Page 26 - Nature Of Space And Time

surface gravity that measures the strength of the gravitational eld on the horizon of
the black hole. The mixing of positive and negative frequencies leads to particle creation.
When I rst studied this e ect in 1973 I expected I would nd a burst of emission
during the collapse but that then the particle creation would die out and one would be
left with a black hole that was truely black. To my great surprise I found that after a

burst during the collapse there remained a steady rate of particle creation and emission.
Moreover, the emission was exactly thermal with a temperature of . This was just what
2
was required to make consistent the idea that a black hole had an entropy proportional
to the area of its event horizon. Moreover, it xed the constant of proportionality to be a

quarter in Planck units, in which G = c = h = 1. This makes the unit of area 10 −66 cm 2
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so a black hole of the mass of the Sun would have an entropy of the order of 10 .This
would re
ect the enormous number of di erent ways in which it could be made.

Black Hole Thermal Radiation

Temperature T =
2

1
Entropy S = A
4

When I made my original discovery of radiation from black holes it seemed a miracle
that a rather messy calculation should lead to emission that was exactly thermal. However,

joint work with Jim Hartle and Gary Gibbons uncovered the deep reason. To explain it I
shall start with the example of the Schwarzschild metric.

Schwarzschild Metric

−1
2M 2M
2
2
2
2
2
ds 2 = − 1 − dt + 1 − dr 2 + r (d +sin d )
r r
This represents the gravitational eld that a black hole would settle down to if it were

non rotating. In the usual r and t coordinates there is an apparent singularity at the
Schwarzschild radius r =2M. However, this is just caused by a bad choice of coordinates.
One can choose other coordinates in which the metric is regular there.

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