Page 13 - Nature Of Space And Time

limit null geodesic l

+
H (S)

round inside a compact set. They will approach a limit null geodesic that will have
no past or future end points in the Cauchy horizon. But if were geodesically complete
the generic energy condition would imply that it would contain conjugate points p and

q. Points on beyond p and q could be joined by a time like curve. But this would be
a contradiction because no two points of the Cauchy horizon can be time like separated.
Therefore either is not geodesically complete and the theorem is proved or the future

Cauchy development of S is not compact.
In the latter case one can show there is a future directed time like curve,
from S that
never leaves the future Cauchy development of S. A rather similar argument shows that

can be extended to the past to a curve that never leaves the past Cauchy development

D (S).
−
Now consider a sequence of point x n on
tending to the past and a similar sequence y n
tending to the future. For each value of n the points x n and y n are time like separated and

are in the globally hyperbolic Cauchy development of S. Thus there is a time like geodesic
of maximum length n from x n to y n .All the n will cross the compact space like surface
S. This means that there will be a time like geodesic in the Cauchy development which is
a limit of the time like geodesics n.Either will be incomplete, in which case the theorem

is proved. Or it will contain conjugate poin because of the generic energy condition. But
in that case n would contain conjugate points for n su ciently large. This would be
a contradiction because the n are supposed to be curves of maximum length. One can
therefore conclude that the spacetime is time like or null geodesically incomplete. In other

words there is a singularity.
The theorems predict singularities in two situations. One is in the future in the

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