Thus,
truly is the
physical
temperature of a black hole, not merely a quantity playing a
role mathematically analogous to temperature in the laws of black
hole mechanics. In this section, we review the status of the
derivation of the Hawking effect and also discuss the closely
related Unruh effect.
The original derivation of the Hawking effect [54] made direct use of the formalism for calculating particle
creation in a curved spacetime that had been developed by
Parker [73] and others. Hawking considered a classical spacetime
describing gravitational collapse to a Schwarzschild black hole.
He then considered a free (i.e., linear) quantum field
propagating in this background spacetime, which is initially in
its vacuum state prior to the collapse, and he computed the
particle content of the field at infinity at late times. This
calculation involves taking the positive frequency mode function
corresponding to a particle state at late times, propagating it
backwards in time, and determining its positive and negative
frequency parts in the asymptotic past. His calculation revealed
that at late times, the expected number of particles at infinity
corresponds to emission from a perfect black body (of finite
size) at the Hawking temperature (Eq. (10
)). It should be noted that this result relies only on the
analysis of quantum fields in the region exterior to the black
hole, and it does not make use of any gravitational field
equations.
The original Hawking calculation can be straightforwardly
generalized and extended in the following ways. First, one may
consider a spacetime representing an arbitrary gravitational
collapse to a black hole such that the black hole ``settles
down'' to a stationary final state satisfying the zeroth law of
black hole mechanics (so that the surface gravity,
, of the black hole final state is constant over its event
horizon). The initial state of the quantum field may be taken to
be any nonsingular state (i.e., any Hadamard state - see,
e.g., [101
]) rather than the initial vacuum state. Finally, it can be
shown [98
] that all aspects of the final state at late times (i.e., not
merely the expected number of particles in each mode) correspond
to black body
thermal radiation emanating from the black hole at temperature
(Eq. (10
)).
It should be noted that no infinities arise in the calculation
of the Hawking effect for a free field, so the results are
mathematically well defined, without any need for regularization
or renormalization. The original derivations [54,
98] made use of notions of ``particles propagating into the black
hole'', but the results for what an observer sees at infinity
were shown to be independent of the ambiguities inherent in such
notions and, indeed, a derivation of the Hawking effect has been
given [44] which entirely avoids the introduction of any notion of
``particles''. However, there remains one significant difficultly
with the Hawking derivation: In the calculation of the
backward-in-time propagation of a mode, it is found that the mode
undergoes a large blueshift as it propagates near the event
horizon, but there is no correspondingly large redshift as the
mode propagates back through the collapsing matter into the
asymptotic past. Indeed, the net blueshift factor of the mode is
proportional to
, where
t
is the time that the mode would reach an observer at infinity.
Thus, within a time of order
of the formation of a black hole (i.e.,
seconds for a one solar mass Schwarzschild black hole), the
Hawking derivation involves (in its intermediate steps) the
propagation of modes of frequency much higher than the Planck
frequency. In this regime, it is difficult to believe in the
accuracy of free field theory - or any other theory known to
mankind.
An approach to investigating this issue was first suggested by
Unruh [92], who noted that a close analog of the Hawking effect occurs for
quantized sound waves in a fluid undergoing supersonic flow. A
similar blueshifting of the modes quickly brings one into a
regime well outside the domain of validity of the continuum fluid
equations. Unruh suggested replacing the continuum fluid
equations with a more realistic model at high frequencies to see
if the fluid analog of the Hawking effect would still occur. More
recently, Unruh investigated models where the dispersion relation
is altered at ultra-high frequencies, and he found no deviation
from the Hawking prediction [93]. A variety of alternative models have been considered by other
researchers [28
,
39
,
62
,
79
,
97
,
40
,
63
]. Again, agreement with the Hawking effect prediction was found
in all cases, despite significant modifications of the theory at
high frequencies.
The robustness of the Hawking effect with respect to modifications of the theory at ultra-high frequency probably can be understood on the following grounds. One may view the backward-in-time propagation of modes as consisting of two stages: a first stage where the blueshifting of the mode brings it into a WKB regime but the frequencies remain well below the Planck scale, and a second stage where the continued blueshifting takes one to the Planck scale and beyond. In the first stage, the usual field theory calculations should be reliable. On the other hand, after the mode has entered a WKB regime, it seems plausible that the kinds of modifications to its propagation laws considered in [93, 28, 39, 62, 79, 97, 40, 63] should not affect its essential properties, in particular the magnitude of its negative frequency part.
Indeed, an issue closely related to the validity of the
original Hawking derivation arises if one asks how a uniformly
accelerating observer in Minkowski spacetime perceives the
ordinary (inertial) vacuum state (see below). The outgoing modes
of a given frequency
as seen by the accelerating observer at proper time
along his worldline correspond to modes of frequency
in a fixed inertial frame. Therefore, at time
one might worry about field-theoretic derivations of what the
accelerating observer would see. However, in this case one can
appeal to Lorentz invariance to argue that what the accelerating
observer sees cannot change with time. It seems likely that one
could similarly argue that the Hawking effect cannot be altered
by modifications of the theory at ultra-high frequencies,
provided that these modifications preserve an appropriate ``local
Lorentz invariance'' of the theory. Thus, there appears to be
strong reasons for believing in the validity of the Hawking
effect despite the occurrence of ultra-high-frequency modes in
the derivation.
There is a second, logically independent result - namely, the
Unruh effect [91] and its generalization to curved spacetime - which also gives
rise to the formula (10
). Although the Unruh effect is mathematically very closely
related to the Hawking effect, it is important to distinguish
clearly between them. In its most general form, the Unruh effect
may be stated as follows (see [64
,
101
] for further discussion): Consider a classical spacetime
that contains a bifurcate Killing horizon,
, so that there is a one-parameter group of isometries whose
associated Killing field,
, is normal to
. Consider a free quantum field on this spacetime. Then there
exists at most one globally nonsingular state of the field which
is invariant under the isometries. Furthermore, in the ``wedges''
of the spacetime where the isometries have timelike orbits, this
state (if it exists) is a KMS (i.e., thermal equilibrium) state
at temperature (10
) with respect to the isometries.
Note that in Minkowski spacetime, any one-parameter group of
Lorentz boosts has an associated bifurcate Killing horizon,
comprised by two intersecting null planes. The unique, globally
nonsingular state which is invariant under these isometries is
simply the usual (``inertial'') vacuum state,
. In the ``right and left wedges'' of Minkowski spacetime defined
by the Killing horizon, the orbits of the Lorentz boost
isometries are timelike, and, indeed, these orbits correspond to
worldlines of uniformly accelerating observers. If we normalize
the boost Killing field,
, so that Killing time equals proper time on an orbit with
acceleration
a, then the surface gravity of the Killing horizon is
. An observer following this orbit would naturally use
to define a notion of ``time translation symmetry''.
Consequently, by the above general result, when the field is in
the inertial vacuum state, a uniformly accelerating observer
would describe the field as being in a thermal equilibrium state
at temperature
as originally discovered by Unruh [91]. A mathematically rigorous proof of the Unruh effect in Minkowski spacetime was given by Bisognano and Wichmann [23] in work motivated by entirely different considerations (and done independently of and nearly simultaneously with the work of Unruh). Furthermore, the Bisognano-Wichmann theorem is formulated in the general context of axiomatic quantum field theory, thus establishing that the Unruh effect is not limited to free field theory.
Although there is a close mathematical relationship between
the Unruh effect and the Hawking effect, it should be emphasized
that these results refer to
different
states of the quantum field. We can divide the late time modes
of the quantum field in the following manner, according to the
properties that they would have in the analytically continued
spacetime [78] representing the asymptotic final stationary state of the black
hole: We refer to modes that would have emanated from the white
hole region of the analytically continued spacetime as ``UP
modes'' and those that would have originated from infinity as
``IN modes''. In the Hawking effect, the asymptotic final state
of the quantum field is a state in which the UP modes of the
quantum field are thermally populated at temperature (10), but the IN modes are unpopulated. This state (usually referred
to as the ``Unruh vacuum'') would be singular on the white hole
horizon in the analytically continued spacetime. On the other
hand, in the Unruh effect and its generalization to curved
spacetimes, the state in question (usually referred to as the
``Hartle-Hawking vacuum'' [52]) is globally nonsingular, and
all
modes of the quantum field in the ``left and right wedges'' are
thermally populated.
The differences between the Unruh and Hawking effects can be seen dramatically in the case of a Kerr black hole. For the Kerr black hole, it can be shown [64] that there does not exist any globally nonsingular state of the field which is invariant under the isometries associated with the Killing horizon, i.e., there does not exist a ``Hartle-Hawking vacuum state'' on Kerr spacetime. However, there is no difficultly with the derivation of the Hawking effect for Kerr black holes, i.e., the ``Unruh vacuum state'' does exist.
It should be emphasized that in the Hawking effect, the
temperature (10) represents the temperature as measured by an observer near
infinity. For any observer following an orbit of the Killing
field,
, normal to the horizon, the locally measured temperature of the
UP modes is given by
where
. In other words, the locally measured temperature of the Hawking
radiation follows the Tolman law. Now, as one approaches the
horizon of the black hole, the UP modes dominate over the IN
modes. Taking Eq. (4
) into account, we see that
as the black hole horizon,
, is approached, i.e., in this limit Eq. (12
) corresponds to the flat spacetime Unruh effect.
Equation (12) shows that when quantum effects are taken into account, a black
hole is surrounded by a ``thermal atmosphere'' whose local
temperature as measured by observers following orbits of
becomes divergent as one approaches the horizon. As we shall see
in the next section, this thermal atmosphere produces important
physical effects on quasi-stationary bodies near the black hole.
On the other hand, it should be emphasized that for a macroscopic
black hole, observers who freely fall into the black hole would
not notice any important quantum effects as they approach and
cross the horizon.
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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 |