In classical general relativity, the matter responsible for
the formation of a black hole propagates into a singularity lying
within the deep interior of the black hole. Suppose that the
matter which forms a black hole possesses quantum correlations
with matter that remains far outside of the black hole. Then it
is hard to imagine how these correlations could be restored
during the process of black hole evaporation unless gross
violations of causality occur. In fact, the semiclassical
analyses of the Hawking process show that, on the contrary,
correlations between the exterior and interior of the black hole
are continually built up as it evaporates (see [101] for further discussion). Indeed, these correlations play an
essential role in giving the Hawking radiation an exactly thermal
character [98].
As already mentioned in subsection
4.1
above, an isolated black hole will ``evaporate'' completely via
the Hawking process within a finite time. If the correlations
between the inside and outside of the black hole are not restored
during the evaporation process, then by the time that the black
hole has evaporated completely, an initial pure state will have
evolved to a mixed state, i.e., ``information'' will have been
lost. In a semiclassical analysis of the evaporation process,
such information loss does occur and is ascribable to the
propagation of the quantum correlations into the singularity
within the black hole. A key unresolved issue in black hole
thermodynamics is whether this conclusion continues to hold in a
complete quantum theory of gravity. On one hand, arguments can be
given [101] that alternatives to information loss - such as the formation
of a high entropy ``remnant'' or the gradual restoration of
correlations during the late stages of the evaporation process -
seem highly implausible. On the other hand, it is commonly
asserted that the evolution of an initial pure state to a final
mixed state is in conflict with quantum mechanics. For this
reason, the issue of whether a pure state can evolve to a mixed
state in the process of black hole formation and evaporation is
usually referred to as the ``
black hole information paradox
''.
There appear to be two logically independent grounds for the claim that the evolution of an initial pure state to a final mixed state is in conflict with quantum mechanics:
With regard to (1), within the semiclassical framework, the
evolution of an initial pure state to a final mixed state in the
process of black hole formation and evaporation can be attributed
to the fact that the final time slice fails to be a Cauchy
surface for the spacetime [101]. No violation of any of the local laws of quantum field theory
occurs. In fact, a closely analogous evolution of an initial pure
state to a final mixed state occurs for a free, massless field in
Minkowski spacetime if one chooses the final ``time'' to be a
hyperboloid rather than a hyperplane [101
]. (Here, the ``information loss'' occurring during the time
evolution results from radiation to infinity rather than into a
black hole.) Indeed, the evolution of an initial pure state to a
final mixed state is naturally accommodated within the framework
of the algebraic approach to quantum theory [101] as well as in the framework of generalized quantum
theory [51].
The main arguments for (2) were given in [11] (see also [42]). However, these arguments assume that the effective evolution
law governing laboratory physics has a ``Markovian'' character,
so that it is purely local in time. As pointed out in [96], one would expect a black hole to retain a ``memory'' (stored
in its external gravitational field) of its energy-momentum, so
it is far from clear that an effective evolution law modeling the
process of black hole formation and evaporation should be
Markovian in nature. Furthermore, even within the Markovian
context, it is not difficult to construct models where rapid
information loss occurs at the Planck scale, but negligible
deviations from ordinary dynamics occur at laboratory
scales [96].
For the above reasons, I do not feel that the issue of whether a pure state evolves to a mixed state in the process of black hole formation and evaporation should be referred to as a ``paradox''. Nevertheless, the resolution of this issue is of great importance: If pure states remain pure, then our basic understanding of black holes in classical and semiclassical gravity will have to undergo significant revision in quantum gravity. On the other hand, if pure states evolve to mixed states in a fully quantum treatment of the gravitational field, then at least the aspect of the classical singularity as a place where ``information can get lost'' must continue to remain present in quantum gravity. In that case, rather than ``smooth out'' the singularities of classical general relativity, one might expect singularities to play a fundamental role in the formulation of quantum gravity [76]. Thus, the resolution of this issue would tell us a great deal about both the nature of black holes and the existence of singularities in quantum gravity.
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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 |