24
infinity. Each point of this triangle corresponds to a two sphere of radius r. r = 0 on the vertical line on the left, which represents the center of symmetry, and r → ∞ on the right of the diagram.
One can easily see from the diagram that every point in Minkowski space is in the
past of future null infinity I+. This means there is no black hole and no event horizon.
However, if one has a spherical body collapsing the diagram is rather different.
singularity
event horizon
black
hole
+
collapsing
body
_
It looks the same in the past but now the top of the triangle has been cut off and replaced by a horizontal boundary. This is the singularity that the Hawking-Penrose theorem predicts.
One can now see that there are points under this horizontal line that are not in the past
of future null infinity I+. In other words there is a black hole. The event horizon, the boundary of the black hole, is a diagonal line that comes down from the top right corner
and meets the vertical line corresponding to the center of symmetry.
One can consider a scalar field φ on this background. If the spacetime were time
independent, a solution of the wave equation, that contained only positive frequencies on
scri minus, would also be positive frequency on scri plus. This would mean that there
would be no particle creation, and there would be no out going particles on scri plus, if
there were no scalar particles initially.
However, the metric is time dependent during the collapse. This will cause a solution
that is positive frequency on I− to be partly negative frequency when it gets to I+.
One can calculate this mixing by taking a wave with time dependence e−iωu on I+ and propagating it back to I−. When one does that one finds that the part of the wave that passes near the horizon is very blue shifted. Remarkably it turns out that the mixing is
independent of the details of the collapse in the limit of late times. It depends only on the 25
surface gravity κ that measures the strength of the gravitational field on the horizon of the black hole. The mixing of positive and negative frequencies leads to particle creation.
When I first studied this effect in 1973 I expected I would find 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 κ . This was just what 2 π
was required to make consistent the idea that a black hole had an entropy proportional
to the area of its event horizon. Moreover, it fixed 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 cm2
so a black hole of the mass of the Sun would have an entropy of the order of 1078. This
would reflect the enormous number of different ways in which it could be made.
Black Hole Thermal Radiation
κ
Temperature T
=
2 π
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
ds 2 = − 1 − 2 M
dt 2 +
1 − 2 M
dr 2 + r 2( dθ 2 + sin2 θdφ 2)
r
r
This represents the gravitational field 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 = 2 M . However, this is just caused by a bad choice of coordinates.