Page 4 - Nature Of Space And Time

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

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