black hole as well as about the center of the circle on which the black hole moves. One has
to adjust the mass to charge ratio of the black hole to make these periods equal. Physically
this means that one chooses the parameters of the black hole so that the temperature of the
black hole is equal to the temperature it sees because it is accelerating.. The temperature
of a magnetically charged black hole tends to zero as the charge tends to the mass in
Planck units. Thus for weak magnetic fields, and hence low acceleration, one can always
match the periods.
Like in the case of pair creation of electrons one can describe pair creation of black
holes by joining the lower half of the imaginary time Euclidean solution to the upper half
of the real time Lorentzian solution.
One can think of the black hole as tunneling through the Euclidean region and emerging
as a pair of oppositely charged black holes that accelerate away from each other pulled
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black hole
accelerating
Lorentzian space
Euclidean space
black hole tunneling through
Euclidean space
apart by the magnetic field. The accelerating black hole solution is not asymptotically flat
because it tends to a uniform magnetic field at infinity. But one can nevertheless use it to
estimate the rate of pair creation of black holes in a local region of magnetic field.
One could imagine that after being created the black holes move far apart into regions
without magnetic field. One could then treat each black hole separately as a black hole
in asymptotically flat space. One could throw an arbitrarily large amount of matter and
information into each hole. The holes would then radiate and lose mass. However, they
couldn’t lose magnetic charge because there are no magnetically charged particles. Thus
they would eventually get back to their original state with the mass slightly bigger than the charge. One could then bring the two holes back together again and let them annihilate
each other. The annihilation process can be regarded as the time reverse of the pair
creation. Thus it is represented by the top half of the Euclidean solution joined to the
bottom half of the Lorentzian solution. In between the pair creation and the annihilation
one can have a long Lorentzian period in which the black holes move far apart, accrete
matter, radiate and then come back together again. But the topology of the gravitational
field will be the topology of the Euclidean Ernst solution. This is S 2 × S 2 minus a point.
One might worry that the Generalized Second Law of Thermodynamics would be
violated when the black holes annihilated because the black hole horizon area would have
disappeared. However it turns out that the area of the acceleration horizon in the Ernst
solution is reduced from the area it would have if there were no pair creation. This is a
rather delicate calculation because the area of the acceleration horizon is infinite in both
cases. Nevertheless there is a well defined sense in which their difference is finite and equal to the black hole horizon area plus the difference in the action of the solutions with and
without pair creation. This can be understood as saying that pair creation is a zero energy
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black hole tunneling through
Euclidean space to annihilate
Euclidean space
matter and information
thrown into black hole
which radiates
Lorentzian space
Euclidean space
black hole tunneling through
Euclidean space to pair create
process; the Hamiltonian with pair creation is the same as the Hamiltonian without. I’m very grateful to Simon Ross and Gary Horovitz for calculating this reduction just in time
for this lecture. It is miracles like this, and I mean the result not that they got it, that
convince me that black hole thermodynamics can’t just be a low energy approximation.
I believe that gravitational entropy won’t disappear even if we have to go to a more
fundemental theory of quantum gravity.
One can see from this thought experiment that one gets intrinsic gravitational en-
tropy and loss of information when the topology of spacetime is different from that of flat
Minkowski space. If the black holes that pair create are large compared to the Planck