### 3.2 Wave fronts in globally hyperbolic spacetimes

In Section 2.2 the notion of wave fronts was discussed in an arbitrary spacetime . It was
mentioned that a wave front can be viewed as a subset of the space of all lightlike geodesics in
. This approach is particularly useful in globally hyperbolic spacetimes, as was demonstrated by
Low [210, 211]. The construction is based on the observations that, if is globally hyperbolic and
is a smooth Cauchy surface, the following is true:
(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.