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.