difference seems to be the addition of one or more integers that label a discrete family of
unstable solutions. It can be shown that there are no more continuous degrees of freedom
22
No Hair Theorem
Stationary black holes are characterised by mass M , angular
momentum J and electric charge Q.
of time independent Einstein-Yang-Mills black holes.
What the no hair theorems show is that a large amount of information is lost when
a body collapses to form a black hole. The collapsing body is described by a very large
number of parameters. There are the types of matter and the multipole moments of the
mass distribution. Yet the black hole that forms is completely independent of the type
of matter and rapidly loses all the multipole moments except the first two: the monopole
moment, which is the mass, and the dipole moment, which is the angular momentum.
This loss of information didn’t really matter in the classical theory. One could say that
all the information about the collapsing body was still inside the black hole. It would be
very difficult for an observer outside the black hole to determine what the collapsing body
was like. However, in the classical theory it was still possible in principle. The observer
would never actually lose sight of the collapsing body. Instead it would appear to slow
down and get very dim as it approached the event horizon. But the observer could still see
what it was made of and how the mass was distributed. However, quantum theory changed
all this. First, the collapsing body would send out only a limited number of photons before
it crossed the event horizon. They would be quite insufficient to carry all the information
about the collapsing body. This means that in quantum theory there’s no way an outside
observer can measure the state of the collapsed body. One might not think this mattered
23
too much because the information would still be inside the black hole even if one couldn’t measure it from the outside. But this is where the second effect of quantum theory on
black holes comes in. As I will show, quantum theory will cause black holes to radiate
and lose mass. Eventually it seems that they will disappear completely, taking with them
the information inside them. I will give arguments that this information really is lost and
doesn’t come back in some form. As I will show, this loss of information would introduce a
new level of uncertainty into physics over and above the usual uncertainty associated with
quantum theory. Unfortunately, unlike Heisenberg’s Uncertainty Principle, this extra level
will be rather difficult to confirm experimentally in the case of black holes. But as I will
argue in my third lecture, there’s a sense in which we may have already observed it in the
measurements of fluctuations in the microwave background.
The fact that quantum theory causes black holes to radiate was first discovered by do-
ing quantum field theory on the background of a black hole formed by collapse. To see how
this comes about it is helpful to use what are normally called Penrose diagrams. However,
I think Penrose himself would agree they really should be called Carter diagrams because
Carter was the first to use them systematically. In a spherical collapse the spacetime won’t
depend on the angles θ and φ. All the geometry will take place in the r- t plane. Because any two dimensional plane is conformal to flat space one can represent the causal structure
by a diagram in which null lines in the r- t plane are at ± 45 degrees to the vertical.
I +
surfaces
(t=constant)
+
centre of
(r = ∞; t = +∞)
symmetry
r = 0
I 0
two spheres
(r=constant)
_ (r =∞; t = _ ∞)
_
I
Let’s start with flat Minkowski space. That has a Carter-Penrose diagram which is a
triangle standing on one corner. The two diagonal sides on the right correspond to the
past and future null infinities I referred to in my first lecture. These are really at infinity but all distances are shrunk by a conformal factor as one approaches past or future null