Page 37 - Nature Of Space And Time

black hole
accelerating

Lorentzian space

Euclidean space
black hole tunneling through
Euclidean space

apart by the magnetic eld. The accelerating black hole solution is not asymptotically
at
because it tends to a uniform magnetic eld at in nity. But one can nevertheless use it to
estimate the rate of pair creation of black holes in a local region of magnetic eld.

One could imagine that after being created the black holes move far apart into regions
without magnetic eld. One could then treat each black hole separately as a black hole
in asymptotically
at 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
2
2
eld will be the topology of the Euclidean Ernst solution. This is S × S 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 in nite in both

cases. Nevertheless there is a well de ned sense in which their di erence is nite and equal
to the black hole horizon area plus the di erence 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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