Criteria for multiple imaging in globally hyperbolic spacetimes

3.1 Criteria for multiple imaging in globally hyperbolic spacetimes

In Section 2.7 we have considered the past light cone of an event $pO$ in an arbitrary spacetime. We have seen that conjugate points (= caustic points) indicate that the past light cone fails to be an immersed submanifold and that cut points indicate that it fails to be an embedded submanifold. In a globally hyperbolic spacetime $(ℳ, g)$ , the following additional statements are true.

(H1) The past light cone of any event $p O$ , together with the vertex ${p } O$ , is closed in $ℳ$ .

(H2) The cut locus of the past light cone of $pO$ is closed in $ℳ$ .

(H3) Let $pS$ be in the cut locus of the past light cone of $pO$ but not in the conjugate locus (= caustic). Then $pS$ can be reached from $pO$ along two different lightlike geodesics. The past light cone of $pO$ has a transverse self-intersection at $pS$ .

(H4) The past light cone of $pO$ is an embedded submanifold if and only if its cut locus is empty.

Analogous results hold, of course, for the future light cone, but the past version is the one that has relevance for lensing. For proofs of these statements see [25 ], Propositions 9.35 and 9.29 and Theorem 9.15, and [268 ], Propositions 13, 14, and 15. According to Statement (H3), a “cut point” indicates a “cut” of two lightlike geodesics. For geodesics in Riemannian manifolds (i.e., in the positive definite case), an analogous statement holds if the Riemannian metric is complete and is known as Poincaré theorem [281 , 350 ]. It was this theorem that motivated the name “cut point”. Note that Statement (H3) is not true without the assumption that $pS$ is not in the caustic. This is exemplified by the swallow-tail point in Figure 25 . However, as points in the caustic of the past light cone of $p O$ can be reached from $pO$ along two “infinitesimally close” lightlike geodesics, the name “cut point” may be considered as justified also in this case.

In addition to Statemens (H1) and (H2) one would like to know whether in globally hyperbolic spactimes the caustic of the past light cone of $pO$ (also known as the past lightlike conjugate locus of $pO$ ) is closed. This question is closely related to the question of whether in a complete Riemannian manifold the conjugate locus of a point is closed. For both questions, the answer was widely believed to be ‘yes’ although actually it is ‘no’. To the surprise of many, Margerin [216 ] constructed Riemannian metrics on the 2-sphere such that the conjugate locus of a point is not closed. Taking the product of such a Riemannian manifold with 2-dimensional Minkowski space gives a globally hyperbolic spacetime in which the caustic of the past light cone of an event is not closed.

In Section 2.8 we gave criteria for the number of past-oriented lightlike geodesics from a point $pO$ (observation event) to a timelike curve $γS$ (worldline of a light source) in an arbitrary spacetime. With Statements (H1), (H2), (H3), and (H4) at hand, the following stronger criteria can be given.

Let $(ℳ, g)$ be globally hyperbolic, fix a point $p O$ and an inextendible timelike curve $γ S$ in $ℳ$ . Then the following is true:

(H5) Assume that $γS$ enters into the chronological past $− I (pO )$ of $pO$ . Then there is a past-oriented lightlike geodesic $λ$ from $pO$ to $γS$ that is completely contained in the boundary of $I− (pO )$ . This geodesic does not pass through a cut point or through a conjugate point before arriving at $γ S$ .

(H6) Assume that $γS$ can be reached from $pO$ along a past-oriented lightlike geodesic that passes through a conjugate point or through a cut point before arriving at $γS$ . Then $γS$ can be reached from $pO$ along a second past-oriented lightlike geodesic.

Statement (H5) was proven in [326 ] with the help of Morse theory. For a more elementary proof see [268 ], Proposition 16. Statement (H5) gives a characterization of the primary image in globally hyperbolic spacetimes. (The primary image is the one that shows the light source at an older age than all other images.) The condition of $γS$ entering into the chronological past of $pO$ is necessary to exclude the case that $pO$ sees no image of $γS$ . Statement (H5) implies that there is a unique primary image unless $γS$ passes through the cut locus of the past light cone of $p O$ . The primary image has even parity. If the weak energy condition is satisfied, the focusing theorem implies that the primary image has magnification factor $≥ 1$ , i.e., that it appears brighter than a source of the same luminosity at the same affine distance and at the same redshift in Minkowski spacetime (recall Sections 2.4 and 2.6, in particular Equation (46 )).

For a proof of Statement (H6) see [268 ], Proposition 17. Statement (H6) says that in a globally hyperbolic spacetime the occurrence of cut points is necessary and sufficient for multiple imaging, and so is the occurrence of conjugate points.