Conjugate points and cut points

2.7 Conjugate points and cut points

In general, the past light cone of an event forms caustics and transverse self-intersections, i.e., it is neither an embedded nor an immersed submanifold. The relevance of this fact in view of lensing was emphasized already in Section 2.1. In the following we demonstrate that caustics and transverse self-intersections of the light cone are related to extremizing properties of lightlike geodesics. A light cone with a caustic and a transverse self-intersection is shown in Figure 25

In this section and in Section 2.8 we use mathematical techniques which are related to the Penrose–Hawking singularity theorems. For background material, see Penrose [261 ], Hawking and Ellis [153 ], O’Neill [247 ], and Wald [341 ].

Recall from Section 2.2 that the caustic of the past light cone of $pO$ is the set of all points where this light cone is not an immersed submanifold. A point $pS$ is in the caustic if a generator $λ$ of the light cone intersects at $p S$ an infinitesimally neighboring generator. In this situation $p S$ is said to be conjugate to $pO$ along $λ$ . The caustic of the past light cone of $pO$ is also called the “past lightlike conjugate locus” of $pO$ .

The notion of conjugate points is related to the extremizing properties of lightlike geodesics in the following way. Let $λ$ be a past-oriented lightlike geodesic with $λ(0) = pO$ . Assume that $pS = λ(s0)$ is the first conjugate point along this geodesic. This means that $pS$ is in the caustic of the past light cone of $pO$ and that $λ$ does not meet the caustic at parameter values between 0 and $s0$ . Then a well-known theorem says that all points $λ (s)$ with $0 < s < s0$ cannot be reached from $pO$ along a timelike curve arbitrarily close to $λ$ , and all points $λ (s)$ with $s > s0$ can. For a proof we refer to Hawking and Ellis [153 ], Proposition 4.5.11 and Proposition 4.5.12. It might be helpful to consult O’Neill [247 ], Chapter 10, Proposition 48, in addition.

Here we have considered a past-oriented lightlike geodesic because this is the situation with relevance to lensing. Actually, Hawking and Ellis consider the time-reversed situation, i.e., with $λ$ future-oriented. Then the result can be phrased in the following way. A material particle may catch up with a light ray $λ$ after the latter has passed through a conjugate point and, for particles staying close to $λ$ , this is impossible otherwise. The restriction to particles staying close to $λ$ is essential. Particles “taking a short cut” may very well catch up with a lightlike geodesic even if the latter is free of conjugate points.

For a discussion of the extremizing property in the global sense, not restricted to timelike curves close to $λ$ , we need the notion of cut points. The precise definition of cut points reads as follows.

As ususal, let $I − (pO)$ denote the chronological past of $pO$ , i.e., the set of all $q ∈ ℳ$ that can be reached from $pO$ along a past-pointing timelike curve. In Minkowski spacetime, the boundary $∂I − (pO)$ of $− I (pO )$ is just the past light cone of $pO$ united with ${pO }$ . In an arbitrary spacetime, this is not true. A lightlike geodesic $λ$ that issues from $pO$ into the past is always confined to the closure of $I− (p ) O$ , but it need not stay on the boundary. The last point on $λ$ that is on the boundary is by definition [46 ] the cut point of $λ$ . In other words, it is exactly the part of $λ$ beyond the cut point that can be reached from $pO$ along a timelike curve. The union of all cut points, along any past-pointing lightlike geodesic $λ$ from $pO$ , is called the cut locus of the past light cone (or the past lightlike cut locus of $pO$ ). For the light cone in Figure 24 this is the curve (actually 2-dimensional) where the two sheets of the light cone intersect. For the light cone in Figure 25 the cut locus is the same set plus the swallow-tail point (actually 1-dimensional). For a detailed discussion of cut points in manifolds with metrics of Lorentzian signature, see [25 ]. For positive definite metrics, the notion of cut points dates back to Poincaré [281 ] and Whitehead [350 ].

For a generator $λ$ of the past light cone of $pO$ , the cut point of $λ$ does not exist in either of the two following cases:

$λ$ always stays on the boundary $− ∂I (pO)$ , i.e., it never loses its extremizing property.
$λ$ is always in $− I (pO)$ , i.e., it fails to be extremizing from the very beginning.

Case 2 occurs, e.g., if there is a closed timelike curve through $pO$ . More precisely, Case 2 is excluded if the past distinguishing condition is satisfied at $pO$ , i.e., if for $q ∈ ℳ$ the implication

holds. If Equation (57

) is true, the following can be shown:

(P1) If, along $λ$ , the point $λ (s)$ is conjugate to $λ (0 )$ , the cut point of $λ$ exists and it comes on or before $λ(s)$ .

(P2) Assume that a point $q$ can be reached from $pO$ along two different lightlike geodesics $λ1$ and $λ2$ from $pO$ . Then the cut point of $λ1$ and of $λ2$ exists and it comes on or before $q$ .

(P3) If the cut locus of a past light cone is empty, this past light cone is an embedded submanifold of $ℳ$ .

For proofs see [268 ]; The proofs can also be found in or easily deduced from [25 ]. Statement (P1) says that conjugate points and cut points are related by the easily remembered rule “the cut point comes first”. Statement (P2) says that a “cut” between two geodesics is indicated by the occurrence of a cut point. However, it does not say that exactly at the cut point a second geodesic is met. Such a stronger statement, which truly justifies the name “cut point”, holds in globally hyperbolic spacetimes (see Section 3.1). Statement (P3) implies that the occurrence of transverse self-intersections of a light cone are always indicated by cut points. Note, however, that transverse self-intersections of the past light cone of $pO$ may occur inside $I− (pO )$ and, thus, far away from the cut locus.

Statement (P1) implies that $− ∂I (pO )$ is an immersed submanifold everywhere except at the cut locus and, of course, at the vertex $pO$ . It is known (see [153 ], Proposition 6.3.1) that $∂I − (pO)$ is achronal (i.e., it is impossible to connect any two of its points by a timelike curve) and thus a 3-dimensional Lipschitz topological submanifold. By a general theorem of Rademacher (see [112 ], Theorem 3.6.1), this implies that $∂I − (p ) O$ is differentiable almost everywhere, i.e., that the cut locus has measure zero in $− ∂I (pO)$ . Note that this argument does not necessarily imply that the cut locus is a “small” subset of $− ∂I (pO )$ . Chruściel and Galloway [57 ] have demonstrated, by way of example, that an achronal subset $𝒜$ of a spacetime may fail to be differentiable on a set that is dense in $𝒜$ . So our reasoning so far does not even exclude the possibility that the cut locus is dense in an open subset of $∂I− (p ) O$ . This possibility can be excluded in globally hyperbolic spacetimes where the cut locus is always a closed subset of $ℳ$ (see Section 3.1). In general, the cut locus need not be closed as is exemplified by Figure 24 .

In Section 2.8 we investigate the relevance of cut points (and conjugate points) for multiple imaging.