Theoretical cosmology likes to make use of a "conformal boundary"
for various spacetime models; this means mapping (via diffeomorphism on
the image) a "physical" spacetime M into a "non-physical" spacetime M',
such that the mapping is conformal (preserved causality) and the image
of M has compact closure in M'--hence, M gets endowed with a boundary,
inheriting geometry from M'. One of the desireable properties of such a
boundary is that it be complete in an appropriate sense. A recent
observation is that this is always possible for a static, spherically
symmetric spacetime (or, at any rate, for physically reasonable ones).

But
a conformal boundary is inherently a very ad hoc construction. Is
there any way to do this "naturally"? The answer is yes: The causal
boundary construction is entirely natural (in a categorical sense), and
it is well understood for static, spherically symmetric spacetimes. The
problem is that the causal boundary comes with a topology, but not a
geometry--and it is geometry that is needed to answer the question about
completeness.

This talk illustrates a method
of imputing a linear connection on the causal boundary of a static,
spherically symmetric spacetime. There is not a unique way to do so,
but all ways in this method agree on completeness.