each other within a finite distance.
This implies that the area of a cross section of the event horizon can never decrease
with time and in general will increase. Moreover if two black holes collide and merge
together the area of the final black hole will be greater than the sum of the areas of the
original black holes.
A2
A3
final black hole
black hole
event horizon
infalling
infalling
matter
matter
two original
A1
A1
black holes
A2
A
≥
2
A1
A
≥
3
A1 + A2
This is very similar to the behavior of entropy according to the Second Law of Thermody-
namics. Entropy can never decrease and the entropy of a total system is greater than the
sum of its constituent parts.
Second Law of Black Hole Mechanics
δA
≥ 0
Second Law of Thermodynamics
δS
≥ 0
The similarity with thermodynamics is increased by what is called the First Law of
Black Hole Mechanics. This relates the change in mass of a black hole to the change in the
area of the event horizon and the change in its angular momentum and electric charge. One
can compare this to the First Law of Thermodynamics which gives the change in internal
energy in terms of the change in entropy and the external work done on the system.
One sees that if the area of the event horizon is analogous to entropy then the quantity
analogous to temperature is what is called the surface gravity of the black hole κ. This is a 17
First Law of Black Hole Mechanics
κ
δE
=
δA + Ω δJ
+ Φ δQ
8 π
First Law of Thermodynamics
δE
=
T δS
+ P δV
measure of the strength of the gravitational field on the event horizon. The similarity with
thermodynamics is further increased by the so called Zeroth Law of Black Hole Mechanics:
the surface gravity is the same everywhere on the event horizon of a time independent
black hole.
Zeroth Law of Black Hole Mechanics
κ is the same everywhere on the horizon of a time independent
black hole.
Zeroth Law of Thermodynamics
T is the same everywhere for a system in thermal equilibrium.
Encouraged by these similarities Bekenstein proposed that some multiple of the area
of the event horizon actually was the entropy of a black hole. He suggested a generalized
Second Law: the sum of this black hole entropy and the entropy of matter outside black
holes would never decrease.
Generalised Second Law
δ( S + cA)
≥ 0
However this proposal was not consistent. If black holes have an entropy proportional to
horizon area they should also have a non zero temperature proportional to surface gravity.
Consider a black hole that is in contact with thermal radiation at a temperature lower
than the black hole temperature. The black hole will absorb some of the radiation but
won’t be able to send anything out, because according to classical theory nothing can get
18
low temperature
thermal radiation
black hole
radiation being absorbed
by black hole
out of a black hole. One thus has heat flow from the low temperature thermal radiation to
the higher temperature black hole. This would violate the generalized Second Law because
the loss of entropy from the thermal radiation would be greater than the increase in black
hole entropy. However, as we shall see in my next lecture, consistency was restored when
it was discovered that black holes are sending out radiation that was exactly thermal.
This is too beautiful a result to be a coincidence or just an approximation. So it seems
that black holes really do have intrinsic gravitational entropy. As I shall show, this is
related to the non trivial topology of a black hole. The intrinsic entropy means that
gravity introduces an extra level of unpredictability over and above the uncertainty usually
associated with quantum theory. So Einstein was wrong when he said “God does not play
dice”. Consideration of black holes suggests, not only that God does play dice, but that
He sometimes confuses us by throwing them where they can’t be seen.
19
20
2. Quantum Black Holes
S. W. Hawking
In my second lecture I’m going to talk about the quantum theory of black holes.