Criteria for multiple imaging

2.8 Criteria for multiple imaging

To investigate whether multiple imaging occurs in a spacetime $(ℳ, g)$ , we choose any point $pO$ (observation event) and any timelike curve $γS$ (wordline of light source) in $ℳ$ . The following cases are possible:

There is no past-pointing lightlike geodesic from $pO$ to $γS$ . Then the observer at $pO$ does not see any image of the light source $γS$ . For instance, this occurs in Minkowski spacetime for an inextendible worldline $γS$ that asymptotically approaches the past light cone of $pO$ .
There is exactly one past-pointing lightlike geodesic from $pO$ to $γS$ . Then the observer at $pS$ sees exactly one image of the light source $γS$ . This is the situation naively taken for granted in pre-relativistic astronomy.
There are at least two but not more than denumerably many past-pointing lightlike geodesics from $p O$ to $γ S$ . Then the observer at $p O$ sees finitely or infinitely many distinct images of $γO$ at his or her celestial sphere.
There are more than denumerably many past-pointing lightlike geodesics from $p$ to $γ$ . This happens, e.g., in rotationally symmetric situations where it gives rise to the so-called “Einstein rings” (see Section 4.3). It also happens, e.g., in plane-wave spacetimes (see Section 5.11).

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 $γS$ can be contained in the past light cone of $pO$ . Thus, if there is a continuous one-parameter family of lightlike geodesics that connect $pO$ and $γO$ , then all family members meet $γS$ at the same point, say $pS$ . 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 $′ pO$ arbitrarily close to $pO$ such that $γS$ does not meet the caustic of the past light cone of $′ pO$ (see, e.g., [268 ], Proposition 10). Hence, by an arbitrarily small perturbation of $pO$ 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 $pO$ in an arbitrary spacetime, the following criteria for multiple imaging hold:

(C1) Let $λ$ be a past-pointing lightlike geodesic from $pO$ and let $pS$ be a point on $λ$ beyond the cut point or beyond the first conjugate point. Then there is a timelike curve $γS$ through $p S$ that can be reached from $p O$ along a second past-pointing lightlike geodesic.

(C2) Assume that at $pO$ the past-distinguishing condition (57 ) is satisfied. If a timelike curve $γS$ can be reached from $pO$ along two different past-pointing lightlike geodesics, at least one of them passes through the cut locus of the past light cone of $pO$ on or before arriving at $γS$ .

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 [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 so-called 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 $Darea$ (recall Section 2.4 and, in particular, Equation (44 )). As $Darea$ vanishes at the vertex $s = 0$ and at each conjugate point, there must be a parameter value $sm$ with $˙ Darea(sm) = 0$ 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 energy-density-plus-shear 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 energy-density-plus-shear along the ray. This line of thought was worked out, under additional assumptions on the spacetime, in [250 ].