is perturbed. To first order, the variations of these quantities in the vacuum case always satisfy
In the original derivation of this law [12], it was required that the perturbation be stationary.
Furthermore, the original derivation made use of the detailed
form of Einstein's equation. Subsequently, the derivation has
been generalized to hold for non-stationary perturbations [84,
60
], provided that the change in area is evaluated at the
bifurcation surface,
, of the unperturbed black hole (see, however, [80] for a derivation of the first law for non-stationary
perturbations that does not require evaluation at the bifurcation
surface). More significantly, it has been shown [60] that the validity of this law depends only on very general
properties of the field equations. Specifically, a version of
this law holds for any field equations derived from a
diffeomorphism covariant Lagrangian,
L
. Such a Lagrangian can always be written in the form
where
denotes the derivative operator associated with
,
denotes the Riemann curvature tensor of
, and
denotes the collection of all matter fields of the theory (with
indices suppressed). An arbitrary (but finite) number of
derivatives of
and
are permitted to appear in
L
. In this more general context, the first law of black hole
mechanics is seen to be a direct consequence of an identity
holding for the variation of the Noether current. The general
form of the first law takes the form
where the ``...'' denote possible additional contributions from long range matter fields, and where
Here
is the binormal to the bifurcation surface
(normalized so that
), and the functional derivative is taken by formally viewing the
Riemann tensor as a field which is independent of the metric in
Eq. (6
). For the case of vacuum general relativity, where
, a simple calculation yields
and Eq. (7) reduces to Eq. (5
).
The close mathematical analogy of the zeroth, first, and
second laws of thermodynamics to corresponding laws of classical
black hole mechanics is broken by the Planck-Nernst form of the
third law of thermodynamics, which states that
(or a ``universal constant'') as
. The analog of this law fails in black hole mechanics - although
analogs of alternative formulations of the third law do appear to
hold for black holes [59] - since there exist extremal black holes (i.e., black holes
with
) with finite
A
. However, there is good reason to believe that the
``Planck-Nernst theorem'' should not be viewed as a fundamental
law of thermodynamics [1] but rather as a property of the density of states near the
ground state in the thermodynamic limit, which happens to be
valid for commonly studied materials. Indeed, examples can be
given of ordinary quantum systems that violate the Planck-Nernst
form of the third law in a manner very similar to the violations
of the analog of this law that occur for black holes [102].
As discussed above, the zeroth and first laws of black hole
mechanics have been formulated in the mathematical setting of
stationary black holes whose event horizons are Killing horizons.
The requirement of stationarity applies to the entire spacetime
and, indeed, for the first law, stationarity of the entire
spacetime is essential in order to relate variations of
quantities defined at the horizon (like
A) to variations of quantities defined at infinity (like
M
and
J). However, it would seem reasonable to expect that the
equilibrium thermodynamic behavior of a black hole would require
only a form of local stationarity at the event horizon. For the
formulation of the first law of black hole mechanics, one would
also then need local definitions of quantities like
M
and
J
at the horizon. Such an approach toward the formulation of the
laws of black hole mechanics has recently been taken via the
notion of an
isolated horizon, defined as a null hypersurface with vanishing shear and
expansion satisfying the additional properties stated in [4]. (This definition supersedes the more restrictive definitions
given, e.g., in [5,
6,
7].) The presence of an isolated horizon does not require the
entire spacetime to be stationary [65]. A direct analog of the zeroth law for stationary event
horizons can be shown to hold for isolated horizons [9
]. In the Einstein-Maxwell case, one can demand (via a choice of
scaling of the normal to the isolated horizon as well as a choice
of gauge for the Maxwell field) that the surface gravity and
electrostatic potential of the isolated horizon be functions of
only its area and charge. The requirement that time evolution be
symplectic then leads to a version of the first law of black hole
mechanics as well as a (in general, non-unique) local notion of
the energy of the isolated horizon [9
]. These results also have been generalized to allow dilaton
couplings [7] and Yang-Mills fields [38,
9
].
In comparing the laws of black hole mechanics in classical
general relativity with the laws of thermodynamics, it should
first be noted that the black hole uniqueness theorems (see,
e.g., [56]) establish that stationary black holes - i.e., black holes ``in
equilibrium'' - are characterized by a small number of
parameters, analogous to the ``state parameters'' of ordinary
thermodynamics. In the corresponding laws, the role of energy,
E, is played by the mass,
M, of the black hole; the role of temperature,
T, is played by a constant times the surface gravity,
, of the black hole; and the role of entropy,
S, is played by a constant times the area,
A, of the black hole. The fact that
E
and
M
represent the same physical quantity provides a strong hint that
the mathematical analogy between the laws of black hole mechanics
and the laws of thermodynamics might be of physical significance.
However, as argued in [12
], this cannot be the case in classical general relativity. The
physical temperature of a black hole is absolute zero (see
subsection
4.1
below), so there can be no physical relationship between
T
and
. Consequently, it also would be inconsistent to assume a
physical relationship between
S
and
A
. As we shall now see, this situation changes dramatically when
quantum effects are taken into account.
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The Thermodynamics of Black Holes
Robert M. Wald http://www.livingreviews.org/lrr-2001-6 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |