neighbouring geodesics
p
meeting at q
Infinitesimally neighbouring null geodesics from p will intersect at q. This means the point q will be conjugate to p along the null geodesic γ joining them. For points on γ beyond the conjugate point q there will be a variation of γ that gives a time like curve from p.
Thus γ can not lie in the boundary of the future of p beyond the conjugate point q. So γ
will have a future end point as a generator of the boundary of the future of p.
The situation with time like geodesics is similar, except that the strong energy con-
dition that is required to make Rablalb non negative for every time like vector la is, as its name suggests, rather stronger. It is still however physically reasonable, at least in an averaged sense, in classical theory. If the strong energy condition holds, and the time like
geodesics from p begin converging again, then there will be a point q conjugate to p.
Finally there is the generic energy condition. This says that first the strong energy
condition holds. Second, every time like or null geodesic encounters some point where
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Strong Energy Condition
Tabvavb
≥ 1 vavaT
2
there is some curvature that is not specially aligned with the geodesic. The generic energy
condition is not satisfied by a number of known exact solutions. But these are rather
special. One would expect it to be satisfied by a solution that was ”generic” in an appro-
priate sense. If the generic energy condition holds, each geodesic will encounter a region
of gravitational focussing. This will imply that there are pairs of conjugate points if one
can extend the geodesic far enough in each direction.
The Generic Energy Condition
1. The strong energy condition holds.
2. Every timelike or null geodesic contains a point where l[ aRb] cd[ elf] lcld 6= 0.
One normally thinks of a spacetime singularity as a region in which the curvature
becomes unboundedly large. However, the trouble with that as a definition is that one
could simply leave out the singular points and say that the remaining manifold was the
whole of spacetime. It is therefore better to define spacetime as the maximal manifold on
which the metric is suitably smooth. One can then recognize the occurrence of singularities
by the existence of incomplete geodesics that can not be extended to infinite values of the
affine parameter.
Definition of Singularity
A spacetime is singular if it is timelike or null geodesically incomplete, but
can not be embedded in a larger spacetime.
This definition reflects the most objectionable feature of singularities, that there can be
particles whose history has a begining or end at a finite time. There are examples in which
geodesic incompleteness can occur with the curvature remaining bounded, but it is thought
that generically the curvature will diverge along incomplete geodesics. This is important if
one is to appeal to quantum effects to solve the problems raised by singularities in classical general relativity.
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Between 1965 and 1970 Penrose and I used the techniques I have described to prove
a number of singularity theorems. These theorems had three kinds of conditions. First
there was an energy condition such as the weak, strong or generic energy conditions. Then
there was some global condition on the causal structure such as that there shouldn’t be
any closed time like curves. And finally, there was some condition that gravity was so
strong in some region that nothing could escape.
Singularity Theorems
1. Energy condition.
2. Condition on global structure.
3. Gravity strong enough to trap a region.
This third condition could be expressed in various ways.
ingoing rays
converging
outgoing rays
outgoing rays
diverging
diverging
Normal closed 2 surface
ingoing and outgoing
rays converging
Closed trapped surface
One way would be that the spatial cross section of the universe was closed, for then there
was no outside region to escape to. Another was that there was what was called a closed
trapped surface. This is a closed two surface such that both the ingoing and out going null
geodesics orthogonal to it were converging. Normally if you have a spherical two surface