a key ingredient rediscovered by Cheeger and Gromoll for use in their proof of the
Riemannian splitting theorem. The first author had studied Busemann functions
during his stay in Bonn in connection with manifolds of negative curvature, and of
course Beem was a student of Busemann himself, familiar with the text Busemann
Since we had decided to take the course of action of exploring the Busemann
function of a timelike geodesic ray, the first author returned once again to Cheeger
and Gromoll
which he had always found a difficult paper to understand.
With the great emphasis on ellipticity of the Laplacian, while the d'Alembertian of
General Relativity is hyperbolic, it also seemed like a daunting task to make any
of this transform to the space-time case. Fortunately, just before we began our
studies, we received an unexpected preprint in the mail which was published later
as Eschenburg and Heintze
At a first glance, it looked like an approach to the
splitting problem that had a good chance of adapting to the space-time setting,
and so with Markvorsen we set to work.
Many standard elementary methods in basic Riemannian geometry, in the con-
text of constructing asymptotic geodesics, rely on the compactness of the set of
unit vectors based at a given point in the manifold. For space-times, however, the
set of future unit timelike vectors based at a point (while closed) is noncompact.
A further difficulty is that a sequence
of unit timelike tangent vectors can never converge to a null vector n, even though
the set of nonspacelike directions in TpM is itself compact, so that
Vn) ---+
direction( n)
is indeed possible.
Here is the basic type of construction that is involved in both the Riemannian
and Lorentzian setting. Let
1: [O,+oo)---+(M,g)
be a unit timelike geodesic ray,
suppose that
d(1(0),1(t)) for all
Take any p in M with p in the chronological past of 1 and any sequence
0 with sn
---+ +oo.
Assuming that
(M, g)
is globally hyperbolic, construct unit
speed maximal timelike geodesic segments
Then in
the Riemannian case, one could turn to the sequence of unit vectors {en' (0)} and
extract a convergent subsequence to define an asymptotic geodesic c to 1 starting
But as we have indicated above, in the space-time case such a convergence is
not guaranteed.
However, already at hand is the limit curve machinery for strongly causal space-
times mentioned in Section 4. So instead, one may let c be a nonspacelike limit
curve of the timelike geodesic segments
Then two issues which must be dealt
with are: (i) why is c timelike rather than null, and (ii) why is c future complete?
Inspired by the somewhat more general approach taken to the asymptotic ge-
odesic construction in Busemann [38] ( cf. Busemann
for apparently the first
appearance of what would later be termed by others the Busemann function), in
Beem, Ehrlich, Markvorsen, and Galloway
the following definition was adopted
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