null geodesic generators can have past end points only on S. It is possible, however, to have spacetimes in which there are generators of the boundary of the future of a set S that never intersect S. Such generators can have no past end point.
A simple example of this is Minkowski space with a horizontal line segment removed.
If the set S lies to the past of the horizontal line, the line will cast a shadow and there will be points just to the future of the line that are not in the future of S. There will be a generator of the boundary of the future of S that goes back to the end of the horizontal 4
+
I (S)
.+
I
generator of (S)
with no end point on S
line removed from
Minkowski space
.+
I
generators of (S)
with past end point on S
S
line. However, as the end point of the horizontal line has been removed from spacetime,
this generator of the boundary will have no past end point. This spacetime is incomplete,
but one can cure this by multiplying the metric by a suitable conformal factor near the
end of the horizontal line. Although spaces like this are very artificial they are important
in showing how careful you have to be in the study of causal structure. In fact Roger
Penrose, who was one of my PhD examiners, pointed out that a space like that I have just
described was a counter example to some of the claims I made in my thesis.
To show that each generator of the boundary of the future has a past end point on
the set one has to impose some global condition on the causal structure. The strongest
and physically most important condition is that of global hyperbolicity.
q
+
_
I (p) ∩ I (q)
p
An open set U is said to be globally hyperbolic if:
1) for every pair of points p and q in U the intersection of the future of p and the past of q has compact closure. In other words, it is a bounded diamond shaped region.
2) strong causality holds on U . That is there are no closed or almost closed time like curves contained in U .
5
p
Σ(t)
every timelike curve
intersects (t)
Σ
The physical significance of global hyperbolicity comes from the fact that it implies
that there is a family of Cauchy surfaces Σ( t) for U . A Cauchy surface for U is a space like or null surface that intersects every time like curve in U once and once only. One can predict what will happen in U from data on the Cauchy surface, and one can formulate a well behaved quantum field theory on a globally hyperbolic background. Whether one can
formulate a sensible quantum field theory on a non globally hyperbolic background is less
clear. So global hyperbolicity may be a physical necessity. But my view point is that one
shouldn’t assume it because that may be ruling out something that gravity is trying to
tell us. Rather one should deduce that certain regions of spacetime are globally hyperbolic
from other physically reasonable assumptions.
The significance of global hyperbolicity for singularity theorems stems from the fol-
lowing.
q
geodesic of
p
maximum length
6
Let U be globally hyperbolic and let p and q be points of U that can be joined by a time like or null curve. Then there is a time like or null geodesic between p and q which maximizes the length of time like or null curves from p to q. The method of proof is to show the space of all time like or null curves from p to q is compact in a certain topology.
One then shows that the length of the curve is an upper semi continuous function on this
space. It must therefore attain its maximum and the curve of maximum length will be a
geodesic because otherwise a small variation will give a longer curve.
p
non-minimal
minimal geodesic
q
geodesic
without conjugate points
geodesic γ
r
point conjugate
to p along γ
q
neighbouring
geodesic