(N1) can be identified
with a sphere bundle over
. The identification is made
by assigning to each lightlike geodesic its tangent line at the
point where it intersects
. As every sphere bundle over
an orientable 3-manifold is trivializable,
is diffeomorphic to
.
(N2) carries a natural
contact structure. (This contact structure is also discussed, in
twistor language, in [262], volume II.)
(N3) The wave fronts are exactly
the Legendre submanifolds of .
Using Statement (N1), the
projection from to
assigns to each wave front
its intersection with
, i.e., an “instantaneous wave front” or
“small wave front” (cf. Section 2.2 for terminology). The points
where this projection has non-maximal rank give the caustic of the
small wave front. According to the general stability results of
Arnold (see [11]), the only
caustic points that are stable with respect to local perturbations
within the class of Legendre submanifolds are cusps and
swallow-tails. By Statement (N3), perturbing within the class of
Legendre submanifolds is the same as perturbing within the class of
wave fronts. For this local stability result the assumption of
global hyperbolicity is irrelevant because every spacelike
hypersurface is a Cauchy surface for an appropriately chosen
neighborhood of any of its points. So we get the result that was
already mentioned in Section 2.2: In an arbitrary spacetime, a
caustic point of an instantaneous wave front is stable if and only
if it is a cusp or a swallow-tail. Here stability refers to
perturbations that keep the metric and the hypersurface fixed and
perturb the wave front within the class of wave fronts. For a
picture of an instantaneous wave front with cusps and a
swallow-tail point, see Figure 28
. In Figure 13
, the caustic points
are neither cusps nor swallow-tails, so the caustic is unstable.