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 oneparameter 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 noncaustic 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 nonzero 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 pastpointing 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 pastpointing lightlike geodesic.
(C2) Assume that at the pastdistinguishing condition (57) is satisfied. If a timelike curve can be reached from along two different pastpointing 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 pastdistinguishing condition is sufficient.) Criteria (C1) and (C2) say that the occurrence of cut points is sufficient and, in pastdistinguishing 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  pastdistinguishing spacetime 
conjugate point  arbitrary spacetime  globally hyperbolic spacetime 






It is well known (see [153], 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 socalled generic condition (see [153] 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
As the Ricci term is related to the energy density via Einstein’s field equation, (58) gives an estimate of energydensityplusshear along the ray. If we observe a multiple imaging situation, and if we know (or assume) that we are in a situation where conjugate points are necessary for multiple imaging, we have thus an estimate on energydensityplusshear along the ray. This line of thought was worked out, under additional assumptions on the spacetime, in [250].http://www.livingreviews.org/lrr20049 
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