Page 35 - Nature Of Space And Time

the t =0 or = 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 rst approximation the probability for pair creation is e −I where

2 m 2
Euclidean action I = :
qE

Pair creation by strong electric elds 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 10 20 times bigger. This means that any electric eld
would be neutralized by electron positron pair creation long before there was a signi cant
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 eld. In this case there would be no competition
from ordinary particle creation because ordinary particles do not carry magnetic charges.
So the magnetic eld could become strong enough that there was a signi cant 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 eld.

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