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.