196 12 The (2+l)-dimensional black hole
properties of black holes in terms of fundamental quantum states.* More-
over, several important questions remain completely open, most notably
the 'information loss paradox'
[126]:
what happens to the information
carried by a quantum field in a pure state which collapses into a black
hole and then disperses as Hawking radiation in, presumably, a mixed
state? It might be hoped that (2+l)-dimensional gravity, which permits
an exact quantum mechanical description of black holes, could suggest
answers to some of these questions.
12.2
The
Lorentzian black hole
If
we
wish to understand the quantum mechanics of the (2+l)-dimensional
black hole, an obvious preliminary question is whether Hawking radiation
occurs. This is a semiclassical question, one that depends on the proper-
ties of quantum fields in a black hole background but does not require
a full treatment of quantum gravity. The starting point for the quan-
tum field theoretical computation is an appropriate two-point function
G(x,x
f
) = (0|(/)(x)(/)(x
/
)|0}, from which such quantities as the expectation
values (OIT^vIO) can be derived. In particular, a Green's function that is
periodic in imaginary time with period /3 is a thermal Green's function
corresponding to a local inverse temperature /?, and such a periodicity can
be interpreted as an indication of Hawking radiation.
The properties of the Green's function G(x,x
/
) can be computed by
brute force, but it is simpler and more instructive to take advantage of a
peculiar property of black holes in 2+1 dimensions, the fact that they are
described by spaces of constant curvature. As we saw in chapter 2, the
(2+l)-dimensional black hole can be represented by the metric
ds
2
=
-N
2
dt
2
+
r
2
(#
2
+ N+dif
+
N~
2
dr
2
(12.3)
with
r
2
J
2
J
r
J
N
2
=
— M A
I N =
f
1
4r
2
' 2r
2
'
where M and J are the mass and angular momentum. But as a space of
constant negative curvature, the BTZ black hole is also locally isometric to
anti-de Sitter (adS) space. In the language of chapter 4, the black hole has
an anti-de Sitter geometric structure, and should be expressible as a set
of adS patches glued together by suitable
<SX(2,
R) x
SL(2,
R) holonomies.
This means that the Green's function G(x,x
f
) can be described in terms
of the much simpler Green's function in adS space.
*
There
has
been some very recent progress
in
finding
a
string theoretical description
of
black hole statistical mechanics
[151,
246].
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