8

PAUL E. EHRLICH AND KEVIN L. EASLEY

nonnegative, then positive Ricci curvature could be produced in an annular region

of the boundary of the convex metric ball, cf. Ehrlich

[50].

Since the distance from

a point prior to the cut locus is nicely related to the index form, rather precise and

detailed calculations and estimates could be made.

In Beem and Ehrlich

[17],

this situation was studied for globally hyperbolic

space-times, where the situation was found to be rather more intricate. Since the

intrinsic metric balls given by the Lorentzian distance function are noncompact

and generally go off to infinity ( cf. Figure 4.4 in

[24]),

the analogous intrinsic

construction of Ehrlich

[50]

could not be employed. Instead, in

[17],

a convex

normal neighborhood

B

centered at p with local coordinates x

=

(x1

,

x

2

, ... ,

Xn)

was employed, and the auxiliary (nonintrinsic) distance function

(3.6)

was used to construct deformations with support in B. As might be expected,

given this combination of intrinsic and auxiliary geometries, it was a more technical

problem to calculate and study the Ricci curvature of the deformed metric. In

contrast to the nicer Riemannian situation, where positive Ricci curvature was

produced in a whole annular neighborhood of the boundary of the convex ball, it

was found in the relativistic setting that positive Ricci curvature could only be

guaranteed near the "north polar cap."

4. The Lorentzian Distance Function and Causal Disconnection

Even if it has been ten years or more since the reader has been the recipient of

a graduate course in Riemannian geometry, she or he will no doubt still recall the

pleasant properties that for a complete Riemannian manifold

(N,

g0

),

the Riemann-

ian distance function d0 is continuous, and moreover, the metric topology induced

by d0 coincides with the given manifold topology. What is even more remarkable

and perhaps less often remembered is that these properties are equally valid for an

arbitrary incomplete Riemannian metric. Furthermore, it is taken for granted that

d0

(p, q)

is finite for all

p, q

E

N.

Let ( M,

g)

be an arbitrary space-time and let

p, q

be two points of M. If there

is no future nonspacelike curve from p to q, set d(p, q)

=

0; if there is such a curve,

let

(4.1) d(p,q)

=

sup{L(c)

I

c: [0, 1]--+ M is a piecewise smooth,

nonspacelike curve with c(O)

=

p and c(1)

=

q}.

Then this defines what some authors term the Lorentzian distance function

(4.2) d

=

d(g): M

x

M--+

[O,+oo]

and other more physically motivated authors term proper time. (Note that unlike

the Riemannian case, (4.1) does not bound the values of d(p,

q)

from above by

L(c) for any selected curve c.) When Beem and the first author surveyed the

scene after completing

[17],

it seemed that time was ripe for a more systematic

exposition of the properties and uses of the space-time distance function. It had

received some discussion in the monographs already published (see Table 1), and

in some research papers. But confusion was found between nonspacelike conjugate

points and nonspacelike cut points in certain aspects of the timelike index theory