of distinctive observationally tested predictions. Not that string theory in its present form is either beautiful or complete.
For these reasons, I shall talk about general relativity in these lectures. I shall con-
centrate on two areas where gravity seems to lead to features that are completely different
from other field theories. The first is the idea that gravity should cause spacetime to have
a begining and maybe an end. The second is the discovery that there seems to be intrinsic
gravitational entropy that is not the result of coarse graining. Some people have claimed
that these predictions are just artifacts of the semi classical approximation. They say that
string theory, the true quantum theory of gravity, will smear out the singularities and will
introduce correlations in the radiation from black holes so that it is only approximately
thermal in the coarse grained sense. It would be rather boring if this were the case. Grav-
ity would be just like any other field. But I believe it is distinctively different, because
it shapes the arena in which it acts, unlike other fields which act in a fixed spacetime
background. It is this that leads to the possibility of time having a begining. It also leads to regions of the universe which one can’t observe, which in turn gives rise to the concept
of gravitational entropy as a measure of what we can’t know.
In this lecture I shall review the work in classical general relativity that leads to these
ideas. In the second and third lectures I shall show how they are changed and extended
2
when one goes to quantum theory. Lecture two will be about black holes and lecture three will be on quantum cosmology.
The crucial technique for investigating singularities and black holes that was intro-
duced by Roger, and which I helped develop, was the study of the global causal structure
of spacetime.
Chronological
future
+
I (p)
.
Null geodesic in (p) which
+
I
does not go back to p and has
no past end point
Point removed
from spacetime
Time
Null geodesics through p
p
.
generating part of
+
I (p)
Space
Define I+( p) to be the set of all points of the spacetime M that can be reached from p by future directed time like curves. One can think of I+( p) as the set of all events that can be influenced by what happens at p. There are similar definitions in which plus is replaced by minus and future by past. I shall regard such definitions as self evident.
.+
I (S)
+
I (S)
q
+
I (S)
q
.+
I (S)
timelike curve
p
All timelike curves from q leave +
I (S)
.+
I (S) can’t be timelike
+
I (S) can’t be spacelike
.
One now considers the boundary ˙
I+( S) of the future of a set S. It is fairly easy to
see that this boundary can not be time like. For in that case, a point q just outside the boundary would be to the future of a point p just inside. Nor can the boundary of the 3
future be space like, except at the set S itself. For in that case every past directed curve from a point q, just to the future of the boundary, would cross the boundary and leave the future of S. That would be a contradiction with the fact that q is in the future of S.
+
I (S)
q
.+
null geodesic segment in I (S)
.+
future end point of generators of I (S)
q
+
I (S)
.+
null geodesic segment in I (S)
One therefore concludes that the boundary of the future is null apart from at S itself.
More precisely, if q is in the boundary of the future but is not in the closure of S there is a past directed null geodesic segment through q lying in the boundary. There may be more than one null geodesic segment through q lying in the boundary, but in that case q will be a future end point of the segments. In other words, the boundary of the future of
S is generated by null geodesics that have a future end point in the boundary and pass into the interior of the future if they intersect another generator. On the other hand, the