Lensing in asymptotically simple and empty spacetimes

3.4 Lensing in asymptotically simple and empty spacetimes

In elementary optics one often considers “light sources at infinity” which are characterized by the fact that all light rays emitted from such a source are parallel to each other. In general relativity, “light sources at infinity” can be defined if one restricts to a special class of spacetimes. These spacetimes, known as “asymptotically simple and empty” are, in particular, globally hyperbolic. Their formal definition, which is due to Penrose [258 ], reads as follows (cf. [153

], p. 222., and [116

], Section 2.3). (Recall that a spacetime is called “strongly causal” if each neighborhood of an event $p$ admits a smaller neighborhood that is intersected by any non-spacelike curve at most once.)

A spacetime $(ℳ, g,)$ is called asymptotically simple and empty if there is a strongly causal spacetime $(ℳ&tidle;, &tidle;g)$ with the following properties:

(S1) $ℳ$ is an open submanifold of $ℳ &tidle;$ with a non-empty boundary $∂ ℳ$ .

(S2) There is a smooth function $Ω :ℳ&tidle; − → ℝ$ such that $ℳ = {p ∈ ℳ&tidle; |Ω (p) > 0}$ , $&tidle; ∂ℳ = {p ∈ ℳ |Ω (p) = 0}$ , $dΩ ⁄= 0$ everywhere on $∂ℳ$ and $2 &tidle;g = Ω g$ on $ℳ$ .

(S3) Every inextendible lightlike geodesic in $ℳ$ has past and future end-point on $∂ℳ$ .

(S4) There is a neighborhood $𝒱$ of $∂ ℳ$ such that the Ricci tensor of $g$ vanishes on $𝒱 ∩ ℳ$ .

Asymptotically simple and empty spacetimes are mathematical models of transparent uncharged gravitating bodies that are isolated from all other gravitational sources. In view of lensing, the transparency condition (S3) is particularly important.

We now summarize some well-known facts about asymptotically simple and empty spacetimes (cf. again [153 ], p. 222, and [116 ], Section 2.3). Every asymptotically simple and empty spacetime is globally hyperbolic. $∂ℳ$ is a $&tidle;g$ -lightlike hypersurface of $&tidle; ℳ$ . It has two connected components, denoted $+ ℐ$ and $− ℐ$ . Each lightlike geodesic in $(ℳ, g)$ has past end-point on $− ℐ$ and future end-point on $ℐ+$ . Geroch [133 ] gave a proof that every Cauchy surface $𝒞$ of an asymptotically simple and empty spacetime has topology $ℝ3$ and that $ℐ ±$ has topology $S2 × ℝ$ . The original proof, which is repeated in [153 ], is incomplete. A complete proof that $𝒞$ must be contractible and that $ℐ ±$ has topology $S2 × ℝ$ was given by Newman and Clarke [239 ] (cf. [238 ]); the stronger statement that $𝒞$ must have topology $ℝ3$ needs the assumption that the Poincaré conjecture is true (i.e., that every compact and simply connected 3-manifold is a 3-sphere). In [239 ] the authors believed that the Poincaré conjecture was proven, but the proof they are refering to was actually based on an error. If the most recent proof of the Poincaré conjecture by Perelman [263 ] (cf. [346 ]) turns out to be correct, this settles the matter.

As $± ℐ$ is a lightlike hypersurface in $ℳ&tidle;$ , it is in particular a wave front in the sense of Section 2.2. The generators of $ℐ ±$ are the integral curves of the gradient of $Ω$ . The generators of $ℐ −$ can be interpreted as the “worldlines” of light sources at infinity that send light into $ℳ$ . The generators of $ℐ+$ can be interpreted as the “worldlines” of observers at infinity that receive light from $ℳ$ . This interpretation is justified by the observation that each generator of $± ℐ$ is the limit curve for a sequence of timelike curves in $ℳ$ .

For an observation event $pO$ inside $ℳ$ and light sources at infinity, lensing can be investigated in terms of the exact lens map (recall Section 2.1), with the role of the source surface $𝒯$ played by $ℐ −$ . (For the mathematical properties of the lens map it is rather irrelevant whether the source surface is timelike, lightlike or even spacelike. What matters is that the arriving light rays meet the source surface transversely.) In this case the lens map is a map $S2 → S2$ , namely from the celestial sphere of the observer to the set of all generators of $ℐ −$ . One can construct it in two steps: First determine the intersection of the past light cone of $pO$ with $ℐ −$ , then project along the generators. The intersections of light cones with $± ℐ$ (“light cone cuts of null infinity”) have been studied in [188 , 187 ].

One can assign a mapping degree (= Brouwer degree = winding number) to the lens map $S2 → S2$ and prove that it must be $±1$ [270 ]. (The proof is based on ideas of [239 , 238 ]. Earlier proofs of similar statements – [187 ], Lemma 1, and [268 ], Theorem 6 – are incorrect, as outlined in [270 ].) Based on this result, the following odd-number theorem can be proven for observer and light source inside $ℳ$ [270 ]: Fix a point $pO$ and a timelike curve $γS$ in an asymptotically simple and empty spacetime $(ℳ, g)$ . Assume that the image of $γS$ is a closed subset of $ℳ&tidle; ∖ ℐ+$ and that $γS$ meets neither the point $pO$ nor the caustic of the past light cone of $p O$ . Then the number of past-pointing lightlike geodesics from $p O$ to $γ S$ in $ℳ$ is finite and odd. The same result can be proven with the help of Morse theory (see Section 3.3).

We will now give an argument to the effect that in an asymptotically simple and empty spacetime the non-occurrence of multiple imaging is rather exceptional. The argument starts from a standard result that is used in the Penrose–Hawking singularity theorems. This standard result, given as Proposition 4.4.5 in [153 ], says that along a lightlike geodesic that starts at a point $pO$ there must be a point conjugate to $pO$ , provided that

the so-called generic condition is satisfied at $pO$ ,
the weak energy condition is satisfied along the geodesic, and
the geodesic can be extended sufficiently far.

The last assumption is certainly true in an asymptotically simple and empty spacetime because there all lightlike geodesics are complete. Hence, the generic condition and the weak energy condition guarantee that every past light cone must have a caustic point. We know from Section 3.1 that this implies multiple imaging for every observer. In other words, the only asymptotically simple and empty spacetimes in which multiple imaging does not occur are non-generic cases (like Minkowski spacetime) and cases where the gravitating bodies have negative energy.

The result that, under the aforementioned conditions, light cones in an asymptotically simple and empty spacetime must have caustic points is due to [164 ]. This paper investigates the past light cones of points on $ℐ+$ and their caustics. These light cones are the generalizations, to an arbitrary asymptotically simple and empty spacetime, of the lightlike hyperplanes in Minkowski spacetime. With their help, the eikonal equation (Hamilton–Jacobi equation) $ij g ∂iS∂jS = 0$ in an asymptotically simple and empty spacetime can be studied in analogy to Minkowski spacetime [125 , 124 ]. In Minkowski spacetime the lightlike hyperplanes are associated with a two-parameter family of solutions to the eikonal equation. In the terminology of classical mechanics such a family is called a complete integral. Knowing a complete integral allows constructing all solutions to the Hamilton–Jacobi equation. In an asymptotically simple and empty spacetime the past light cones of points on $+ ℐ$ give us, again, a complete integral for the eikonal equation, but now in a generalized sense, allowing for caustics. These past light cones are wave fronts, in the sense of Section 2.2, and cannot be represented as surfaces $S = constant$ near caustic points. The way in which all other wave fronts can be determined from knowledge of this distinguished family of wave fronts is detailed in [124 ]. The distinguished family of wave fronts gives a natural choice for the space of trial maps in the Frittelli–Newman variational principle which was discussed in Section 2.9.