A celebrated early example of ( 4.3) is the Topological Sphere Theorem of Rie-
mannian geometry of the 1950s and 1960s: suppose that a complete, simply con-
nected Riemannian manifold admits a metric whose sectional curvatures vary be-
tween 1/4 and 1, but are always strictly greater than 1/4. Then the manifold M
must be homeomorphic to the standard round sphere of the same dimension as the
given manifold.
In contrast with the Riemannian situation, we have already remarked in Sec-
tion 1 that many standard examples of space-times fail to be nonspacelike geodesi-
cally complete. As we thought about ( 4.3) and certain of the singularity theorems
already published in General Relativity from
the viewpoint of differential geometry,
the following contrasting pattern (4.4) emerged:
(4.4) curvature inequality (gravitation is attractive)
physical or geometric assumption
the existence of an incomplete timelike or nonspacelike geodesic.
As we were working on the concept of causal disconnection, we also had as a
motivation that some cosmological and physical models are not globally hyperbolic,
but are only strongly causal. Thus we wanted to establish a formalism in the more
general strongly causal setting for which (a) the space-time distance function may
be less tractable than for globally hyperbolic space-times, and also (b) the tool
of maximal nonspacelike geodesic connectability is not available. (Later, in the
context of the Lorentzian splitting theorem in the timelike geodesically complete
case, Newman
and Galloway and Horta
would find the use of almost max-
imizers essential.) We were also motivated by the theory of the end structure for
Riemannian or topological manifolds in establishing this concept. Strong causality
did at least have the virtue that convergence in the limit curve sense and conver-
gence in the
0 topology on curves were closely related, as well as the fact that the
upper semi-continuity of arc length in the
topology fits in well with the lower
semi-continuity of the space-time distance function
Beem, Ehrlich and Easley
Section 3.3].
As the concept was stated in
p. 283:
4.1. A space-time (M,g) is said to be causally disconnected by
a compact set K if there exist two infinite sequences {Pn} and { qn} diverging to
infinity such that for each n, Pn
qn, Pn
qn, and all future directed nonspacelike
curves from Pn to qn meet K.
This definition as finally formulated had the virtue that all Lorentzian metrics
in the conformal class C(M,g) are causally disconnected if any one of them is,
unlike the original formulation in Beem and Ehrlich
which also assumed the
finite distance condition
(4.5) 0 d(pn, qn) +oo for all n.
This finiteness assumption made it possible to use a simpler limit curve con-
struction procedure, but unfortunately this earlier version of causal disconnection
was only conformally invariant in the case that ( M,
was globally hyperbolic, so
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