If Case 3 or 4 occurs, astronomers speak of multiple imaging. We first demonstrate that
Case 4 is
exceptional. It is easy to prove (see, e.g., [268], Proposition 12)
that no finite segment of the timelike curve
can be contained in the past light cone of
. Thus, if there is a continuous one-parameter family
of lightlike geodesics that connect
and
, then all family members meet
at the same point, say
. This point must be
in the caustic of the light cone because through all non-caustic
points there is only a discrete number of generators. One can
always find a point
arbitrarily close to
such that
does not meet the caustic of
the past light cone of
(see, e.g., [268
], Proposition 10).
Hence, by an arbitrarily small perturbation of
one can always destroy a Case 4 situation. One
may interpret this result as saying that Case 4 situations have
zero probability. This is, indeed, true as long as we consider
point sources (worldlines). The observed rings and arcs refer to
extended sources (worldtubes) which are close to the caustic
(recall Section 2.5). Such situations occur
with non-zero probability.
We will now show how multiple imaging is related
to the notion of cut points (recall Section 2.7). For any point in an arbitrary spacetime, the following criteria
for multiple imaging hold:
(C1) Let be a past-pointing
lightlike geodesic from
and let
be a point on
beyond the cut point or
beyond the first conjugate point. Then there is a timelike curve
through
that can be reached from
along a second past-pointing lightlike geodesic.
(C2) Assume that at the past-distinguishing condition (57
) is satisfied. If a
timelike curve
can be reached from
along two different past-pointing lightlike
geodesics, at least one of them passes through the cut locus of the
past light cone of
on or before arriving at
.
For proofs see [267] or [268
]. (In [267]
Criterion (C2) is formulated with the strong
causality condition, although the past-distinguishing condition is
sufficient.) Criteria (C1) and (C2) say that
the occurrence of cut points is sufficient and, in
past-distinguishing spacetimes, also necessary for multiple
imaging. The occurrence of conjugate points is sufficient but, in
general, not necessary for multiple imaging (see Figure 24
for an example
without conjugate points where multiple imaging occurs). In
Section 3.1 we will see that in globally
hyperbolic spacetimes conjugate points are necessary for multiple
imaging. So we have the following diagram:
|
|
|
|
|
|
Occurrence of: | Sufficient for multiple imaging in: | Necessary for multiple imaging in: |
|
|
|
cut point | arbitrary spacetime | past-distinguishing spacetime |
conjugate point | arbitrary spacetime | globally hyperbolic spacetime |
|
|
|
|
|
|
It is well known (see [154], in particular
Proposition 4.4.5) that, under conditions which are to be
considered as fairly general from a physical point of view, a
lightlike geodesic must either be incomplete or contain a pair of
conjugate points. These “fairly general conditions” are, e.g., the
weak energy condition and the so-called generic condition
(see [154
] for details). This
result implies the occurrence of conjugate points and, thus, of
multiple imaging, for a large class of spacetimes.
The occurrence of conjugate points has an
important consequence in view of the focusing equation for the area
distance (recall Section 2.4 and, in particular,
Equation (44
)). As
vanishes at the vertex
and at each
conjugate point, there must be a parameter value
with
between the vertex and the
first conjugate point. An elementary evaluation of the focusing
equation (44
) then implies