6
PAUL E. EHRLICH AND KEVIN L. EASLEY
After serving
as
Associate Chairman under Chairman Joseph Zemmer, Beem
headed to Berkeley in the early 1970s while the first author was in Paris and Bonn.
At Berkeley, R. Sachs and
K.
Wu would probably have been collaborating on their
Bulletin survey article and influential monograph General Relativity for Math-
ematicians [83], referred to in Table
1.
Beem told the first author that he ap-
proached Sachs and asked him for suggestions about suitable open research ques-
tions. From his time in Berkeley, [12] and [13] would result.
Before discussing these results, we must first review a wonderful way in which
semi-Riemannian manifolds differ from definite metric manifolds, from the view-
point of the geodesic equation. When considering perturbations of Riemannian and
semi-Riemannian metrics, attention is often restricted to conformal changes of the
metric.
If
0
M
-+
(0, +oo) is a given smooth function, then
(3.1)
is called a conformal change of metric. (The factor of 2 produces pleasant curvature
and connection formulae.) Note that in the space-time case, the null, timelike and
spacelike tangent vectors for both
g
and g are the same; hence the basic causality
conditions like chronological, strongly causal, globally hyperbolic, etc., hold for
both (M,g) and (M,g) simultaneously.
A second aspect, absent for the definite case for which there are no null vectors,
is that null geodesics for (M,
g)
remain null pregeodesics for (M, g). To see this,
write the conformal factor in the form
(3.2)
Then the Levi-Civita connections V' and V' for (M,g) and (M,g) are related by
(3.3)
V'
x
Y
=
V'
x
Y
+
X(f)Y
+
Y(f)X- g(X, Y) grad(!).
Especially, if X is a null vector field on M (so that g(X, X)= 0), then
(3.4)
Y'xX
=
Y'xX
+
2X(f)X.
Thus, if (3 is a null geodesic on (M,
g)
and V' /3' (3'
=
0, it follows that
(3.5)
V' /3'(3'
=
2(3'(t)(f)f3'(t)
and it is known that if such an equation holds, then (3 may be reparametrized to be
a null geodesic of (M,g). For Riemannian manifolds, by contrast, the last gradient
term in equation (3.3) will not vanish if the gradient is nonzero, and hence geodesics
never persist under general conformal changes of metric.
With this background established, we can report that in Beem [13], an ex-
ample was given employing a conformal change involving an infinite product of
factors. These factors were supported on suitably chosen subsets of a globally hy-
perbolic space-time to preserve timelike and spacelike geodesic completeness, but to
produce null geodesic incompleteness for the deformed space-time metric. Hence,
combining [13] with the examples of these earlier authors, no two types of geodesic
completeness (or incompleteness) imply the third type. Beem liked to phrase this
as
follows:
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