Clearly, a light cone is a special case of a
wave front. One gets this special case by choosing for 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 [180]. 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 on
. One chooses a parameter value
and parametrizes each generator
affinely such that
and
depends smoothly on the foot-point in
. This gives the wave front as the image of a map
Near a non-caustic point, a wave front is a
hypersurface where
satisfies the Hamilton-Jacobi 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
(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
has non-maximal
rank. For the case
, 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.
|
Friedrich and Stewart [118] 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 [150] where the
Lagrangian rather than the Legendrian formalism was used. However,
the main result of this paper (Theorem 4.4 of [150]) 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
. As an alternative
terminology, one calls the intersection of a (“big”) wave front
with a hypersurface
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
. 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 [210
]. In this article,
the author views wave fronts as subsets of the space
of all lightlike geodesics in
. General properties of this space
are derived in earlier articles by Low [208, 209
] (also see Penrose
and Rindler [262
], volume II,
where the space
is treated in twistor language). Low
considers, in particular, the case of a globally hyperbolic
spacetime [210
]; 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 [33]. 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.,
[226] for an overview). In
his millennium essay on future perspectives of gravitational
lensing, Blandford [34]
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 [128].