rather than a 3+1, there is no reason to include a surface term on the horizon because the
metric is regular there. Leaving out the surface term on the horizon reduces the action by
one quarter the area of the horizon, which is just the intrinsic gravitational entropy of the black hole.
The fact that the entropy of black holes is connected with a topological invariant,
the Euler number, is a strong argument that it will remain even if we have to go to a
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more fundemental theory. This idea is anathema to most particle physicists who are a very conservative lot and want to make everything like Yang-Mills theory. They agree that
the radiation from black holes seems to be thermal and independent of how the hole was
formed if the hole is large compared to the Planck length. But they would claim that when
the black hole loses mass and gets down to the Planck size, quantum general relativity will
break down and all bets will be off. However, I shall describe a thought experiment with
black holes in which information seems to be lost yet the curvature outside the horizons
always remains small.
It has been known for some time that one can create pairs of positively and negatively
charged particles in a strong electric field. One way of looking at this is to note that in
flat Euclidean space a particle of charge q such as an electron would move in a circle in a uniform electric field E. One can analytically continue this motion from the imaginary time τ to real time t. One gets a pair of positively and negatively charged particles accelerating away from each other pulled apart by the electric field.
world line
world line
of electron
of positron
t = 0
Minkowski space
Electric Field
world line of electron
τ = 0
Euclidean space
The process of pair creation is described by chopping the two diagrams in half along
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the t = 0 or τ = 0 lines. One then joins the upper half of the Minkowski space diagram to the lower half of the Euclidean space diagram.
electron and positron
accelerating in electric
field
Minkowski space
Euclidean space
electron tunneling through
Euclidean space
This gives a picture in which the positively and negatively charged particles are really the
same particle. It tunnels through Euclidean space to get from one Minkowski space world
line to the other. To a first approximation the probability for pair creation is e−I where 2 πm 2
Euclidean action I =
.
qE
Pair creation by strong electric fields has been observed experimentally and the rate agrees
with these estimates.
Black holes can also carry electric charges so one might expect that they could also be
pair created. However the rate would be tiny compared to that for electron positron pairs
because the mass to charge ratio is 1020 times bigger. This means that any electric field
would be neutralized by electron positron pair creation long before there was a significant
probability of pair creating black holes. However there are also black hole solutions with
magnetic charges. Such black holes couldn’t be produced by gravitational collapse because
there are no magnetically charged elementary particles. But one might expect that they
could be pair created in a strong magnetic field. In this case there would be no competition
from ordinary particle creation because ordinary particles do not carry magnetic charges.
So the magnetic field could become strong enough that there was a significant chance of
creating a pair of magnetically charged black holes.
In 1976 Ernst found a solution that represented two magnetically charged black holes
accelerating away from each other in a magnetic field.
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charged black hole
accelerating in magnetic field
t = 0
Lorentzian space
black hole
τ = 0
Euclidean space
If one analytically continues it to imaginary time one has a picture very like that of the
electron pair creation. The black hole moves on a circle in a curved Euclidean space just
like the electron moves in a circle in flat Euclidean space. There is a complication in the
black hole case because the imaginary time coordinate is periodic about the horizon of the