A BEEMIAN SAMPLER: 1966-2002
9
as explained in Hawking and Ellis [64] (indeed, nonspacelike cut points had not
yet been formulated in the literature). An intrinsic Morse index theorem for null
geodesic segments in arbitrary space-times had not yet been published, despite
several works such as Uhlenbeck [87] and Woodhouse [93], among others.
Working some of these issues out was accomplished in Beem and Ehrlich [18],
[19], [20], and in greater detail in the First Edition of Global Lorentzian Geom-
etry [21]. In place of the complete metric of Riemannian geometry, what emerged
was an interesting interplay between the causal properties of the given space-time
and the continuity (and other properties) of the Lorentzian distance function. Philo-
sophically, this aspect emerges since
d(p, q)
0 iff
q
E
J+(p).
For example, at the
one extreme of
totally vicious space-times,
the Lorentzian distance always takes on
the value +oo ( cf. p. 137 in [24]). Less drastically, if (
M, g)
contains a closed time-
like curve passing through
p,
then
d(p,
q)
=
+oo for all q
E
J+(p).
In an allied vein,
a space-time
(M, g)
is chronological iff its distance function vanishes identically on
the diagonal
!:!.(M)
=
{(p,p) : p EM}.
In general, the Lorentzian distance function is only lower semi-continuous. The
strength of the additional property of upper semi-continuity is seen in the result
that if a distinguishing space-time has a continuous distance function, then it is
causally continuous.
At the other extreme from totally vicious space-times in the hierarchy of
causality are globally hyperbolic space-times. In some sense, these space-times
share many of the properties of complete Riemannian manifolds. For instance, the
Lorentzian distance function of a globally hyperbolic space-time is both continu-
ous and finite-valued. (Indeed, it may be shown that a strongly causal space-time
( M,
g)
is globally hyperbolic iff all Lorentz metrics
g'
in the conformal class C ( M,
g)
also have finite-valued distance functions
d(g').)
Second, for globally hyperbolic
space-times, Seifert [85] and others had established the important working tool of
maximal nonspacelike geodesic connectability:
given any
p,
q
E
M with
p
~
q, there
exists a nonspacelike geodesic segment
c :
[0, 1]---+
(M,g)
with c(O)
=
p,
c(1)
=
q,
and
L(c)
=
d(p, q),
in exact analogy with property (5) of the Hopf-Rinow Theorem
stated in Section 1. In view of the last several properties of the distance function,
the theories of the timelike and null cut loci were studied for globally hyperbolic
space-times in Beem and Ehrlich [18].
Now let us turn to the main aim of this section, the discussion of causally
disconnected space-times. Here is a general pattern which is common in global
Riemannian geometry:
(4.3) complete Riemannian metric
and
curvature inequality
implies
topological or geometric conclusion.
Previous Page Next Page