  A change of spatial topology in a causal, compact spacetime cannot occur when
the metric is globally Lorentzian. One can however construct a causal metric
from a Riemannian metric and a Morse function on the background cobordism
manifold, which is Lorentzian almost everywhere except that it is degenerate at
each critical point of the function. We investigate causal structure in the
neighbourhood of such a degeneracy, when the auxiliary Riemannian metric is
taken to be Cartesian flat in appropriate coordinates. For these geometries, we
verify Borde and Sorkin's conjecture that causal discontinuity occurs if and
only if the Morse index is 1 or n-1.
