Wave fronts

2.2 Wave fronts

Wave fronts are related to light rays as solutions of the Hamilton–Jacobi equation are related to solutions of Hamilton’s equations in classical mechanics. For the case at hand (i.e., vacuum light propagation in an arbitrary spacetime, corresponding to the Hamiltonian $ℋ = 1gij(x)pipj 2$ ), a wave front is a subset of the spacetime that can be constructed in the following way:

Choose a spacelike 2-surface $𝒮$ that is orientable.
At each point of $𝒮$ , choose a lightlike direction orthogonal to $𝒮$ that depends smoothly on the foot-point. (You have to choose between two possibilities.)
Take all lightlike geodesics that are tangent to the chosen directions. These lightlike geodesics are called the generators of the wave front, and the wave front is the union of all generators.

Clearly, a light cone is a special case of a wave front. One gets this special case by choosing for $𝒮$ an appropriate (small) sphere. Any wave front is the envelope of all light cones with vertices on the wave front. In this sense, general-relativistic wave fronts can be constructed according to the Huygens principle.

In the context of general relativity the notion of wave fronts was introduced by Kermack, McCrea, and Whittaker [179 ]. For a modern review article see, e.g., Ehlers and Newman [93 ].

A coordinate representation for a wave front can be given with the help of (local) coordinates $(u1, u2)$ on $𝒮$ . One chooses a parameter value $s 0$ and parametrizes each generator $λ$ affinely such that $λ (s0) ∈ 𝒮$ and $˙ λ(s0)$ depends smoothly on the foot-point in $𝒮$ . This gives the wave front as the image of a map

For light cones we may choose spherical coordinates, $(u1 = Ψ, u2 = Θ )$ , (cf. Equation (4

) with fixed $τ$ ). Near $s = s 0$ , map (6

) is an embedding, i.e., the wave front is a submanifold. Orthogonality to $𝒮$ of the initial vectors $˙ λ (s0)$ assures that this submanifold is lightlike. Farther away from $𝒮$ , however, the wave front need not be a submanifold. The caustic of the wave front is the set of all points where the map (6

) is not an immersion, i.e., where its differential has rank $< 3$ . As the derivative with respect to $s$ is always non-zero, the rank can be $3 − 1$ (caustic point of multiplicity one, astigmatic focusing) or $3 − 2$ (caustic point of multiplicity two, anastigmatic focusing). In the first case, the cross-section of an “infinitesimally thin” bundle of generators collapses to a line, in the second case to a point (see Section 2.3). For the case that the wave front is a light cone with vertex $pO$ , caustic points are said to be conjugate to $pO$ along the respective generator. For an arbitrary wave front, one says that a caustic point is conjugate to any spacelike 2-surface in the wave front. In this sense, the terms “conjugate point” and “caustic point” are synonymous. Along each generator, caustic points are isolated (see Section 2.3) and thus denumerable. Hence, one may speak of the first caustic, the second caustic, and so on. At all points where the caustic is a manifold, it is either spacelike or lightlike. For instance, the caustic of the Schwarzschild light cone in Figure 12

is a spacelike curve; in the spacetime of a transparent string, the caustic of the light cone consists of two lightlike 2-manifolds that meet in a spacelike curve (see Figure 25

Near a non-caustic point, a wave front is a hypersurface $S = constant$ where $S$ satisfies the Hamilton–Jacobi equation

In the terminology of optics, Equation (7

) is called the eikonal equation.

At caustic points, a wave front typically forms cuspidal edges or vertices whose geometry might be arbitrarily complicated, even locally. If one restricts to caustics which are stable against perturbations in a certain sense, then a local classification of caustics is possible with the help of Arnold’s singularity theory of Lagrangian or Legendrian maps. Full details of this theory can be found in [11 ]. For a readable review of Arnold’s results and its applications to wave fronts in general relativity, we refer again to [93 ]. In order to apply Arnold’s theory to wave fronts, one associates each wave front with a Legendrian submanifold in the projective cotangent bundle over $ℳ$ (or with a Lagrangian submanifold in an appropriately reduced bundle). A caustic point of the wave front corresponds to a point where the differential of the projection from the Legendrian submanifold to $ℳ$ has non-maximal rank. For the case $dim (ℳ ) = 4$ , which is of interest here, Arnold has shown that there are only five types of caustic points that are stable with respect to perturbations within the class of all Legendrian submanifolds. They are known as fold, cusp, swallow-tail, pyramid, and purse (see Figure 2 ). Any other type of caustic is unstable in the sense that it changes non-diffeomorphically if it is perturbed within the class of Legendrian submanifolds.

Figure 2: Wave fronts that are locally stable in the sense of Arnold. Each picture shows the projection into 3-space of a wave-front, locally near a caustic point. The projection is made along the integral curves of a timelike vector field. The qualitative features are independent of which timelike vector field is chosen. In addition to regular, i.e., non-caustic, points $(A ) 1$ , there are five kinds of stable points, known as fold $(A2 )$ , cusp $(A3 )$ , swallow-tail $(A4 )$ , pyramid $− (D 4 )$ , and purse $+ (D 4 )$ . The $Ak$ and $Dk$ notation refers to a relation to exceptional groups (see [11

]). The picture is taken from [149

Fold singularities of a wave front form a lightlike 2-manifold in spacetime, on a sufficiently small neighborhood of any fold caustic point. The second picture in Figure 2 shows such a “fold surface”, projected to 3-space along the integral curves of a timelike vector field. This projected fold surface separates a region covered twice by the wave front from a region not covered at all. If the wave front is the past light cone of an observation event, and if one restricts to light sources with worldlines in a sufficiently small neighborhood of a fold caustic point, there are two images for light sources on one side and no images for light sources on the other side of the fold surface. Cusp singularities of a wave front form a spacelike curve in spacetime, again locally near any cusp caustic point. Such a curve is often called a “cusp ridge”. Along a cusp ridge, two fold surfaces meet tangentially. The third picture in Figure 2 shows the situation projected to 3-space. Near a cusp singularity of a past light cone, there is local triple-imaging for light sources in the wedge between the two fold surfaces and local single-imaging for light sources outside this wedge. Swallow-tail, pyramid, and purse singularities are points where two or more cusp ridges meet with a common tangent, as illustrated by the last three pictures in Figure 2 .

Friedrich and Stewart [117 ] have demonstrated that all caustic types that are stable in the sense of Arnold can be realized by wave fronts in Minkowski spacetime. Moreover, they stated without proof that, quite generally, one gets the same stable caustic types if one allows for perturbations only within the class of wave fronts (rather than within the larger class of Legendrian submanifolds). A proof of this statement was claimed to be given in [149 ] where the Lagrangian rather than the Legendrian formalism was used. However, the main result of this paper (Theorem 4.4 of [149 ]) is actually too weak to justify this claim. A different version of the desired stability result was indeed proven by another approach. In this approach one concentrates on an instantaneous wave front, i.e., on the intersection of a wave front with a spacelike hypersurface $𝒞$ . As an alternative terminology, one calls the intersection of a (“big”) wave front with a hypersurface $𝒞$ that is transverse to all generators a “small wave front”. Instantaneous wave fronts are special cases of small wave fronts. The caustic of a small wave front is the set of all points where the small wave front fails to be an immersed 2-dimensional submanifold of $𝒞$ . If the spacetime is foliated by spacelike hypersurfaces, the caustic of a wave front is the union of the caustics of its small (= instantaneous) wave fronts. Such a foliation can always be achieved locally, and in several spacetimes of interest even globally. If one identifies different slices with the help of a timelike vector field, one can visualize a wave front, and in particular a light cone, as a motion of small (= instantaneous) wave fronts in 3-space. Examples are shown in Figures 13 , 18 , 19 , 27 , and 28 . Mathematically, the same can be done for non-spacelike slices as long as they are transverse to the generators of the considered wave front (see Figure 30 for an example). Turning from (big) wave fronts to small wave fronts reduces the dimension by one. The only caustic points of a small wave front that are stable in the sense of Arnold are cusps and swallow-tails. What one wants to prove is that all other caustic points are unstable with respect to perturbations of the wave front within the class of wave fronts, keeping the metric and the slicing fixed. For spacelike slicings (i.e., for instantaneous wave fronts), this was indeed demonstrated by Low [211 ]. In this article, the author views wave fronts as subsets of the space $𝒩$ of all lightlike geodesics in $(ℳ, g)$ . General properties of this space $𝒩$ are derived in earlier articles by Low [209 , 210 ] (also see Penrose and Rindler [262 ], volume II, where the space $𝒩$ is treated in twistor language). Low considers, in particular, the case of a globally hyperbolic spacetime [211 ]; he demonstrates the desired stability result for the intersections of a (big) wave front with Cauchy hypersurfaces (see Section 3.2). As every point in an arbitrary spacetime admits a globally hyperbolic neighborhood, this local stability result is universal. Figure 28 shows an instantaneous wave front with cusps and a swallow-tail point. Figure 13 shows instantaneous wave fronts with caustic points that are neither cusps nor swallow-tails; hence, they must be unstable with respect to perturbations of the wave front within the class of wave fronts.

It is to be emphasized that Low’s work allows to classify the stable caustics of small wave fronts, but not directly of (big) wave fronts. Clearly, a (big) wave front is a one-parameter family of small wave fronts. A qualitative change of a small wave front, in dependence of a parameter, is called a “metamorphosis” in the English literature and a “perestroika” in the Russian literature. Combining Low’s results with the theory of metamorphoses, or perestroikas, could lead to a classsification of the stable caustics of (big) wave fronts. However, this has not been worked out until now.

Wave fronts in general relativity have been studied in a long series of articles by Newman, Frittelli, and collaborators. For some aspects of their work see Sections 2.9 and 3.4. In the quasi-Newtonian approximation formalism of lensing, the classification of caustics is treated in great detail in the book by Petters, Levine, and Wambsganss [275 ]. Interesting related mateial can also be found in Blandford and Narayan [34 ]. For a nice exposition of caustics in ordinary optics see Berry and Upstill [28 ].

A light source that comes close to the caustic of the observer’s past light cone is seen strongly magnified. For a point source whose worldline passes exactly through the caustic, the ray-optical treatment even gives an infinite brightness (see Section 2.6). If a light source passes behind a compact deflecting mass, its brightness increases and decreases in the course of time, with a maximum at the moment of closest approach to the caustic. Such microlensing events are routinely observed by monitoring a large number of stars in the bulge of our Galaxy, in the Magellanic Clouds, and in the Andromeda Galaxy (see, e.g., [227 ] for an overview). In his millennium essay on future perspectives of gravitational lensing, Blandford [33 ] mentioned the possibility of observing a chosen light source strongly magnified over a period of time with the help of a space-born telescope. The idea is to guide the spacecraft such that the worldline of the light source remains in (or close to) the one-parameter family of caustics of past light cones of the spacecraft over a period of time. This futuristic idea of “caustic surfing” was mathematically further discussed by Frittelli and Petters [127 ].