nonnegative, then positive Ricci curvature could be produced in an annular region
of the boundary of the convex metric ball, cf. Ehrlich
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
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
the analogous intrinsic
construction of Ehrlich
could not be employed. Instead, in
a convex
normal neighborhood
centered at p with local coordinates x
, ... ,
was employed, and the auxiliary (nonintrinsic) distance function
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
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
(p, q)
is finite for all
p, q
Let ( M,
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,
(4.1) d(p,q)
c: [0, 1]--+ M is a piecewise smooth,
nonspacelike curve with c(O)
p and c(1)
Then this defines what some authors term the Lorentzian distance function
(4.2) d
d(g): M
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,
from above by
L(c) for any selected curve c.) When Beem and the first author surveyed the
scene after completing
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
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