THEOREM 3 .1. Timelike geodesic completeness, null geodesic completeness,
and spacelike geodesic completeness are logically inequivalent.
In a somewhat related area, Beem also studied another important issue in
space-time geodesic geometry at the time. A well-known result of Nomizu and
asserts that an arbitrary Riemannian metric for a smooth manifold
can be made geodesically complete by a conformal change of metric. We were all
exposed to at least a statement of this result in any basic differential geometry
course! On the other hand, the situation for space-times had been seen to be
more complicated. Misner
4] gave a 2-dimensional null geodesically incomplete
example which could not be made complete by any conformal change (since the
incomplete null geodesics were future trapped in a compact set and would remain
future trapped null pre geodesics under any conformal change of metric). Seifert [86]
showed that if ( M,
were stably causal, then a conformal change of metric could be
made which would produce future nonspacelike geodesic completeness. Clarke [46]
showed that a strongly causal space-time could be made null geodesically complete.
So that is the setting for Beem's paper
on "Conformal changes and geodesic
In this paper, Beem formulated what he termed "Condition N" for nonimpris-
DEFINITION 3.2. The causal space-time (M,g) will satisfy Condition N if, for
each compact subset K of M, there is no future inextendible nonspacelike curve
x(t) which is totally future imprisoned in K.
Here it should be noted that while in Riemannian geometry attention is often
simply restricted to geodesics, typically in General Relativity one has to consider
all nonspacelike curves, not just geodesics. Here also the curve x(t) is said to be
totally future imprisoned in K if there exists
so that x ( t)
K for all
By making a sequence of conformal changes related to a compact exhaustion
and taking the infinite product of those functions for the final conformal factor,
Beem established
THEOREM 3.3. Let (M,g) be a causal space-time which satisfies Condition N.
There is some conformal factor
such that (M,
g) is null and timelike geodesi-
cally complete.
As a corollary, it followed that if
(M, g)
were distinguishing, strongly causal,
stably causal or globally hyperbolic, then Condition N held, so that (
M, g)
be made nonspacelike geodesically complete by a conformal change of metric. In
Beem and Powell [33] an interesting study was made of Condition N for doubly
warped products.
Since conformal changes are being discussed, comments may also be made in
this section about the first collaboration of Beem and the first author in
formal deformations, Ricci curvature and energy conditions on globally hyperbolic
space-times." The first author's thesis research was originally motivated by efforts
to better understand a result of T. Aubin
which considered the question of
deforming a Riemannian metric of nonnegative Ricci curvature and all Ricci cur-
vatures positive at some point, to a metric of everywhere positive Ricci curvature.
Looking at
led to the study of local convex deformations: conformal de-
formations of a given Riemannian metric expressed in terms of the distance to the
boundary of a convex metric ball. It was found that if the given Ricci curvature was
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