A BEEMIAN SAMPLER: 1966-2002
11
condition (4.5) was later dropped. As pointed out on Figure 8.1 of
[24],
non-globally
hyperbolic space-times may be causally disconnected.
By using local distance functions related to a compact exhaustion {
Bn}
of
M,
almost maximizers related to the local distance functions, and taking limits (thanks
to strong causality), the following result was obtained.
THEOREM
4.2. Let (M,
g)
denote a strongly causal space-time which is causally
disconnected by a compact set K. Then M contains a nonspacelike geodesic line
which intersects
K.
Here, a line is a past and future inextendible nonspacelike geodesic which real-
izes the Lorentzian distance function between every pair of its points. Now, unlike
the familiar situation in the global Riemannian geometry of complete Riemannian
manifolds, no assertion is being made here that the line is geodesically complete,
only that it is inextendible.
With this result in hand, a singularity theorem fitting the pattern of (4.4) could
be obtained:
THEOREM
4.3. Let (M,g) be a chronological space-time of dimension greater
than or equal to three which is causally disconnected. If (M, g) satisfies the time-
like convergence condition and the generic condition, then (M,
g)
is nonspacelike
geodesically incomplete.
Lurking in the background is a well-used result in causality theory of General
Relativity that if ( M, g) is a chronological space-time such that each inextendible
null geodesic has a pair of conjugate points, then (M,g) is strongly causal (cf.
[24],
p. 467). In the statement of Theorem 4.3, we find two of the curvature
conditions traditionally imposed in this branch of General Relativity. The first,
the timelike convergence condition, is simply stated as Ric( v, v)
2::
0 for all timelike
tangent vectors v (hence, by continuity, the same condition holds for all nonspacelike
tangent vectors). The second condition is somewhat more mysterious to differential
geometers, and on a first pass can be paraphrased as the assertion that every
inextendible nonspacelike geodesic has some suitable nonzero sectional curvature.
Indeed, formulating this condition in various ways easier for differential geometers
to understand was done in Beem and Ehrlich
[21]
and Beem, Ehrlich and Easley
[24];
cf. Beem and Parker
[32]
among others for a discussion of the physical aspects
of this condition.
In any event, the generic condition in the more unfriendly language of tensor
calculus is discussed in the first author's second favorite passage in Hawking and
Ellis
([64],
p. 101), where K denotes the tangent vector to the null geodesic under
consideration:
"As in the timelike case, this condition will be satisfied for
a null geodesic which passes through some matter provided that
the matter is not pure radiation (energy-momentum tensor type
II of §4.3) and moving in the direction of the geodesic tangent
vector
K.
It will be satisfied in empty space if the null geodesic
contains some point where the Weyl tensor is non-zero and where
K does not lie in one of the directions (there are at most four such
directions) at that point for which
Kc Kd K[aCb]cd[eK!]
=
0. It
therefore seems reasonable to assume that in a physically realistic
solution every timelike or null geodesic will contain a point at which
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