It seems to lead to a new level of unpredictability in physics over and above the usual
uncertainty associated with quantum mechanics. This is because black holes appear to
have intrinsic entropy and to lose information from our region of the universe. I should say
that these claims are controversial: many people working on quantum gravity, including
almost all those that entered it from particle physics, would instinctively reject the idea
that information about the quantum state of a system could be lost. However they have
had very little success in showing how information can get out of a black hole. Eventually
I believe they will be forced to accept my suggestion that it is lost, just as they were forced to agree that black holes radiate, which was against all their preconceptions.
I should start by reminding you about the classical theory of black holes. We saw in
the last lecture that gravity is always attractive, at least in normal situations. If gravity had been sometimes attractive and sometimes repulsive, like electro-dynamics, we would
never notice it at all because it is about 1040 times weaker. It is only because gravity always has the same sign that the gravitational force between the particles of two macroscopic
bodies like ourselves and the Earth add up to give a force we can feel.
The fact that gravity is attractive means that it will tend to draw the matter in the
universe together to form objects like stars and galaxies. These can support themselves for
a time against further contraction by thermal pressure, in the case of stars, or by rotation
and internal motions, in the case of galaxies. However, eventually the heat or the angular
momentum will be carried away and the object will begin to shrink. If the mass is less
than about one and a half times that of the Sun the contraction can be stopped by the
degeneracy pressure of electrons or neutrons. The object will settle down to be a white
dwarf or a neutron star respectively. However, if the mass is greater than this limit there
is nothing that can hold it up and stop it continuing to contract. Once it has shrunk to a
certain critical size the gravitational field at its surface will be so strong that the light cones will be bent inward as in the diagram on the following page. I would have liked to draw
you a four dimensional picture. However, government cuts have meant that Cambridge
university can afford only two dimensional screens. I have therefore shown time in the
vertical direction and used perspective to show two of the three space directions. You can
see that even the outgoing light rays are bent towards each other and so are converging
rather than diverging. This means that there is a closed trapped surface which is one of
the alternative third conditions of the Hawking-Penrose theorem.
21
r=0 singularity
r = 2M
event
horizon
trapped
surface
surface
interior
of star
of star
If the Cosmic Censorship Conjecture is correct the trapped surface and the singularity
it predicts can not be visible from far away. Thus there must be a region of spacetime
from which it is not possible to escape to infinity. This region is said to be a black hole.
Its boundary is called the event horizon and it is a null surface formed by the light rays
that just fail to get away to infinity. As we saw in the last lecture, the area of a cross
section of the event horizon can never decrease, at least in the classical theory. This, and
perturbation calculations of spherical collapse, suggest that black holes will settle down to a stationary state. The no hair theorem, proved by the combined work of Israel, Carter,
Robinson and myself, shows that the only stationary black holes in the absence of matter
fields are the Kerr solutions. These are characterized by two parameters, the mass M and the angular momentum J . The no hair theorem was extended by Robinson to the case
where there was an electromagnetic field. This added a third parameter Q, the electric charge. The no hair theorem has not been proved for the Yang-Mills field, but the only