A BEEMIAN SAMPLER:
1966~2002
17
for the concept of a nonspacelike asymptotic geodesic ray in which the point x cor-
responding to the point p above was allowed to vary in the limit construction:
DEFINITION
6 .1. A future
co~my
to "( from x will be a causal curve starting
at x which is future inextendible and is the limit curve of a sequence of maximal
length timelike geodesic segments from Xn to "f(rn) for two sequences {xn},
{rn}
with Xn--+ x and rn--+
+oo.
To cope with the technicalities discussed above, the concept of the timelike
co-my condition was also formulated.
DEFINITION
6.2. The globally hyperbolic space-time
(M, g)
satisfies the time-
like co-my condition for the timelike line "( : (
-oo, +oo)
--+ (
M,
g)
if, for each
x
in
I("()
=
I+("()
n
I-("(),
all future and past co-rays to "( from x are timelike.
Perhaps it is time to reveal the analytic definition of the Busemann function
corresponding to the future timelike geodesic ray "fl[o,+oo):
(6.2)
(b,)+(x)
=
lim
(r- d(x,"((r))).
r-oo
As mentioned in Section 4, the space-time distance function is generally less
tractable than the Riemannian distance function. Hence, even issues such as conti-
nuity are less obvious. However, it was established in
[25]
that the timelike co-ray
condition implied the continuity of the Busemann functions on
I("().
Moreover,
making the stronger hypothesis that all timelike sectional curvatures were nonpos-
itive, it was established that the timelike co-ray condition holds on all of
I("(),
so
that each of b+, b-, and B
=
b+
+
b- is continuous on I("(). Thanks to the aid of the
powerful Toponogov Theorem for globally hyperbolic space-times with nonpositive
timelike sectional curvatures, established in Harris
[62], [63],
it was also possible to
prove that all past and future timelike co-rays to the given timelike geodesic line
were complete. Hence, under the timelike sectional curvature hypothesis rather
than the more desirable Ricci curvature hypothesis, one had what the first author
liked to think of as "large scale control of the geometry on all of I ("f)." From this,
one could obtain the splitting of
I("()
as a metric product
(I("(), g)= (JR
X
H, -dt2 +h)
where
(H, h)
was any level set of the Busemann function in the induced metric.
(In Riemannian geometry, the corresponding level sets are called "horospheres.")
Finally, by inextendibility arguments, one deduced that
I("()
=
M.
What are some geometric issues hidden in the proofs involved in the B
=
b+ +b-
theory? Let"( be a complete timelike line as above and let
p
E
I("(). Form a future
timelike co-ray
c1
to "fl[o,+oo) and form a past timelike co-ray
c2
to "fl[o,-oo)' both
starting at
p.
Then the biggest geometric issue is, why does it happen that
(6.3)
so that c1 and c2 join together at p to form a smooth geodesic? Secondly, why is the
geodesic globally maximal? Once these things have been established, then one can
view the factor lR of the splitting as being formed geometrically by the collection
of all of these asymptotic past and future rays to "( fitting together properly and H
as any level set of the Busemann function.
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