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in Minkowski space the ingoing null geodesics are converging but the outgoing ones are diverging. But in the collapse of a star the gravitational field can be so strong that the
light cones are tipped inwards. This means that even the out going null geodesics are
converging.
The various singularity theorems show that spacetime must be time like or null
geodesically incomplete if different combinations of the three kinds of conditions hold.
One can weaken one condition if one assumes stronger versions of the other two. I shall
illustrate this by describing the Hawking-Penrose theorem. This has the generic energy
condition, the strongest of the three energy conditions. The global condition is fairly weak, that there should be no closed time like curves. And the no escape condition is the most
general, that there should be either a trapped surface or a closed space like three surface.
H (S)
+
D (S)
+
every past directed
q
timelike curve from q
intersects S
S
For simplicity, I shall just sketch the proof for the case of a closed space like three
surface S. One can define the future Cauchy development D+( S) to be the region of points q from which every past directed time like curve intersects S. The Cauchy development is the region of spacetime that can be predicted from data on S. Now suppose that the future Cauchy development was compact. This would imply that the Cauchy development
would have a future boundary called the Cauchy horizon, H+( S). By an argument similar to that for the boundary of the future of a point the Cauchy horizon will be generated by
null geodesic segments without past end points.
However, since the Cauchy development is assumed to be compact, the Cauchy horizon
will also be compact. This means that the null geodesic generators will wind round and
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limit null geodesic λ
H (S)
+
round inside a compact set. They will approach a limit null geodesic λ that will have no past or future end points in the Cauchy horizon. But if λ were geodesically complete the generic energy condition would imply that it would contain conjugate points p and q. Points on λ 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 λ 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, γ from S that never leaves the future Cauchy development of S. A rather similar argument shows that γ can be extended to the past to a curve that never leaves the past Cauchy development D−( S).
Now consider a sequence of point xn on γ tending to the past and a similar sequence yn tending to the future. For each value of n the points xn and yn are time like separated and are in the globally hyperbolic Cauchy development of S. Thus there is a time like geodesic of maximum length λn from xn to yn. All the λn will cross the compact space like surface S. This means that there will be a time like geodesic λ in the Cauchy development which is a limit of the time like geodesics λn. Either λ 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 λn would contain conjugate points for n sufficiently large. This would be a contradiction because the λn 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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H (S)
+
point at infinity
D (S)
+
timelike curve γ
S
_
D (S)
point at infinity
_
H (S)
limit geodesic λ
yn
xn
gravitational collapse of stars and other massive bodies. Such singularities would be an
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end of time, at least for particles moving on the incomplete geodesics. The other situation