Wave fronts in globally hyperbolic spacetimes

3.2 Wave fronts in globally hyperbolic spacetimes

In Section 2.2 the notion of wave fronts was discussed in an arbitrary spacetime $(ℳ, g)$ . It was mentioned that a wave front can be viewed as a subset of the space $𝒩$ of all lightlike geodesics in $(ℳ, g)$ . 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 $(ℳ, g)$ is globally hyperbolic and $𝒞$ is a smooth Cauchy surface, the following is true:

(N1) $𝒩$ can be identified with a sphere bundle over $𝒞$ . The identification is made by assigning to each lightlike geodesic its tangent line at the point where it intersects $𝒞$ . As every sphere bundle over an orientable 3-manifold is trivializable, $𝒩$ is diffeomorphic to $𝒞 × S2$ .

(N2) $𝒩$ 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 $𝒩$ .

Using Statement (N1), the projection from $𝒩$ to $𝒞$ assigns to each wave front its intersection with $𝒞$ , 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.