In spite of their high idealization, plane gravitational waves are interesting mathematical models for studying the lensing effect of gravitational waves. In particular, the focusing effect of plane gravitational waves on light rays can be studied quite explicitly, without any weak-field or small-angle approximations. This focusing effect is reflected by an interesting light cone structure.
The basic features with relevance to lensing can
be summarized in the following way. If the profile functions and
are differentiable, and the coordinates
range over
, the spacetime with
the metric (156
) is geodesically
complete [92].
With the exception of the integral curves of
, all inextendible lightlike geodesics contain a pair
of conjugate points. Let
be the first conjugate point
along a past-oriented lightlike geodesic from an observation event
. Then the first caustic of the past light cone of
is a parabola through
. (It depends on the
profile functions
and
whether or not there are more
caustics, i.e., second, third, etc. conjugate points.) This
parabola is completely contained in a hyperplane
. All light rays through
, with the exception
of the integral curve of
, pass through this parabola.
In other words, the entire sky of
, with the exception
of one point, is focused into a curve (see Figure 29
). This astigmatic focusing effect of plane
gravitational waves was discovered by Penrose [259
] who worked out the
details for “sufficiently weak sandwich waves”. (The name “sandwich
wave” refers to the case that
and
are different from zero only in a finite interval
.) Full proofs of the above statements,
for arbitrary profile functions
and
, were given by Ehrlich and Emch [94, 95]
(cf. [25
], Chapter 13).
The latter authors also demonstrate that plane gravitational wave
spacetimes are causally continuous but not causally simple. This
strengthens Penrose’s observation [259
] that they are not
globally hyperbolic. (For the hierarchy of causality notions
see [25].) The generators of
the light cone leave the boundary of the chronological past
when they reach the caustic. Thus, the
above-mentioned parabola is also the cut locus of the past light
cone. By the general results of Section 2.8, the occurrence of a cut locus
implies that there is multiple imaging in the plane-wave spacetime.
The number of images depends on the profile functions. We may
choose the profile functions such that there is no second caustic.
(The “sufficiently weak sandwich waves” considered by
Penrose [259
] are of this kind.)
Then Figure 29
demonstrates that an
appropriately placed worldline (close to the caustic) intersects
the past light cone exactly twice, so there is double-imaging.
Thus, the plane waves demonstrate that the number of images need
not be odd, even in the case of a geodesically complete spacetime
with trivial topology.
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One often considers profile functions and
with Dirac-delta-like singularities
(“impulsive gravitational waves”). Then a mathematically rigorous
treatment of the geodesic equation, and of the geodesic deviation
equation, is delicate because it involves operations on
distributions which are not obviously well-defined. For a detailed
mathematical study of this situation see [310, 193].
Garfinkle [132] discovered an interesting example for a pp-wave which is singular on a 2-dimensional worldsheet. This exact solution of Einstein’s vacuum field equation can be interpreted as a wave that travels along a cosmic string. Lensing in this spacetime was numerically discussed by Vollick and Unruh [340].
The vast majority of work on lensing by gravitational waves is done in the weak-field approximation. For the exact treatment and in the weak-field approximation one may use Kovner’s version of Fermat’s principle (see Section 2.9), which has the advantage that it allows for time-dependent situations. Applications of this principle to gravitational waves have been worked out in the original article by Kovner [187] and by Faraoni [110, 111].