Gravitational Lensing from a Spacetime Perspective
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Abstract
The theory of gravitational lensing is reviewed from a spacetime perspective, without quasiNewtonian approximations. More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature. It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary generalrelativistic spacetime. It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images. The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others.
1 Introduction
In its most general sense, gravitational lensing is a collective term for all effects of a gravitational field on the propagation of electromagnetic radiation, with the latter usually described in terms of rays. According to general relativity, the gravitational field is coded in a metric of Lorentzian signature on the 4dimensional spacetime manifold, and the light rays are the lightlike geodesics of this spacetime metric. From a mathematical point of view, the theory of gravitational lensing is thus the theory of lightlike geodesics in a 4dimensional manifold with a Lorentzian metric.

Multiple quasars.
The gravitational field of a galaxy (or a cluster of galaxies) bends the light from a distant quasar in such a way that the observer on Earth sees two or more images of the quasar.

Rings.
An extended light source, like a galaxy or a lobe of a galaxy, is distorted into a closed or almost closed ring by the gravitational field of an intervening galaxy. This phenomenon occurs in situations where the gravitational field is almost rotationally symmetric, with observer and light source close to the axis of symmetry. It is observed primarily, but not exclusively, in the radio range.

Arcs.
Distant galaxies are distorted into arcs by the gravitational field of an intervening cluster of galaxies. Here the situation is less symmetric than in the case of rings. The effect is observed in the optical range and may produce “giant luminous arcs”, typically of a characteristic blue color.

Microlensing.
When a light source passes behind a compact mass, the focusing effect on the light leads to a temporal change in brightness (energy flux). This microlensing effect is routinely observed since the early 1990s by monitoring a large number of stars in the bulge of our Galaxy, in the Magellanic Clouds and in the Andromeda galaxy. Microlensing has also been observed on quasars.

Image distortion by weak lensing.
In cases where the distortion effect on galaxies is too weak for producing rings or arcs, it can be verified with statistical methods. By evaluating the shape of a large number of background galaxies in the field of a galaxy cluster, one can determine the surface mass density of the cluster. By evaluating fields without a foreground cluster one gets information about the largescale mass distribution.
Observational aspects of gravitational lensing and methods of how to use lensing as a tool in astrophysics are the subject of the Living Review by Wambsganss [343]. There the reader may also find some notes on the history of lensing.
The present review is meant as complementary to the review by Wambsganss. While all the theoretical methods reviewed in [343] rely on quasiNewtonian approximations, the present review is devoted to the theory of gravitational lensing from a spaectime perspective, without such approximations. Here the terminology is as follows: “Lensing from a spacetime perspective” means that light propagation is described in terms of lightlike geodesics of a generalrelativistic spacetime metric, without further approximations. (The term “nonperturbative lensing” is sometimes used in the same sense.) “QuasiNewtonian approximation” means that the generalrelativistic spacetime formalism is reduced by approximative assumptions to essentially Newtonian terms (Newtonian space, Newtonian time, Newtonian gravitational field). The quasiNewtonian approximation formalism of lensing comes in several variants, and the relation to the exact formalism is not always evident because sometimes plausibility and adhoc assumptions are implicitly made. A common feature of all variants is that they are “weakfield approximations” in the sense that the spacetime metric is decomposed into a background (“spacetime without the lens”) and a small perturbation of this background (“gravitational field of the lens”). For the background one usually chooses either Minkowski spacetime (isolated lens) or a spatially flat RobertsonWalker spacetime (lens embedded in a cosmological model). The background then defines a Euclidean 3space, similar to Newtonian space, and the gravitational field of the lens is similar to a Newtonian gravitational field on this Euclidean 3space. Treating the lens as a small perturbation of the background means that the gravitational field of the lens is weak and causes only a small deviation of the light rays from the straight lines in Euclidean 3space. In its most traditional version, the formalism assumes in addition that the lens is “thin”, and that the lens and the light sources are at rest in Euclidean 3space, but there are also variants for “thick” and moving lenses. Also, modifications for a spatially curved RobertsonWalker background exist, but in all variants a nontrivial topological or causal structure of spacetime is (explicitly or implicitly) excluded. At the center of the quasiNewtonian formalism is a “lens equation” or “lens map”, which relates the position of a “lensed image” to the position of the corresponding “unlensed image”. In the most traditional version one considers a thin lens at rest, modeled by a Newtonian gravitational potential given on a plane in Euclidean 3space (“lens plane”). The light rays are taken to be straight lines in Euclidean 3space except for a sharp bend at the lens plane. For a fixed observer and light sources distributed on a plane parallel to the lens plane (“source plane”), the lens map is then a map from the lens plane to the source plane. In this way, the geometric spacetime setting of general relativity is completely covered behind a curtain of approximations, and one is left simply with a map from a plane to a plane. Details of the quasiNewtonian approximation formalism can be found not only in the abovementioned Living Review [343], but also in the monographs of Schneider, Ehlers, and Falco [299] and Petters, Levine, and Wambsganss [275].

Didactical.
The theoretical foundations of lensing can be properly formulated only in terms of the full formalism of general relativity. Working out examples with strong curvature and with nontrivial causal or topological structure demonstrates that, in principle, lensing situations can be much more complicated than suggested by the quasiNewtonian formalism.

Methodological.
General theorems on lensing (e.g., criteria for multiple imaging, characterizations of caustics, etc.) should be formulated within the exact spacetime setting of general relativity, if possible, to make sure that they are not just an artifact of approximative assumptions. For those results which do not hold in arbitrary spacetimes, one should try to find the precise conditions on the spacetime under which they are true.

Practical.
There are some situations of astrophysical interest to which the quasiNewtonian formalism does not apply. For instance, near a black hole light rays are so strongly bent that, in principle, they can make arbitrarily many turns around the hole. Clearly, in this situation it is impossible to use the quasiNewtonian formalism which would treat these light rays as small perturbations of straight lines.

The basic equations and all relevant techniques that are needed for calculating the position, the shape, and the brightness of images in an arbitrary generalrelativistic spacetime are reviewed. Part of this material is wellestablished since decades, like the Sachs equations for the optical scalars (Section 2.3), which are of crucial relevance for calculating distance measures (Section 2.4), image distortion (Section 2.5), and the brightness of images (Section 2.6). It is included here to keep the review selfcontained. Other parts refer to more recent developments which are far from being fully explored, like the exact lens map (Section 2.1) and variational techniques (Section 2.9). Specifications and simplifications are possible for spacetimes with symmetries. The case of spherically symmetric and static spacetimes is treated in greater detail (Section 4.3).

General theorems on lensing in arbitrary spacetimes, or in certain classes of spacetimes, are reviewed. Some of these results are of a local character, like the classification of locally stable caustics (Section 2.2). Others are related to global aspects, like the criteria for multiple imaging in terms of conjugate points and cut points (Sections 2.7 and 2.8). The global theorems can be considerably strengthened if one restricts to globally hyperbolic spacetimes (Section 3.1) or, more specifically, to asymptotically simple and empty spacetimes (Section 3.4). The latter may be viewed as spacetime models for isolated transparent lenses. Also, in globally hyperbolic spacetimes Morse theory can be used for investigating whether the total number of images is finite or infinite, even or odd (Section 3.3). In a spherically symmetric and static spacetime, the occurrence of an infinite sequence of images is related to the occurrence of a “light sphere” (circular lightlike geodesics), like in the Schwarzschild spacetime at r = 3m (Section 4.3).

Several examples of spacetimes are considered, where the lightlike geodesics and, thus, the lensing features can be calculated explicitly. The examples are chosen such that they illustrate the general results. Therefore, in many parts of the review the reader will find suggestions to look at pictures in the example section. The best known and astrophysically most relevant examples are the Schwarzschild spacetime (Section 5.1), the Kerr spacetime (Section 5.8) and the spacetime of a straight string (Section 5.10). Schwarzschild black hole lensing and Kerr black hole lensing was intensively investigated already in the 1960s, 1970s, and 1980s, with astrophysical applications concentrating on observable features of accretion disks. More recently, the increasing evidence that there is a black hole at the center of our Galaxy (and probably at the center of most galaxies) has led to renewed and intensified interest in black hole lensing (see Sections 5.1 and 5.8). This is a major reason for the increasing number of articles on lensing beyond the quasiNewtonian approximation. (It is, of course, true that this number is still small in comparison to the huge number of all articles on lensing; see [298, 244] for extensive lensing bibliographies.)

Wave optics.
In the electromagnetic theory, light is described by wavelike solutions to Maxwell’s equations. The rayoptical treatment used throughout this review is the standard highfrequency approximation (geometric optics approximation) of the electromagnetic theory for light propagation in vacuum on a generalrelativistic spacetime (see, e.g., [225], § 22.5 or [299], Section 3.2). (Other notions of vacuum light rays, based on a different approximation procedure, have been occasionally suggested [217], but will not be considered here. Also, results specific to spacetime dimensions other than four or to gravitational theories other than Einstein’s are not covered.) For most applications to lensing the rayoptical treatment is valid and appropriate. An exception, where waveoptical corrections are necessary, is the calculation of the brightness of images if a light source comes very close to the caustic of the observer’s light cone (see Section 2.6).

Light propagation in matter.
If light is directly influenced by a medium, the light rays are no longer the lightlike geodesics of the spacetime metric. For an isotropic nondispersive medium, they are the lightlike geodesics of another metric which is again of Lorentzian signature. (This “optical metric” was introduced by Gordon [143]. For a rigourous derivation, starting from Maxwell’s equation in an isotropic nondispersive medium, see Ehlers [88].) Hence, the formalism used throughout this review still applies to this situation after an appropriate reinterpretation of the metric. In anisotropic or dispersive media, however, the light rays are not the lightlike geodesics of a Lorentzian metric. There are some lensing situations where the influence of matter has to be taken into account. For instance., for the deflection of radio signals by our Sun the influence of the plasma in the Solar corona (to be treated as a dispersive medium) is very well measurable. However, such situations will not be considered in this review. For light propagation in media on a generalrelativistic spacetime, see [269] and references cited therein.

Kinetic theory.
As an alternative to the (geometric optics approximation of) electromagnetic theory, light can be treated as a photon gas, using the formalism of kinetic theory. This has relevance, e.g., for the cosmic background radiation. For basic notions of generalrelativistic kinetic theory see, e.g., [89]. Apart from some occasional remarks, kinetic theory will not be considered in this review.

Derivation of the quasiNewtonian formalism.
It is not satisfacory if the quasiNewtonian formalism of lensing is set up with the help of adhoc assumptions, even if the latter look plausible. From a methodological point of view, it is more desirable to start from the exact spacetime setting of general relativity and to derive the quasiNewtonian lens equation by a welldefined approximation procedure. In comparison to earlier such derivations [299, 294, 303] more recent effort has led to considerable improvements. For lenses embedded in a cosmological model, see Pyne and Birkinshaw [283] who consider lenses that need not be thin and may be moving on a RobertsonWalker background (with positive, negative, or zero spatial curvature). For the noncosmological situation, a Lorentz covariant approximation formalism was derived by Kopeikin and Schäfer [185]. Here Minkowski spacetime is taken as the background, and again the lenses need not be thin and may be moving.
2 Lensing in Arbitrary Spacetimes
By a spacetime we mean a 4dimensional manifold \({\mathcal M}\) with a (C^{ ∞ }, if not otherwise stated) metric tensor field g of signature (+, +, +, −) that is timeoriented. The latter means that the nonspacelike vectors make up two connected components in the entire tangent bundle, one of which is called “futurepointing” and the other one “pastpointing”. Throughout this review we restrict to the case that the light rays are freely propagating in vacuum, i.e., are not influenced by mirrors, refractive media, or any other impediments. The light rays are then the lightlike geodesics of the spacetime metric. We first summarize results on the lightlike geodesics that hold in arbitrary spacetimes. In Section 3 these results will be specified for spacetimes with conditions on the causal structure and in Section 4 for spacetimes with symmetries.
2.1 Light cone and exact lens map
In an arbitrary spacetime (\({\mathcal M}\), g), what an observer at an event p_{O} can see is determined by the lightlike geodesics that issue from p_{O} into the past. Their union gives the past light cone of p_{O}. This is the central geometric object for lensing from the spacetime perspective. For a point source with worldline γ_{S}, each pastoriented lightlike geodesic λ from p_{O} to γ_{S} gives rise to an image of γ_{S} on the observer’s sky. One should view any such λ as the central ray of a thin bundle that is focused by the observer’s eye lens onto the observer’s retina (or by a telescope onto a photographic plate). Hence, the intersection of the past light cone with the worldline of a point source (or with the worldtube of an extended source) determines the visual appearance of the latter on the observer’s sky.
Writing map (4) explicitly requires solving the lightlike geodesic equation. This is usually done, using standard index notation, in the Lagrangian formalism, with the Lagrangian \({\mathcal L} = {1 \over 2}{g_{ij}}(x){{\dot x}^i}{{\dot x}^j}\), or in the Hamiltonian formalism, with the Hamiltonian \({\mathcal H} = {1 \over 2}{g^{ij}}(x){p_i}{p_j}\). A nontrivial example where the solutions can be explicitly written in terms of elementary functions is the string spacetime of Section 5.10. Somewhat more general, although still very special, is the situation that the lightlike geodesic equation admits three independent constants of motion in addition to the obvious one g^{ ij }(x)p_{ i }p_{ j } = 0. If, for any pair of the four constants of motion, the Poisson bracket vanishes (“complete integrability”), the lightlike geodesic equation can be reduced to firstorder form, i.e., the light cone can be written in terms of integrals over the metric coefficients. This is true, e.g., in spherically symmetric and static spacetimes (see Section 4.3).
Having parametrized the past light cone of the observation event p_{O} in terms of (s, w), or more specifically in terms of (s, Ψ, Θ), one may set up an exact lens map. This exact lens map is analogous to the lens map of the quasiNewtonian approximation formalism, as far as possible, but it is valid in an arbitrary spacetime without approximation. In the quasiNewtonian formalism for thin lenses at rest, the lens map assigns to each point in the lens plane a point in the source plane (see, e.g., [299, 275, 343]). When working in an arbitrary spacetime without approximations, the observer’s sky \({{\mathcal S}_{\rm{O}}}\) is an obvious substitute for the lens plane. As a substitute for the source plane we choose a 3dimensional submanifold \({\mathcal T}\) with a prescribed ruling by timelike curves. We assume that \({\mathcal T}\) is globally of the form \({\mathcal Q} \times {\mathbb R}\), where the points of the 2manifold \({\mathcal Q}\) label the timelike curves by which \({\mathcal T}\) is ruled. These timelike curves are to be interpreted as the worldlines of light sources. We call any such \({\mathcal T}\) a source surface. In a nutshell, choosing a source surface means choosing a twoparameter family of light sources.
The exact lens map was introduced by Frittelli and Newman [123] and further discussed in [91, 90]. The following global aspects of the exact lens map were investigated in [270]. First, in general the lens map is not defined on all of \({{\mathcal S}_{\rm{O}}}\) because not all pastoriented lightlike geodesics that start at p_{O} necessarily meet \({\mathcal T}\). Second, in general the lens map is multivalued because a lightlike geodesic might meet \({\mathcal T}\) several times. Third, the lens map need not be differentiable and not even continuous because a lightlike geodesic might meet \({\mathcal T}\) tangentially. In [270], the notion of a simple lensing neighborhood is introduced which translates the statement that a deflector is transparent into precise mathematical language. It is shown that the lens map is globally welldefined and differentiable if the source surface is the boundary of such a simple lensing neighborhood, and that for each light source that does not meet the caustic of the observer’s past light cone the number of images is finite and odd. This result applies, as a special case, to asymptotically simple and empty spacetimes (see Section 3.4).
2.2 Wave fronts
 1.
Choose a spacelike 2surface S that is orientable.
 2.
At each point of \({\mathcal S}\), choose a lightlike direction orthogonal to \({\mathcal S}\) that depends smoothly on the footpoint. (You have to choose between two possibilities.)
 3.
Take all lightlike geodesics that are tangent to the chosen directions. These lightlike geodesics are called the generators of the wave front, and the wave front is the union of all generators.
In the context of general relativity the notion of wave fronts was introduced by Kermack, McCrea, and Whittaker [180]. For a modern review article see, e.g., Ehlers and Newman [93].
Fold singularities of a wave front form a lightlike 2manifold in spacetime, on a sufficiently small neighborhood of any fold caustic point. The second picture in Figure 2 shows such a “fold surface”, projected to 3space along the integral curves of a timelike vector field. This projected fold surface separates a region covered twice by the wave front from a region not covered at all. If the wave front is the past light cone of an observation event, and if one restricts to light sources with worldlines in a sufficiently small neighborhood of a fold caustic point, there are two images for light sources on one side and no images for light sources on the other side of the fold surface. Cusp singularities of a wave front form a spacelike curve in spacetime, again locally near any cusp caustic point. Such a curve is often called a “cusp ridge”. Along a cusp ridge, two fold surfaces meet tangentially. The third picture in Figure 2 shows the situation projected to 3space. Near a cusp singularity of a past light cone, there is local tripleimaging for light sources in the wedge between the two fold surfaces and local singleimaging for light sources outside this wedge. Swallowtail, pyramid, and purse singularities are points where two or more cusp ridges meet with a common tangent, as illustrated by the last three pictures in Figure 2.
Friedrich and Stewart [118] have demonstrated that all caustic types that are stable in the sense of Arnold can be realized by wave fronts in Minkowski spacetime. Moreover, they stated without proof that, quite generally, one gets the same stable caustic types if one allows for perturbations only within the class of wave fronts (rather than within the larger class of Legendrian submanifolds). A proof of this statement was claimed to be given in [150] where the Lagrangian rather than the Legendrian formalism was used. However, the main result of this paper (Theorem 4.4 of [150]) is actually too weak to justify this claim. A different version of the desired stability result was indeed proven by another approach. In this approach one concentrates on an instantaneous wave front, i.e., on the intersection of a wave front with a spacelike hypersurface \({\mathcal C}\). As an alternative terminology, one calls the intersection of a (“big”) wave front with a hypersurface \({\mathcal C}\) that is transverse to all generators a “small wave front”. Instantaneous wave fronts are special cases of small wave fronts. The caustic of a small wave front is the set of all points where the small wave front fails to be an immersed 2dimensional submanifold of \({\mathcal C}\). If the spacetime is foliated by spacelike hypersurfaces, the caustic of a wave front is the union of the caustics of its small (= instantaneous) wave fronts. Such a foliation can always be achieved locally, and in several spacetimes of interest even globally. If one identifies different slices with the help of a timelike vector field, one can visualize a wave front, and in particular a light cone, as a motion of small (= instantaneous) wave fronts in 3space. Examples are shown in Figures 13, 18, 19, 27, and 28. Mathematically, the same can be done for nonspacelike slices as long as they are transverse to the generators of the considered wave front (see Figure 30 for an example). Turning from (big) wave fronts to small wave fronts reduces the dimension by one. The only caustic points of a small wave front that are stable in the sense of Arnold are cusps and swallowtails. What one wants to prove is that all other caustic points are unstable with respect to perturbations of the wave front within the class of wave fronts, keeping the metric and the slicing fixed. For spacelike slicings (i.e., for instantaneous wave fronts), this was indeed demonstrated by Low [210]. In this article, the author views wave fronts as subsets of the space \({\mathcal N}\) of all lightlike geodesics in (\({\mathcal M},g\)). General properties of this space \({\mathcal N}\) are derived in earlier articles by Low [208, 209] (also see Penrose and Rindler [262], volume II, where the space \({\mathcal N}\) is treated in twistor language). Low considers, in particular, the case of a globally hyperbolic spacetime [210]; he demonstrates the desired stability result for the intersections of a (big) wave front with Cauchy hypersurfaces (see Section 3.2). As every point in an arbitrary spacetime admits a globally hyperbolic neighborhood, this local stability result is universal. Figure 28 shows an instantaneous wave front with cusps and a swallowtail point. Figure 13 shows instantaneous wave fronts with caustic points that are neither cusps nor swallowtails; hence, they must be unstable with respect to perturbations of the wave front within the class of wave fronts.
It is to be emphasized that Low’s work allows to classify the stable caustics of small wave fronts, but not directly of (big) wave fronts. Clearly, a (big) wave front is a oneparameter family of small wave fronts. A qualitative change of a small wave front, in dependence of a parameter, is called a “metamorphosis” in the English literature and a “perestroika” in the Russian literature. Combining Low’s results with the theory of metamorphoses, or perestroikas, could lead to a classsification of the stable caustics of (big) wave fronts. However, this has not been worked out until now.
Wave fronts in general relativity have been studied in a long series of articles by Newman, Frittelli, and collaborators. For some aspects of their work see Sections 2.9 and 3.4. In the quasiNewtonian approximation formalism of lensing, the classification of caustics is treated in great detail in the book by Petters, Levine, and Wambsganss [275]. Interesting related mateial can also be found in Blandford and Narayan [33]. For a nice exposition of caustics in ordinary optics see Berry and Upstill [28].
A light source that comes close to the caustic of the observer’s past light cone is seen strongly magnified. For a point source whose worldline passes exactly through the caustic, the rayoptical treatment even gives an infinite brightness (see Section 2.6). If a light source passes behind a compact deflecting mass, its brightness increases and decreases in the course of time, with a maximum at the moment of closest approach to the caustic. Such microlensing events are routinely observed by monitoring a large number of stars in the bulge of our Galaxy, in the Magellanic Clouds, and in the Andromeda Galaxy (see, e.g., [226] for an overview). In his millennium essay on future perspectives of gravitational lensing, Blandford [34] mentioned the possibility of observing a chosen light source strongly magnified over a period of time with the help of a spaceborn telescope. The idea is to guide the spacecraft such that the worldline of the light source remains in (or close to) the oneparameter family of caustics of past light cones of the spacecraft over a period of time. This futuristic idea of “caustic surfing” was mathematically further discussed by Frittelli and Petters [128].
2.3 Optical scalars and Sachs equations
For the calculation of distance measures, of image distortion, and of the brightness of images one has to study the Jacobi equation (= equation of geodesic deviation) along lightlike geodesics. This is usually done in terms of the optical scalars which were introduced by Sachs et al. [172, 292]. Related background material on lightlike geodesic congruences can be found in many textbooks (see, e.g., Wald [341], Section 9.2). In view of applications to lensing, a particularly useful exposition was given by Seitz, Schneider and Ehlers [303]. In the following the basic notions and results will be summarized.
2.3.1 Infinitesimally thin bundles
2.3.2 Sachs basis
2.3.3 Jacobi matrix
2.3.4 Shape parameters
2.3.5 Optical scalars
2.3.6 Conservation law
2.3.7 Infinitesimally thin bundles with vertex
The results of this section are the basis for Sections 2.4, 2.5, and 2.6.
2.4 Distance measures
2.4.1 Affine distance
There is a unique affine parametrization s ↦ λ(s) for each lightlike geodesic through the observation event p_{O} such that λ(0) = p_{O} and \(g\left({\dot \lambda (0),{U_O}} \right) = 0\). Then the affine parameter s itself can be viewed as a distance measure. This affine distance has the desirable features that it increases monotonously along each ray and that it coincides in an infinitesimal neighborhood of p_{O} with Euclidean distance in the rest system of U_{O}. The affine distance depends on the 4velocity U_{O} of the observer but not on the 4velocity U_{S} of the light source. It is a mathematically very convenient notion, but it is not an observable. (It can be operationally realized in terms of an observer field whose 4velocities are parallel along the ray. Then the affine distance results by integration if each observer measures the length of an infinitesimally short part of the ray in his rest system. However, in view of astronomical situations this is a purely theoretical construction.) The notion of affine distance was introduced by Kermack, McCrea, and Whittaker [180].
2.4.2 Travel time
As an alternative distance measure one can use the travel time. This requires the choice of a time function, i.e., of a function t that slices the spacetime into spacelike hypersurfaces t = constant. (Such a time function globally exists if and only if the spacetime is stably causal; see, e.g., [154], p. 198.) The travel time is equal to t(p_{O}) − t(p_{S}), for each p_{S} on the past light cone of p_{O}. In other words, the intersection of the light cone with a hypersurface t = constant determines events of equal travel time; we call these intersections “instantaneous wave fronts” (recall Section 2.2). Examples of instantaneous wave fronts are shown in Figures 13, 18, 19, 27, and 28. The travel time increases monotonously along each ray. Clearly, it depends neither on the 4velocity U_{O} of the observer nor on the 4velocity U_{S} of the light source. Note that the travel time has a unique value at each point of p_{O}’s past light cone, even at events that can be reached by two different rays from p_{O}. Near p_{O} the travel time coincides with Euclidean distance in the observer’s rest system only if U_{O} is perpendicular to the hypersurface t = constant with dt(U_{O}) = 1. (The latter equation is true if along the observer’s world line the time function t coincides with proper time.) The travel time is not directly observable. However, travel time differences are observable in multipleimaging situations if the intrinsic luminosity of the light source is timedependent. To illustrate this, think of a light source that flashes at a particular instant. If the flash reaches the observer’s wordline along two different rays, the proper time difference Δτ_{O} of the two arrival events is directly measurable. For a time function t that along the observer’s worldline coincides with proper time, this observed time delay Δτ_{O} gives the difference in travel time for the two rays. In view of applications, the measurement of time delays is of great relevance for quasar lensing. For the double quasar 0957+561 the observed time delay Δτ_{O} is about 417 days (see, e.g., [275], p. 149).
2.4.3 Redshift
2.4.4 Angular diameter distances
2.4.5 Area distance
2.4.6 Corrected luminosity distance
2.4.7 Luminosity distance
2.4.8 Parallax distance
In an arbitrary spacetime, we fix an observation event p_{O} and the observer’s 4velocity U_{O}. We consider a pastoriented lightlike geodesic λ parametrized by affine distance, λ(0) = p_{O} and \(g\left({\dot \lambda (0),{U_{\rm{O}}}} \right) = 1\). To a light source passing through the event λ(s) we assign the (averaged) parallax distance D_{par}(s) = −θ(0)^{−1}, where θ is the expansion of an infinitesimally thin bundle with vertex at λ(s). This definition follows [172]. Its relevance in view of cosmology was discussed in detail by Rosquist [289]. D_{par} can be measured by performing the standard trigonometric parallax method of elementary Euclidean geometry, with the observer at p_{O} and an assistant observer at the perimeter of the bundle, and then averaging over all possible positions of the assistant. Note that the method refers to a bundle with vertex at the light source, i.e., to light rays that leave the light source simultaneously. (Averaging is not necessary if this bundle is circular.) D_{par} depends on the 4velocity of the observer but not on the 4velocity of the light source. To within firstorder approximation near the observer it coincides with affine distance (recall Equation (32)). For the potential obervational relevance of D_{par} see [289], and [299], p. 509.
In view of lensing, D_{+}, D_{−}, and D_{lum} are the most important distance measures because they are related to image distortion (see Section 2.5) and to the brightness of images (see Section 2.6). In spacetimes with many symmetries, these quantities can be explicitly calculated (see Section 4.1 for conformally flat spactimes, and Section 4.3 for spherically symmetric static spacetimes). This is impossible in a spacetime without symmetries, in particular in a realistic cosmological model with inhomogeneities (“clumpy universe”). Following Kristian and Sachs [190], one often uses series expansions with respect to s. For statistical considerations one may work with the focusing equation in a FriedmannRobertsonWalker spacetime with average density (see Section 4.1), or with a heuristically modified focusing equation taking clumps into account. The latter leads to the socalled DyerRoeder distance [86, 87] which is discussed in several textbooks (see, e.g., [299]). (For preDyerRoeder papers on optics in cosmological models with inhomogeneities, see the historical notes in [174].) As overdensities have a focusing and underdensities have a defocusing effect, it is widely believed (following [344]) that after averaging over sufficiently large angular scales the FriedmannRobertsonWalker calculation gives the correct distanceredshift relation. However, it was argued by Ellis, Bassett, and Dunsby [99] that caustics produced by the lensing effect of overdensities lead to a systematic bias towards smaller angular sizes (“shrinking”). For a spherically symmetric inhomogeneity, the effect on the distanceredshift relation can be calculated analytically [230]. For thorough discussions of light propagation in a clumpy universe also see Pyne and Birkinshaw [283], and Holz and Wald [161].
2.5 Image distortion
It is recommendable to change from the ϵ determined this way to \(\varepsilon =  \bar \epsilon\). This transformation corresponds to replacing the Jacobi matrix D by its inverse. The original quantity ϵ(s) gives the true shape of objects at affine distance s that show a circular image on the observer’s sky. The new quantity ε(s) gives the observed shape for objects at affine distance s that actually have a circular crosssection. In other words, if a (small) spherical body at affine distance s is observed, the ellipticity of its image on the observer’s sky is given by ε(s).
By Equations (50, 51), ϵ vanishes along the entire ray if and only if the shear σ vanishes along the entire ray. By Equations (26, 33), the shear vanishes along the entire ray if and only if the conformal curvature term ψ_{0} vanishes along the entire ray. The latter condition means that \(K = \dot \lambda\) is tangent to a principal null direction of the conformal curvature tensor (see, e.g., Chandrasekhar [54]). At a point where the conformal curvature tensor is not zero, there are at most four different principal null directions. Hence, the distortion effect vanishes along all light rays if and only if the conformal curvature vanishes everywhere, i.e., if and only if the spacetime is conformally flat. This result is due to Sachs [296]. An alternative proof, based on expressions for image distortions in terms of the exponential map, was given by Hasse [149].
For any observer, the distortion measure \(\varepsilon =  \bar \epsilon\) is defined along every light ray from every point of the observer’s worldline. This gives ε as a function of the observational coordinates (s, Ψ, Θ, τ) (recall Section 2.1, in particular Equation (4)). If we fix τ and s, ε is a function on the observer’s sky. (Instead of s, one may choose any of the distance measures discussed in Section 2.4, provided it is a unique function of s.) In spacetimes with sufficiently many symmetries, this function can be explicitly determined in terms of integrals over the metric function. This will be worked out for spherically symmetric static spacetimes in Section 4.3. A general consideration of image distortion and example calculations can also be found in papers by Frittelli, Kling and Newman [121, 120]. Frittelli and Oberst [127] calculate image distortion by a “thick gravitational lens” model within a spacetime setting.
The distortion effect is routinely observed since the mid1980s in the form of arcs and (radio) rings (see [299, 275, 343] for an overview). In these cases a distant galaxy appears strongly elongated in one direction. Such strong elongations occur near a caustic point of multiplicity one where ε → ∞. In the case of rings and (long) arcs, the entire bundle cannot be treated as infinitesimally thin, i.e., a theoretical description of the effect requires an integration. For the idealized case of a point source, images in the form of (1dimensional) rings on the observer’s sky occur in cases of rotational symmetry and are usually called “Einstein rings” (see Section 4.3). The rings that are actually observed show extended sources in situations close to rotational symmetry.
For the majority of galaxies that are not distorted into arcs or rings, there is a “weak lensing” effect on the apparent shape that can be investigated statistically. The method is based on the assumption that there is no prefered direction in the universe, i.e., that the axes of (approximately spheroidal) galaxies are randomly distributed. So, without a distortion effect, the axes of galaxy images should make a randomly distributed angle with the (Ψ, Θ) grid on the observer’s sky. Any deviation from a random distribution is to be attributed to a distortion effect, produced by the gravitational field of intervening masses. With the help of the quasiNewtonian approximation, this method has been elaborated into a sophisticated formalism for determining mass distributions, projected onto the plane perpendicular to the line of sight, from observed image distortions. This is one of the most important astrophysical tools for detecting (dark) matter. It has been used to determine the mass distribution in galaxies and galaxy clusters, and more recently observations of image distortions produced by largescale structure have begun (see [22] for a detailed review). From a methodological point of view, it would be desirable to analyse this important line of astronomical research within a spacetime setting. This should give prominence to the role of the conformal curvature tensor.
Another interesting way of observing weak image distortions is possible for sources that emit linearly polarized radiation. (This is true for many radio galaxies. Polarization measurements are also relevant for stronglensing situations; see Schneider, Ehlers, and Falco [299], p. 82 for an example.) The method is based on the geometric optics approximation of Maxwell’s theory. In this approximation, the polarization vector is parallel along each ray between source and observer [88] (cf., e.g., [225], p. 577). We may, thus, use the polarization vector as a realization of the Sachs basis vector E_{1}. If the light source is a spheroidal celestial body (e.g., an elliptic galaxy), it is reasonable to assume that at the light source the polarization direction is aligned with one of the axes, i.e., 2χ(s)/π ∈ ℤ. A distortion effect is verified if the observed polarization direction is not aligned with an axis of the image, 2χ(0)/π ∉ ℤ. It is to be emphasized that the deviation of the polarization direction from the elongation axis is not the result of a rotation (the bundles under consideration have a vertex and are, thus, twistfree) but rather of successive shearing processes along the ray. Also, the effect has nothing to do with the rotation of an observer field. It is a pure conformal curvature effect. Related misunderstandings have been clarified by Panov and Sbytov [254, 255]. The distortion effect on the polarization plane has, so far, not been observed. (Panov and Sbytov [254] have clearly shown that an effect observed by Birch [31], even if real, cannot be attributed to distortion.) Its future detectability is estimated, for distant radio sources, in [318].
2.6 Brightness of images
D_{lum} can be explicitly calculated in spacetimes where the Jacobi fields along lightlike geodesics can be explicitly determined. This is true, e.g., in spherically symmetric and static spacetimes where the extremal angular diameter distances D_{+} and D_{−} can be calculated in terms of integrals over the metric coefficients. The resulting formulas are given in Section 4.3 below. Knowledge of D_{+} and D_{−} immediately gives the area distance D_{area} via Equation (41). D_{area} together with the redshift determines D_{lum} via Equation (48). Such an explicit calculation is, of course, possible only for spacetimes with many symmetries.
By Equation (48), the zeros of D_{lum} coincide with the zeros of D_{area}, i.e., with the caustic points. Hence, in the rayoptical treatment a point source is infinitely bright (magnitude m = −∞) if it passes through the caustic of the observer’s past light cone. A waveoptical treatment shows that the energy flux at the observer is actually bounded by diffraction. In the quasiNewtonian approximation formalism, this was demonstrated by an explicit calculation for light rays deflected by a spheroidal mass by Ohanian [245] (cf. [299], p. 220). Quite generally, the rayoptical calculation of the energy flux gives incorrect results if, for two different light paths from the source worldline to the observation event, the time delay is smaller than or approximately equal to the coherence time. Then interference effects give rise to frequencydependent corrections to the energy flux that have to be calculated with the help of wave optics. In multipleimaging situations, the time delay decreases with decreasing mass of the deflector. If the deflector is a cluster of galaxies, a galaxy, or a star, interference effects can be ignored. Gould [145] suggested that they could be observable if a deflector of about 10^{−15} Solar masses happens to be close to the line of sight to a gammaray burster. In this case, the angleseparation between the (unresolvable) images would be of the order 10^{−15} arcseconds (“femtolensing”). Interference effects could make a frequencydependent imprint on the total intensity. Ulmer and Goodman [328] discussed related effects for deflectors of up to 10^{−11} Solar masses. Femtolensing has not been observed so far. However, it is an interesting future perspective for lensing effects where wave optics has to be taken into account. This would give practical relevance to the theoretical work of Herlt and Stephani [156, 157] who calculated gravitational lensing on the basis of wave optics in the Schwarzschild spacetime.
As an example for the calculation of the brightness of images we consider the Schwarzschild spactime (see Figure 17).
2.7 Conjugate points and cut points
In general, the past light cone of an event forms caustics and transverse selfintersections, 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 selfintersections of the light cone are related to extremizing properties of lightlike geodesics. A light cone with a caustic and a transverse selfintersection is shown in Figure 25.
In this section and in Section 2.8 we use mathematical techniques which are related to the PenroseHawking singularity theorems. For background material, see Penrose [261], Hawking and Ellis [154], O’Neill [247], and Wald [341].
Recall from Section 2.2 that the caustic of the past light cone of p_{O} is the set of all points where this light cone is not an immersed submanifold. A point p_{S} 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 p_{O} along λ. The caustic of the past light cone of p_{O} is also called the “past lightlike conjugate locus” of p_{O}.
The notion of conjugate points is related to the extremizing properties of lightlike geodesics in the following way. Let λ be a pastoriented lightlike geodesic with λ(0) = p_{O}. Assume that p_{S} = λ(s_{0}) is the first conjugate point along this geodesic. This means that p_{S} is in the caustic of the past light cone of p_{O} and that λ does not meet the caustic at parameter values between 0 and s_{0}. Then a wellknown theorem says that all points λ(s) with 0 < s < s_{0} cannot be reached from p_{O} along a timelike curve arbitrarily close to λ, and all points λ(s) with s > s_{0} can. For a proof we refer to Hawking and Ellis [154], 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 pastoriented lightlike geodesic because this is the situation with relevance to lensing. Actually, Hawking and Ellis consider the timereversed situation, i.e., with λ futureoriented. 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^{−}(p_{O}) denote the chronological past of p_{O}, i.e., the set of all \(q \in {\mathcal M}\) that can be reached from p_{O} along a pastpointing timelike curve. In Minkowski spacetime, the boundary ∂I^{−}(p_{O}) of I^{∂}(p_{O}) is just the past light cone of p_{O} united with {p_{O}}. In an arbitrary spacetime, this is not true. A lightlike geodesic λ that issues from p_{O} 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 p_{O} along a timelike curve. The union of all cut points, along any pastpointing lightlike geodesic λ from p_{O}, is called the cut locus of the past light cone (or the past lightlike cut locus of p_{O}). For the light cone in Figure 24 this is the curve (actually 2dimensional) 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 swallowtail point (actually 1dimensional). 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 [280] and Whitehead [350].
 1.
λ always stays on the boundary ∂I^{−}(p_{O}), i.e., it never loses its extremizing property.
 2.
λ is always in I^{−}(p_{O}), i.e., it fails to be extremizing from the very beginning.
 (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 p_{O} along two different lightlike geodesics λ_{1} and λ_{2} from p_{O}. Then the cut point of λ_{1} and of λ_{2} 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 \({\mathcal M}\).
Statement (P1) implies that dI^{−}(p_{O}) is an immersed submanifold everywhere except at the cut locus and, of course, at the vertex p_{O}. It is known (see [154], Proposition 6.3.1) that ∂I^{−}(p_{O}) is achronal (i.e., it is impossible to connect any two of its points by a timelike curve) and thus a 3dimensional Lipschitz topological submanifold. By a general theorem of Rademacher (see [113], 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^{−}(p_{O}). Note that this argument does not necessarily imply that the cut locus is a “small” subset of ∂I^{−}(p_{O}). Chruściel and Galloway [57] have demonstrated, by way of example, that an achronal subset \({\mathcal A}\) of a spacetime may fail to be differentiable on a set that is dense in \({\mathcal A}\). 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 \({\mathcal M}\) (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.
2.8 Criteria for multiple imaging
 1.
There is no pastpointing lightlike geodesic from p_{O} to γ_{S}. Then the observer at p_{O} does not see any image of the light source γ_{S}. For instance, this occurs in Minkowski spacetime for an inextendible worldline γ_{S} that asymptotically approaches the past light cone of p_{O}.
 2.
There is exactly one pastpointing lightlike geodesic from p_{O} to γ_{S}. Then the observer at p_{S} sees exactly one image of the light source γ_{S}. This is the situation naively taken for granted in prerelativistic astronomy.
 3.
There are at least two but not more than denumerably many pastpointing lightlike geodesics from p_{O} to γ_{S}. Then the observer at p_{O} sees finitely or infinitely many distinct images of γ_{O} at his or her celestial sphere.
 4.
There are more than denumerably many pastpointing lightlike geodesics from p to γ. This happens, e.g., in rotationally symmetric situations where it gives rise to the socalled “Einstein rings” (see Section 4.3). It also happens, e.g., in planewave spacetimes (see Section 5.11).
 (C1)
Let λ be a pastpointing lightlike geodesic from p_{O} and let p_{S} be a point on λ beyond the cut point or beyond the first conjugate point. Then there is a timelike curve γ_{S} through p_{S} that can be reached from p_{O} along a second pastpointing lightlike geodesic.
 (C2)
Assume that at p_{O} the pastdistinguishing condition (57) is satisfied. If a timelike curve γ_{S} can be reached from p_{O} along two different pastpointing lightlike geodesics, at least one of them passes through the cut locus of the past light cone of p_{O} on or before arriving at γ_{S}.
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 [154], 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 [154] for details). This result implies the occurrence of conjugate points and, thus, of multiple imaging, for a large class of spacetimes.
2.9 Fermat’s principle for light rays
It is often advantageous to characterize light rays by a variational principle, rather than by a differential equation. This is particularly true in view of applications to lensing. If we have chosen a point p_{O} (observation event) and a timelike curve γ_{S} (worldline of light source) in spacetime \({\mathcal M}\), we want to determine all pastpointing lightlike geodesics from p_{O} to γ_{S}. When working with a differential equation for light rays, we have to calculate all light rays issuing from p_{O} into the past, and to see which of them meet γ_{S}. If we work with a variational principle, we can restrict to curves from p_{O} to γ_{S} at the outset.
Among all ways to move from p_{O} to γ_{S} in the pastpointing (or futurepointing) direction at the speed of light, the actual light rays choose those paths that make the arrival time stationary.
This formulation of Fermat’s principle was suggested by Kovner [187]. The crucial idea is to refer to the arrival time which is given only along the curve γ_{S}, and not to some kind of global time which in an arbitrary spacetime does not even exist. The proof that the solution curves of Kovner’s variational principle are, indeed, exactly the lightlike geodesics was given in [264]. The proof can also be found, with a slight restriction on the spacetime that simplifies matters considerably, in [299]. An alternative version, based on making \({{\mathcal L}_{{p_{\rm{O}}},\gamma {\rm{s}}}}\) into a Hilbert manifold, is given in [266].
 (A1)
If along λ there is no point conjugate to p_{O}, λ is a strict local minimum of T.
 (A2)
If λ passes through a point conjugate to p_{O} before arriving at γ, it is a saddle of T.
 (A3)
If λ reaches the first point conjugate to p_{O} exactly on its arrival at γ_{S}, it may be a local minimum or a saddle but not a local maximum.
The advantage of Kovner’s version of Fermat’s principle is that it works in an arbitrary spacetime. In particular, the spacetime need not be stationary and the light source may arbitrarily move around (at subluminal velocity, of course). This allows applications to dynamical situations, e.g., to lensing by gravitational waves (see Section 5.11). If the spacetime is stationary or conformally stationary, and if the light source is at rest, a purely spatial reformulation of Fermat’s principle is possible. This more specific version of Femat’s principle is known since decades and has found various applications to lensing (see Section 4.2). A more sophisticated application of Fermat’s principle to lensing theory is to put up a Morse theory in order to prove theorems on the possible number of images. In its strongest version, this approach has to presuppose a globally hyperbolic spacetime and will be reviewed in Section 3.3.
For a generalization of Kovner’s version of Fermat’s principle to the case that observer and light source have a spatial extension (see [272]).
An alternative variational principle was introduced by Frittelli and Newman [123] and evaluated in [124, 12]. While Kovner’s principle, like the classical Fermat principle, is a varional principle for rays, the FrittelliNewman principle is a variational principle for wave fronts. (For the definition of wave fronts see Section 2.2.) Although Frittelli and Newman call their variational principle a version of Fermat’s principle, it is actually closer to the classical Huygens principle than to the classical Fermat principle. Again, one fixes p_{O} and γ_{S} as above. To define the trial maps, one chooses a set \({\mathcal W}({p_{\rm{O}}})\) of wave fronts, such that for each lightlike geodesic through p_{O} there is exactly one wave front in \({\mathcal W}({p_{\rm{O}}})\) that contains this geodesic. Hence, \({\mathcal W}({p_{\rm{O}}})\) is in onetoone correspondence to the lightlike directions at p_{O} and thus to the 2sphere. Now let \({\mathcal W}({p_{\rm{O}}},{\gamma _{\rm{s}}})\) denote the set of all wave fronts in \({\mathcal W}({p_{\rm{O}}})\) that meet γ_{S}. We can then define the arrival time functional \(T:{\mathcal W}({p_{\rm{O}}},{\gamma _{\rm{s}}}) \rightarrow {\mathbb R}\) by assigning to each wave front the parameter value at which it intersects γ_{S}. There are some cases to be excluded to make sure that T is defined on an open subset of \({\mathcal W}({p_{\rm{O}}}) \simeq {S^2}\), singlevalued and differentiable. If this is the case, one finds that T is stationary at \(W \in {\mathcal W}({p_{\rm{O}}})\) if and only if W contains a lightlike geodesic from p_{O} to γ_{S}. Thus, to each image of γ_{S} on the sky of po there corresponds a critical point of T. The great technical advantage of the FrittelliNewman principle over the Kovner principle is that T is defined on a finite dimensional manifold, directly to be identified with (part of) the observer’s celestial sphere. The arrival time T in the FrittelliNewman approach is directly analogous to the “Fermat potential” in the quasiNewtonian formalism which is discussed, e.g., in [299]. In view of applications, a crucial point is that the space \({\mathcal W}({p_{\rm{O}}})\) is a matter of choice; there are many wave fronts which have one light ray in common. There is a natural choice, e.g., in asymptotically simple spacetimes (see Section 3.4).
Frittelli, Newman, and collaborators have used their variational principle in combination with the exact lens map (recall Section 2.1) to discuss thick and thin lens models from a spacetime perspective [ 24, 12 ]. Methods from differential topology or global analysis, e.g., Morse theory, have not yet been applied to the FrittelliNewman principle.
3 Lensing in Globally Hyperbolic Spacetimes
In a globally hyperbolic spacetime, considerably stronger statements on qualitative lensing features can be made than in an arbitrary spacetime. This includes, e.g., multiple imaging criteria in terms of cut points or conjugate points, and also applications of Morse theory. The value of these results lies in the fact that they hold in globally hyperbolic spacetimes without symmetries, where lensing cannot be studied by explicitly integrating the lightlike geodesic equation.
The most convenient formal definition of global hyperbolicity is the following. In a spacetime (\({\mathcal M},g\)), a subset \({\mathcal C}\) of \({\mathcal M}\) is called a Cauchy surface if every inextendible causal (i.e., timelike or lightlike) curve intersects \({\mathcal C}\) exactly once. A spacetime is globally hyperbolic if and only if it admits a Cauchy surface. The name globally hyperbolic refers to the fact that for hyperbolic differential equations, like the wave equation, existence and uniqueness of a global solution is guaranteed for initial data given on a Cauchy surface. For details on globally hyperbolic spacetimes see, e.g., [154, 25]. It was demonstrated by Geroch [133] that every gobally hyperbolic spacetime admits a continuous function \(t:{\mathcal M} \rightarrow {\mathbb R}\) such that t^{−1}(t_{0}) is a Cauchy surface for every t_{0} ∈ ℝ. A complete proof of the fact that such a Cauchy time function can be chosen differentiable was given much later by Bernal and Sánchez [27, 26]. The topology of a globally hyperbolic spacetime is determined by the topology of any of its Cauchy surfaces, \({\mathcal M} \simeq {\mathcal C} \times {\mathbb R}\). Note, however, that the converse is not true because \({{\mathcal C}_1} \times {\mathbb R}\) may be homeomorphic (and even diffeomorphic) to \({{\mathcal C}_2} \times {\mathbb R}\) without \({{\mathcal C}_1}\) being homeomorphic to \({{\mathcal C}_2}\). For instance, one can construct a globally hyperbolic spacetime with topology ℝ^{4} that admits a Cauchy surface which is not homeomorphic to ℝ^{3} [238].
In view of applications to lensing the following observation is crucial. If one removes a point, a worldline (timelike curve), or a world tube (region with timelike boundary) from an arbitrary spacetime, the resulting spacetime cannot be globally hyperbolic. Thus, restricting to globally hyperbolic spacetimes excludes all cases where a deflector is treated as nontransparent by cutting its world tube from spacetime (see Figure 24 for an example). Note, however, that this does not mean that globally hyperbolic spacetimes can serve as models only for transparent deflectors. First, a globally hyperbolic spacetime may contain “nontransparent” regions in the sense that a light ray may be trapped in a spatially compact set. Second, the region outside the horizon of a (Schwarzschild, Kerr, …) black hole is globally hyperbolic.
3.1 Criteria for multiple imaging in globally hyperbolic spacetimes
 (H1)
The past light cone of any event p_{O}, together with the vertex {p_{O}}, is closed in \({\mathcal M}\).
 (H2)
The cut locus of the past light cone of p_{O} is closed in \({\mathcal M}\).
 (H3)
Let p_{S} be in the cut locus of the past light cone of p_{O} but not in the conjugate locus (= caustic). Then p_{S} can be reached from p_{O} along two different lightlike geodesics. The past light cone of p_{O} has a transverse selfintersection at p_{S}.
 (H4)
The past light cone of p_{O} is an embedded submanifold if and only if its cut locus is empty.
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 p_{O} (also known as the past lightlike conjugate locus of p_{O}) 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 [215] constructed Riemannian metrics on the 2sphere such that the conjugate locus of a point is not closed. Taking the product of such a Riemannian manifold with 2dimensional 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 pastoriented lightlike geodesics from a point p_{O} (observation event) to a timelike curve γ_{S} (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.
 (H5)
Assume that γ_{S} enters into the chronological past I^{−} (p_{O}) of p_{O}. Then there is a pastoriented lightlike geodesic λ from p_{O} to γ_{S} that is completely contained in the boundary of I^{−}(p_{O}). This geodesic does not pass through a cut point or through a conjugate point before arriving at γ_{S}.
 (H6)
Assume that γ_{S} can be reached from po along a pastoriented lightlike geodesic that passes through a conjugate point or through a cut point before arriving at γ_{S}. Then γ_{S} can be reached from p_{O} along a second pastoriented lightlike geodesic.
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.
3.2 Wave fronts in globally hyperbolic spacetimes
 (N1)
\({\mathcal N}\) can be identified with a sphere bundle over \({\mathcal C}\). The identification is made by assigning to each lightlike geodesic its tangent line at the point where it intersects \({\mathcal C}\). As every sphere bundle over an orientable 3manifold is trivializable, \({\mathcal N}\) is diffeomorphic to \({\mathcal C} \times {S^2}\).
 (N2)
\({\mathcal N}\) carries a natural contact structure. (This contact structure is also discussed, in twistor language, in [262], volume II.)
 (N3)
The wave fronts are exactly the Legendre submanifolds of \({\mathcal N}\).
3.3 Fermat’s principle and Morse theory in globally hyperbolic spacetimes
In an arbitrary spacetime, the pastoriented lightlike geodesics from a point p_{O} (observation event) to a timelike curve γ_{S} (worldline of light source) are the solutions of a variational principle (Kovner’s version of Fermat’s principle; see Section 2.9). Every solution of this variational principle corresponds to an image on p_{O}’s sky of γ_{S}. Determining the number of images is the same as determining the number of solutions to the variational problem. If the variational functional satisfies some technical conditions, the number of solutions to the variational principle can be related to the topology of the space of trial paths. This is the content of Morse theory. In the case at hand, the “technical conditions” turn out to be satisfied in globally hyperbolic spacetimes.
To briefly review Morse theory, we consider a differentiable function \(F:{\mathcal X} \rightarrow {\mathbb R}\) on a real manifold \({\mathcal X}\). Points where the differential of F vanishes are called critical points of F. A critical point is called nondegenerate if the Hessian of F is nondegenerate at this point. F is called a Morse function if all its critical points are nondegenerate. In applications to variational problems, \({\mathcal X}\) is the space of trial maps, F is the functional to be varied, and the critical points of F are the solutions to the variational problem. The nondegeneracy condition guarantees that the character of each critical point — local minimum, local maximum, or saddle — is determined by the Hessian of F at this point. The index of the Hessian is called the Morse index of the critical point. It is defined as the maximal dimension of a subspace on which the Hessian is negative definite. At a local minimum the Morse index is zero, at a local maximum it is equal to the dimension of \({\mathcal X}\).
Palais and Smale [251, 252] realized that the Morse inequalities and the Morse relations are also true for a Morse function F on a noncompact and possibly infinitedimensional Hilbert manifold, provided that F is bounded below and satisfies a technical condition known as Condition C or PalaisSmale condition. In that case, the N_{ k } and B_{ k } need not be finite.
The standard application of Morse theory is the geodesic problem for Riemannian (i.e., positive definite) metrics: given two points in a Riemannian manifold, to find the geodesics that join them. In this case F is the “energy functional” (squaredlength functional). Varying the energy functional is related to varying the length functional like Hamilton’s principle is related to Maupertuis’ principle in classical mechanics. For the space \({\mathcal X}\) one chooses, in the PalaisSmale approach [251], the H^{1}curves between the given two points. (An H^{ n }curve is a curve with locally squareintegrable nth derivative). This is an infinitedimensional Hilbert manifold. It has the same homotopy type (and thus the same Betti numbers) as the loop space of the Riemannian manifold. (The loop space of a connected topological space is the space of all continuous curves joining any two fixed points.) On this Hilbert manifold, the energy functional is always bounded from below, and its critical points are exactly the geodesics between the given endpoints. A critical point (geodesic) is nondegenerate if the two endpoints are not conjugate to each other, and its Morse index is the number of conjugate points in the interior, counted with multiplicity (“Morse index theorem”). The PalaisSmale condition is satisfied if the Riemannian manifold is complete. So one has the following result: Fix any two points in a complete Riemannian manifold that are not conjugate to each other along any geodesic. Then the Morse inequalities (59) and the Morse relation (60) are true, with N_{ k } denoting the number of geodesics with Morse index k between the two points and B_{ k } denoting the kth Betti number of the loop space of the Riemannian manifold. The same result is achieved in the original version of Morse theory [229] (cf. [224]) by choosing for \({\mathcal X}\) the space of broken geodesics between the two given points, with N break points, and sending N → ∞ at the end.
 (M1)
p_{O} is a point and γ_{S} is a timelike curve in a globally hyperbolic spacetime (\({\mathcal M},g\)).
 (M2)
γ_{S} does not meet the caustic of the past light cone of p_{O}.
 (M3)
Every continuous curve from p_{O} to γ_{S} can be continuously deformed into a pastoriented lightlike curve, with all intermediary curves starting at p_{O} and terminating on γ_{S}.
 (R1)
If \({\mathcal M}\) is not contractible to a point, there are infinitely many images. This follows from Equation (59) because for the loop space of a noncontractible space either B_{0} is infinite or almost all B_{ k } are different from zero [304].
 (R2)
If \({\mathcal M}\) is contractible to a point, the total number of images is infinite or odd. This follows from Equation (60) because in this case the loop space of \({\mathcal M}\) is contractible to a point, so all Betti numbers B_{ k } vanish with the exception of B_{0} = 1. As a consequence, Equation (60) can be written as N_{+} − N_{−} = 1, where N_{+} is the number of images with even parity (geodesics with even Morse index) and N_{−} is the number of images with odd parity (geodesics with odd Morse index), hence N_{+} + N_{−} = 2N_{−} + 1.
3.3.1 Black hole spacetimes
Let (\({\mathcal M},g\)) be the domain of outer communication of the Kerr spacetime, i.e., the region between the (outer) horizon and infinity (see Section 5.8). Then the assumption of global hyperbolicity is satisfied and \({\mathcal M}\) is not contractible to a point. Statement (M3) is satisfied if γ_{S} is inextendible and approaches neither the horizon nor (past lightlike) infinity for t → −∞. (This can be checked with the help of an analytical criterion that is called the “metric growth condition” in [327].) If, in addition Statement (M2) is satisfied, the reasoning of Statement (R1) applies. Hence, a Kerr black hole produces infinitely many images. The same argument can be applied to black holes with (electric, magnetic, YangMills, …) charge.
3.3.2 Asymptotically simple and empty spacetimes
As discussed in Section 3.4, asymptotically simple and empty spacetimes are globally hyperbolic and contractible to a point. They can be viewed as models of isolated transparent gravitational lenses. Statement (M3) is satisfied if γ_{S} is inextendible and bounded away from past lightlike infinity ℐ^{}. If, in addition, Statement (M2) is satisfied, Statement (R2) guarantees that the number of images is infinite or odd. If it were infinite, we had as the limit curve a pastinextendible lightlike geodesic that would not go out to ℐ^{}, in contradiction to the definition of asymptotic simplicity. So the number of images must be finite and odd. The same oddnumber theorem can also be proven with other methods (see Section 3.4).
In this way Morse theory provides us with precise mathematical versions of the statements “A black hole produces infinitely many images” and “An isolated transparent gravitational lens produces an odd number of images”. When comparing this theoretical result with observations one has to be aware of the fact that some images might be hidden behind the deflecting mass, some might be too faint for being detected, and some might be too close together for being resolved.
In conformally stationary spacetimes, with γ_{S} being an integral curve of the conformal Killing vector field, a simpler version of Fermat’s principle and Morse theory can be used (see Section 4.2).
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. [154], p. 222., and [117], 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 nonspacelike curve at most once.)
 (S1)
\({\mathcal M}\) is an open submanifold of \({\tilde {\mathcal M}}\) with a nonempty boundary \(\partial {\mathcal M}\).
 (S2)
There is a smooth function \(\Omega:\tilde {\mathcal M} \rightarrow {\mathbb R}\) such that \({\mathcal M} = \{p \in \tilde {\mathcal M}\vert \Omega (p) > 0\}, \partial {\mathcal M} = \{p \in \tilde {\mathcal M}\vert \Omega (p) = 0\}\), dΩ ≠ 0 everywhere on \(\partial {\mathcal M}\) and \(\tilde g = {\Omega ^2}g\) on \({\mathcal M}\).
 (S3)
Every inextendible lightlike geodesic in M has past and future endpoint on \(\partial {\mathcal M}\).
 (S4)
There is a neighborhood \({\mathcal V}\) of \(\partial {\mathcal M}\) such that the Ricci tensor of g vanishes on \({\mathcal V} \cap {\mathcal M}\).
We now summarize some wellknown facts about asymptotically simple and empty spacetimes (cf. again [154], p. 222, and [117], Section 2.3). Every asymptotically simple and empty spacetime is globally hyperbolic. \(\partial {\mathcal M}\) is a \({\tilde g}\)lightlike hypersurface of \({\tilde {\mathcal M}}\). It has two connected components, denoted ℐ^{+} and ℐ^{}. Each lightlike geodesic in (\({\mathcal M},g\)) has past endpoint on ℐ^{} and future endpoint on ℐ_{+}roch [134] gave a proof that every Cauchy surface \({\mathcal C}\) of an asymptotically simple and empty spacetime has topology ℝ^{3} and that ℐ_{±} has topology S^{2} × ℝ. The original proof, which is repeated in [154], is incomplete. A complete proof that \({\mathcal C}\) must be contractible and that ℐ_{±} has topology S^{2} × ℝ was given by Newman and Clarke [238] (cf. [237]); the stronger statement that C must have topology ℝ^{3} needs the assumption that the Poincaré conjecture is true (i.e., that every compact and simply connected 3manifold is a 3sphere). In [238] 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 \({\tilde {\mathcal M}}\), 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 \({\mathcal M}\). The generators of ℐ_{+} can be interpreted as the “worldlines” of observers at infinity that receive light from \({\mathcal M}\). This interpretation is justified by the observation that each generator of ℐ_{±} is the limit curve for a sequence of timelike curves in \({\mathcal M}\).
For an observation event p_{O} inside \({\mathcal M}\) 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 \({\mathcal T}\) 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 S^{2} → S^{2}, 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 p_{O} with ℐ_{}, then project along the generators. The intersections of light cones with ℐ_{±} (“light cone cuts of null infinity”) have been studied in [189, 188].
One can assign a mapping degree (= Brouwer degree = winding number) to the lens map S^{2} → S^{2} and prove that it must be ±1 [270]. (The proof is based on ideas of [238, 237]. Earlier proofs of similar statements — [188], Lemma 1, and [268], Theorem 6 — are incorrect, as outlined in [270].) Based on this result, the following oddnumber theorem can be proven for observer and light source inside \({\mathcal M}\) [270]: Fix a point p_{O} and a timelike curve γ_{S} in an asymptotically simple and empty spacetime (\({\mathcal M},g\)). Assume that the image of γ_{S} is a closed subset of ℐ_{+} and that γ_{S} meets neither the point p_{O} nor the caustic of the past light cone of p_{O}. Then the number of pastpointing lightlike geodesics from p_{O} to γ_{S} in \({\mathcal M}\) is finite and odd. The same result can be proven with the help of Morse theory (see Section 3.3).
 1.
the socalled generic condition is satisfied at p_{O},
 2.
the weak energy condition is satisfied along the geodesic, and
 3.
the geodesic can be extended sufficiently far.
The result that, under the aforementioned conditions, light cones in an asymptotically simple and empty spacetime must have caustic points is due to [165]. 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 (HamiltonJacobi equation) g^{ ij }∂_{ i }S∂_{ j }S = 0 in an asymptotically simple and empty spacetime can be studied in analogy to Minkowski spacetime [126, 125]. In Minkowski spacetime the lightlike hyperplanes are associated with a twoparameter 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 HamiltonJacobi 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 [125]. The distinguished family of wave fronts gives a natural choice for the space of trial maps in the FrittelliNewman variational principle which was discussed in Section 2.9.
4 Lensing in Spacetimes with Symmetry
4.1 Lensing in conformally flat spacetimes
By definition, a spacetime is conformally flat if the conformal curvature tensor (=Weyl tensor) vanishes. An equivalent condition is that every point admits a neighborhood that is conformal to an open subset of Minkowski spacetime. As a consequence, conformally flat spacetimes have the same local conformal symmetry as Minkowski spacetime, that is they admit 15 independent conformal Killing vector fields. The global topology, however, may be different from the topology of Minkowski spacetime. The class of conformally flat spacetimes includes all (kinematic) RobertsonWalker spacetimes. Other physically interesting examples are some (generalized) interior Schwarzschild solutions and some pure radiation spacetimes. All conformally flat solutions to Einstein’s field equation with a perfect fluid or an electromagnetic field are known (see [311], Section 37.5.3).
If a spacetime is globally conformal to an open subset of Minkowski spacetime, the past light cone of every event is an embedded submanifold. Hence, multiple imaging cannot occur (recall Section 2.8). For instance, multiple imaging occurs in spatially closed but not in spatially open RobertsonWalker spacetimes. In any conformally flat spacetime, there is no image distortion, i.e., a sufficiently small sphere always shows a circular outline on the observer’s sky (recall Section 2.5). Correspondingly, every infinitesimally thin bundle of light rays with a vertex is circular, i.e., the extremal angular diameter distances D_{+} and D_{−} coincide (recall Section 2.4). In addition, D_{+} = D_{−} also coincides with the area distance D_{area}, at least up to sign. D_{+} = D_{−} changes sign at every caustic point. As D_{+} has a zero if and only if D_{−} has a zero, all caustic points of an infinitesimally thin bundle with vertex are of multiplicity two (anastigmatic focusing), so all images have even parity.
The geometry of light bundles can be studied directly in terms of the Jacobi equation (= equation of geodesic deviation) along lightlike geodesics. For a detailed investigation of the latter in conformally flat spacetimes, see [273]. The more special case of FriedmannLemaîtreRobertsonWalker spacetimes (with dust, radiation, and cosmological constant) is treated in [101]. For bundles with vertex, one is left with one scalar equation for D_{+} = D_{−} = ±D_{area}, that is the focusing equation (44) with σ = 0. This equation can be explicitly integrated for FriedmannRobertsonWalker spacetimes (dust without cosmological constant). In this way one gets, for the standard observer field in such a spacetime, relations between redshift and (area or luminosity) distance in closed form [219]. There are generalizations for a RobertsonWalker universe with dust plus cosmological constant [178] and dust plus radiation plus cosmological constant [71]. Similar formulas can be written for the relation between age and redshift [322].
4.2 Lensing in conformally stationary spacetimes
Conformally stationary spacetimes are models for gravitational fields that are timeindependent up to an overall conformal factor. (The timedependence of the conformal factor is important, e.g., if cosmic expansion is to be taken into account.) This is a reasonable model assumption for many, though not all, lensing situations of interest. It allows describing light rays in a 3dimensional (spatial) formalism that will be outlined in this section. The class of conformally stationary spacetimes includes spherically symmetric and static spacetimes (see Sections 4.3) and axisymmetric stationary spacetimes (see Section 4.4). Also, conformally flat spacetimes (see Section 4.1) are conformally stationary, at least locally. A physically relevant example where the conformalstationarity assumption is not satisfied is lensing by a gravitational wave (see Section 5.11).
If \({{\hat \phi}_\mu} = {\partial _\mu}h\), where h is a function of x = (x^{1}, x^{2}, x^{3}), we can change the time coordinate according to t ↦ t + h(x), thereby transforming \({{\hat \phi}_\mu}d{x^\mu}\) to zero, i.e., making the surfaces t = constant orthogonal to the tlines. This is the conformally static case. Also, Equation (61) includes the stationary case (f independent of t) and the static case (\({{\hat \phi}_\mu} = {\partial _\mu}h\) and f independent of t).
Fermat’s principle in static spacetimes dates back to Weyl [347] (cf. [206, 319]). The stationary case was treated by Pham Mau Quan [284], who even took an isotropic medium into account, and later, in a more elegant presentation, by Brill [42]. These versions of Fermat’s principle are discussed in several textbooks on general relativity (see, e.g., [225, 116, 312] for the static and [200] for the stationary case). A detailed discussion of the conformally stationary case can be found in [265]. Fermat’s principle in conformally stationary spacetimes was used as the starting point for deriving the lens equation of the quasiNewtonian apporoximation formalism by Schneider [297] (cf. [299]). As an alternative to the name “Fermat metric” (used, e.g., in [116, 312, 265]), the names “optical metric” (see, e.g., [141, 79]) and “optical reference geometry” (see, e.g., [4]) are also used.
In the conformally static case, one can apply the standard Morse theory for Riemannian geodesics to the Fermat metric \({\hat g}\) to get results on the number of \({\hat g}\)geodesics joining two points in space. This immediately gives results on the number of lightlike geodesics joining a point in spacetime to an integral curve of W = ∂_{ t }. Completeness of the Fermat metric corresponds to global hyperbolicity of the spacetime metric. The relevant techniques, and their generalization to (conformally) stationary spacetimes, are detailed in a book by Masiello [218]. (Note that, in contrast to standard terminology, Masiello’s definition of a stationary spacetime includes the assumption that the hypersurfaces t = constant are spacelike.) The resulting Morse theory is a special case of the Morse theory for Fermat’s principle in globally hyperbolic spacetimes (see Section 3.3). In addition to Morse theory, other standard methods from Riemannian geometry have been applied to the Fermat metric, e.g., convexity techniques [139, 140].
Extremizing the functional (67) is formally analogous to Maupertuis’ principle for a particle in a scalar potential on flat space, which is discussed in any book on classical mechanics. Dropping the assumption that the Fermat oneform is a differential, but still requiring the Fermat metric to be conformal to the Euclidean metric, corresponds to introducing an additional vector potential. This form of the opticalmechanical analogy, for light rays in stationary spacetimes whose Fermat metric is conformal to the Euclidean metric, is discussed, e.g., in [7].
Conformally stationary spacetimes can be characterized by another interesting property. Let W be a timelike vector field in a spacetime and fix three observers whose worldlines are integral curves of W. Then the angle under which two of them are seen by the third one remains constant in the course of time, for any choice of the observers, if and only if W is proportional to a conformal Killing vector field. For a proof see [151].
4.3 Lensing in spherically symmetric and static spacetimes
4.3.1 Redshift and Fermat geometry
4.3.2 Index of refraction and embedding diagrams
4.3.3 Light cone
4.3.4 Exact lens map
4.3.4.1 Distance measures, image distortion and brightness of images
4.3.5 Caustics of light cones
4.4 Lensing in axisymmetric stationary spacetimes
Variational techniques related to Fermat’s principal in stationary spacetimes are detailed in a book by Masiello [218]. Note that, in contrast to standard terminology, Masiello’s definition of stationarity includes the assumption that the surfaces t = constant are spacelike.
For a rotating body with an equatorial plane (i.e., with reflectional symmetry), the Fermat metric of the equatorial plane can be represented by an embedding diagram, in analogy to the spherically symmetric static case (recall Figure 11). However, one should keep in mind that in the nonstatic case the lightlike geodesics do not correspond to the geodesics of \({\hat g}\) but are affected, in addition, by a sort of Coriolis force produced by \({\hat \phi}\). For a review on embedding diagrams, including several examples (see [160]).
5 Examples
5.1 Schwarzschild spacetime
5.1.1 Historical notes
Shortly after the discovery of the Schwarzschild metric by Schwarzschild [302] and independently by Droste [80], basic features of its lightlike geodesics were found by Flamm [114], Hilbert [158], and Weyl [347]. Detailed studies of its timelike and lightlike geodesics were made by Hagihara [146] and Darwin [72, 73]. For a fairly complete list of the pre1979 literature on Schwarzschild geodesics see Sharp [306]. All modern textbooks on general relativity include a section on Schwarzschild geodesics, but not all of them go beyond the weakfield approximation. For a particularly detailed exposition see Chandrasekhar [54].
5.1.2 Redshift and Fermat geometry
5.1.3 Index of refraction and embedding diagrams
5.1.4 Lensing by a Schwarzschild black hole
 1.
by showing the visual appearance of some background pattern as distorted by the black hole [66, 296, 233] (only primary images, i = 1, are considered), and
 2.
by showing the visual appearence of an accretion disk around the black hole [211, 130, 15, 14] (higherorder images are taken into account).
5.1.5 Lensing by a nontransparent Schwarzschild star
To model a nontransparent star of radius r_{*} one has to restrict the exterior Schwarzschild metric to the region r > r_{*}. Lightlike geodesics terminate when they arrive at r = r_{*}. The star’s radius cannot be smaller than 2m unless it is allowed to be timedependent. The qualitative features of lensing depend on whether r_{*} is bigger than 3m. Stars with 2m < r_{*} ≤ 3m are called ultracompact [166]. Their existence is speculative. The lensing properties of an ultracompact star are the same as that of a Schwarzschild black hole of the same mass, for observer and light source in the region r > r_{*}. In particular, the apparent angular radius δ on the observer’s sky of an ultracompact star is given by the escape cone of Figure 14. Also, an ultracompact star produces the same infinite sequence of images of each light source as a black hole. For r_{*} > 3m, only finitely many of the images survive because the other lightlike geodesics are blocked. A nontransparent star has a finite focal length r_{f} > 2m in the sense that parallel light from infinity is focused along a line that extends from radius value r_{f} to infinity. r_{f} depends on m and on r_{*}. For the values of our Sun one finds r_{f} = 550 au (1 au = 1 astronomical unit = average distance from the Earth to the Sun). An observer at r ≥ r_{f} can observe strong lensing effects of the Sun on distant light sources. The idea of sending a spacecraft to r ≥ r_{f} was occasionally discussed in the literature [103, 234, 326]. The lensing properties of a nontransparent Schwarzschild star have been illustrated by showing the appearance of the star’s surface to a distant observer. For r_{*} bigger than but of the same order of magnitude as 3m, this has relevance for neutron stars (see [352, 256, 129, 287, 221,]). r_{*} may be chosen timedependent, e.g., to model a nontransparent collapsing star. A star starting with r_{*} > 2m cannot reach r = 2m in finite Schwarzschild coordinate time t (though in finite proper time of an observer at the star’s surface), i.e., for a collapsing star one has r_{*}(t) ↦ 2m for t → ∞. To a distant observer, the total luminosity of a freely (geodesically) collapsing star is attenuated exponentially, \(L(t) \propto \exp ( t(3\sqrt 3 m)  1)\). This formula was first derived by Podurets [279] with an incorrect factor 2 under the exponent and corrected by Ames and Thorne [8]. Both papers are based on kinetic photon theory (Liouville’s equation). An alternative derivation of the luminosity formula, based on the optical scalars, was given by Dwivedi and Kantowski [84]. Ames and Thorne also calculated the spectral distribution of the radiation as a function of time and position on the apparent disk of the star. All these analyses considered radiation emitted at an angle < π/2 against the normal of the star as measured by a static (Killing) observer. Actually, one has to refer not to a static observer but to an observer comoving with the star’s surface. This modification was worked out by Lake and Roeder [198].
5.1.6 Lensing by a transparent Schwarzschild star
To model a transparent star of radius r_{*} one has to join the exterior Schwarzschild metric at r = r_{*} to an interior (e.g., perfect fluid) metric. Lightlike geodesics of the exterior Schwarzschild metric are to be joined to lightlike geodesics of the interior metric when they arrive at r = r_{*}. The radius r_{*} of the star can be timeindependent only if r_{*} > 2m. For 2m < r_{*} ≤ 3m (ultracompact star), the lensing properties for observer and light source in the region r > r_{*} differ from the black hole case only by the possible occurrence of additional images, corresponding to light rays that pass through the star. Inside such a transparent ultracompact star, there is at least one stable photon sphere, in addition to the unstable one at r = 3m outside the star (cf. [153]). In principle, there may be arbitrarily many photon spheres [177]. For r_{*} > 3m, the lensing properties depend on whether there are light rays trapped inside the star. For a perfect fluid with constant density, this is not the case; the resulting spacetime is then asymptotically simple, i.e., all inextendible light rays come from infinity and go to infinity. General results (see Section 3.4) imply that then the number of images must be finite and odd. The light cone in this exteriorplusinterior Schwarzschild spacetime is discussed in detail by Kling and Newman [183]. (In this paper the authors constantly refer to their interior metric as to a “dust” where obviously a perfect fluid with constant density is meant.) Effects on light rays issuing from the star’s interior have been discussed already earlier by Lawrence [203]. The “escape cones”, which are shown in Figure 14 for the exterior Schwarzschild metric have been calculated by Jaffe [167] for points inside the star. The focal length of a transparent star with constant density is smaller than that of a nontransparent star of the same mass and radius. For the mass and the radius of our Sun, one finds 30 au for the transparent case, in contrast to the abovementioned 550 au for the nontransparent case [234]. Radiation from a spherically symmetric homogeneous dust star that collapses to a black hole is calculated in [305], using kinetic theory. An inhomogeneous spherically symmetric dust configuration may form a naked singularity. The redshift of light rays that travel from such a naked singularity to a distant observer is discussed in [83].
5.1.7 Lensing by a Schwarzschild white hole
To get a Schwarzschild white hole one joins at r = 2m the static Schwarzschild region 2m < r < ∞ to the nonstatic Schwarzschild region 0 < r < 2m at r = 2m in such a way that outgoing light rays can cross this surface but ingoing cannot. The appearance of light sources in the region r < 2m to an observer in the region r > 2m is discussed in [112, 232, 81, 196, 197].
5.2 Kottler spacetime
In the following we consider the Kottler metric with a constant m > 0 and we ignore the region r < 0 for which the singularity at r = 0 is naked, for any value of Λ. For Λ < 0, there is one horizon at a radius r_{H} with 0 < r_{H} < 2m; the staticity condition e^{ f }^{(r)} > 0 is satisfied on the region r_{H} < r < ∞. For 0 < Λ < (3m)^{−2}, there are two horizons at radii r_{H1} and r_{H2} with 2m < r_{H1} < 3m < r_{H2}; the staticity condition e^{ f }^{(r)} > 0 is satisfied on the region r_{H1} < r < r_{H2}. For Λ > (3m)^{−2} there is no horizon and no static region. At the horizon(s), the Kottler metric can be analytically extended into nonstatic regions. For Λ < 0, the resulting global structure is similar to the Schwarzschild case. For 0 < Λ < (3m)^{−2}, the resulting global structure is more complex (see [195]). The extreme case Λ = (3m)^{−2} is discussed in [278].
For any value of Λ, the Kottler metric has a light sphere at r = 3m. Escape cones and embedding diagrams for the Fermat geometry (optical geometry) can be found in [314, 160] (cf. Figures 14 and 11 for the Schwarzschild case). Similarly to the Schwarzschild spacetime, the Kottler spacetime can be joined to an interior perfectfluid metric with constant density. Embedding diagrams for the Fermat geometry (optical geometry) of the exteriorplusinterior spacetime can be found in [315]. The dependence on Λ of the light bending is discussed in [194]. For the optical appearance of a Kottler white hole see [196]. The shape of infinitesimally thin light bundles in the Kottler spacetime is determined in [85].
5.3 ReissnerNordström spacetime
 1.
0 ≤ e^{2} ≤ m^{2}; in this case the staticity condition e^{ f }^{(r)} > 0 is satisfied on the regions \(0 < r < m  \sqrt {{m^2}  {e^2}}\) and \(m + \sqrt {{m^2}  {e^2}} < r < \infty\), i.e., there are two horizons.
 2.
m^{2} < e^{2}; then the staticity condition e^{ f }^{(r)} > 0 is satisfied on the entire region 0 < r < ∞, i.e., there is no horizon and the singularity at r = 0 is naked.
By switching to isotropic coordinates, one can describe light propagation in the ReissnerNordström metric by an index of refraction (see, e.g., [105]). The resulting Fermat geometry (optical geometry) is discussed, in terms of embedding diagrams for the blackhole case and for the nakedsingularity case, in [191, 3] (cf. [160]). The visual appearance of a background, as distorted by a ReissnerNordström black hole, is calculated in [222]. Lensing by a charged neutron star, whose exterior is modeled by the ReissnerNordström metric, is the subject of [68, 69]. The lensing properties of a ReissnerNordström black hole are qualitatively (though not quantitatively) the same as that of a Schwarzschild black hole. The reason is the following. For a ReissnerNordström black hole, the metric coefficient R(r) has one local minimum and no other extremum between horizon and infinity, just as in the Schwarzschild case (recall Figure 9). The minimum of R(r) indicates an unstable light sphere towards which light rays can spiral asymptotically. The existence of this minimum, and of no other extremum, was responsible for all qualitative features of Schwarzschild lensing. Correspondingly, Figures 15, 16, and 17 also qualitatively illustrate lensing by a ReissnerNordström black hole. In particular, there is an infinite sequence of images for each light source, corresponding to an infinite sequence of light rays whose limit curve asymptotically spirals towards the light sphere. One can consider the “strongfield limit” [39, 37] of lensing for a ReissnerNordström black hole, in analogy to the Schwarzschild case which is indicated by the asymptotic straight line in the middle graph of Figure 15. Bozza [37] investigates whether quantitative features of the “strongfield limit”, e.g., the slope of the asymptotic straight line, can be used to distinguish between different black holes. For the ReissnerNordström black hole, image positions and magnifications have been calculated in [96], and travel times have been calculated in [290]. In both cases, the authors use the “almost exact lens map” of Virbhadra and Ellis [337] (recall Section 4.3) and analytical methods of Bozza et al. [39, 37, 40].
5.4 MorrisThorne wormholes
Lensing by the Ellis wormhole was discussed in [55]; in this paper the authors identified the region r > 0 with the region r < 0 and they developed a scattering formalism, assuming that observer and light source are in the asymptotic region. Lensing by the Ellis wormhole was also discussed in [271] in terms of the exact lens map. The resulting features are qualitatively very similar to the Schwarzschild case, with the radius values r = −∞, r = 0, r = ∞ in the wormhole case corresponding to the radius values r = 2m, r = 3m, r = ∞ in the Schwarzschild case. With this correspondence, Figures 15, 16, and 17 qualitatively illustrate lensing by the Ellis wormhole. More generally, the same qualitative features occur whenever the metric function R(r) has one minimum and no other extrema, as in Figure 9.
If observer and light source are on the same side of the wormhole’s neck, and if only light rays in the asymptotic region are considered, lensing by a wormhole can be studied in terms of the quasiNewtonian approximation formalism [182]. However, as wormholes are typically associated with negative energy densities [227, 228], the usual assumption of the quasiNewtonian approximation formalism that the mass density is positive cannot be maintained. This observation has raised some interest in lensing by negative masses, in particular in the question of whether negative masses can be detected by their (“microlensing”) effect on the energy flux from sources passing behind them. So far, related calculations [64, 293] have been done only in the quasiNewtonian approximation formalism.
5.5 BarriolaVilenkin monopole
The metric (115) was briefly mentioned as an example for a conical singularity by Sokolov and Starobinsky [308]. Barriola and Vilenkin [21] realized that this metric can be used as a model for monopoles that might exist in the universe, resulting from breaking a global \({\mathcal O}(3)\) symmetry. They also discussed the question of whether such monopoles could be detected by their lensing properties which were characterized on the basis of some approximative assumptions (cf. [82]). However, such approximative assumptions are actually not necessary. The metric (115) has the nice property that the geodesics can be written explicitly in terms of elementary functions. This allows to write down explicit expressions for image positions and observables such as angular diameter distances, luminosity distances, image distortion, etc. (see [271]). Note that because of the deficit angle the metric (115) is not asymptotically flat in the usual sense. (It is “quasiasymptotically flat” in the sense of [243].) For this reason, the “almost exact lens map” of Virbhadra and Ellis [337] (see Section 4.3), is not applicable to this case, at least not without modification.
5.6 JanisNewmanWinicour spacetime
5.7 Boson and fermion stars
Spherically symmetric static solutions of Einstein’s field equation coupled to a scalar field may be interpreted as (uncharged, nonrotating) boson stars if they are free of singularities. Because of the latter condition, the WymanNewmanJanis metric (see Section 5.6) does not describe a boson star. The theoretical concept of boson stars goes back to [179, 291]. The analogous idea of a fermion star, with the scalar field replaced by a spin 1/2 (neutrino) field, is even older [216]. Until today there is no observational evidence for the existence of either a boson or a fermion star. However, they are considered, e.g., as hypothetical candidates for supermassive objects at the center of galaxies (see [301, 324] for the boson and [335, 325] for the fermion case). For the supermassive object at the center of our own galaxy, evidence points towards a black hole, but the possibility that it is a boson or fermion star cannot be completely excluded so far.
Exact solutions that describe boson or fermion stars have been found only numerically (in 3 + 1 dimensions). For this reason there is no boson star model for which the lightlike geodesics could be studied analytically. Numerical studies of lensing have been carried out by Dabrowski and Schunck [70] for a transparent spherically symmetric static maximal boson star, and by Bilić, Nikolić, and Viollier [30] for a transparent spherically symmetric static maximal fermion star. For the case of a fermionfermion star (two components) see [171]. In all three articles the authors use the “almost exact lens map” of Virbhadra and Ellis (see Section 4.3) which is valid for observer and light source in the asymptotic region and almost aligned. Dąbrowski and Schunck [70] also discuss how the alignment assumption can be dropped. The lensing features found in [70] for the boson star and in [30] for the fermion star have several similarities. In both cases, there is a tangential caustic and a radial caustic (recall Figure 8 for terminology). A (point) source on the tangential caustic (i.e., on the axis of symmetry through the observer) is seen as a (1dimenional) Einstein ring plus a (point) image in the center. If the (point) source is moved away from the axis the Einstein ring breaks into two (point) images, so there are three images altogether. Two of them merge and vanish if the radial caustic is crossed. So the qualitative lensing features are quite different from a Schwarzschild black hole with (theoretically) infinitely many images (see Section 5.1). The essential difference is that in the case of a boson or fermion star there are no circular lightlike geodesics towards which light rays could asymptotically spiral.
5.8 Kerr spacetime
5.8.1 Historical notes
The Kerr metric was found by Kerr [181]. The coordinate representation (118) is due to Boyer and Lindquist [36]. The literature on lightlike (and timelike) geodesics of the Kerr metric is abundant (for an overview of the pre1979 literature, see Sharp [306]). Detailed accounts on Kerr geodesics can be found in the books by Chandrasekhar [54] and O’Neill [248].
5.8.2 Fermat geometry
5.8.3 First integrals for lightlike geodesics
5.8.4 Light cone
5.8.5 Lensing by a Kerr black hole

the infinite sequence of images must have an accumulation point on the observer’s sky, by compactness, and

the lightlike geodesic with this initial direction cannot go to infinity or to the horizon, by assumption on γ_{S}.
5.8.5.1 Notes on Kerr naked singularities and on the KerrNewman spacetime
The Kerr metric with a > m describes a naked singularity and is considered as unphysical by most authors. Its lightlike geodesics have been studied in [47, 49] (cf. [54], p. 375). The KerrNewman spacetime (charged Kerr spacetime) is usually thought to be of little astrophysical relevance because the net charge of celestial bodies is small. For the lightlike geodesics in this spacetime the reader may consult [48, 50].
5.9 Rotating disk of dust
The stationary axisymmetric spacetime around a rigidly rotating disk of dust was first studied in terms of a numerical solution to Einstein’s field equation by Bardeen and Wagoner [18, 19]. The exact solution was found much later by Neugebauer and Meinel [236]. It is discussed, e.g., in [235]. The metric cannot be written in terms of elementary functions because it involves the solution to an ultraelliptic integral equation. It depends on a parameter μ which varies between zero and μ_{c} = 4.62966 …. For small μ one gets the Newtonian approximation, for μ → μ_{c} the extreme Kerr metric (a = m) is approached. The lightlike geodesics in this spacetime have been studied numerically and the appearance of the disk to a distant observer has been visualized [345]. It would be desirable to support these numerical results with exact statements. From the known properties of the metric, only a few qualitative lensing features of the disk can be deduced. As Minkowski spacetime is approached for μ → 0, the spacetime must be asymptotically simple and empty as long as μ is sufficiently small. (This is true, of course, only if the disk is treated as transparent.) The general results of Section 3.4 imply that in this case the gravitational field of the disk produces finitely many images of each light source, and that the number of images is odd, provided that the worldline of the light source is pastinextendible and does not go out to past lightlike infinity. For larger values of μ, this is no longer true. For μ > 0.5 there are two counterrotating circular lightlike geodesics in the equatorial plane, a stable one at a radius \({{\tilde \rho}_1}\) inside the disk and an unstable one at a radius \({{\tilde \rho}_2}\) outside the disk. (This follows from [10] where it is shown that for μ > 0.5 timelike counterrotating circular geodesics do not exist in a radius interval [\({{\tilde \rho}_1},{{\tilde \rho}_2}\)]. The boundary values of this interval give the radii of lightlike circular geodesics.) The existence of circular light rays has the consequence that the number of images must be infinite; this is obviously true if light source and observer are exactly on the spatial track of such a circular light ray and, by continuity, also in a neighborhood. For a better understanding of lensing by the disk of dust it is desirable to investigate, for each value of μ and each event p_{O}: Which pastoriented lightlike geodesics that issue from p_{O} go out to infinity and which are trapped? Also, it is desirable to study the light cones and their caustics.
5.10 Straight spinning string
Cosmic strings (and other topological defects) are expected to exist in the universe, resulting from a phase transition in the early universe (see, e.g., [334] for a detailed account). So far, there is no direct observational evidence for the existence of strings. In principle, they could be detected by their lensing effect (see [295] for observations of a recent candidate and [164] for a discussion of the general perspective). Basic lensing features for various string configurations are briefly summarized in [9]. Here we consider the simple case of a straight string that is isolated from all other masses. This is one of the most attractive examples for investigating lensing from the spacetime perspective without approximations. In particular, studying the light cones in this metric is an instructive exercise. The geodesic equation is completely integrable, and the geodesics can even be written explicitly in terms of elementary functions.
5.10.1 Historical notes
With a = 0, the metric (133) and its geodesics were first studied by Marder [213, 214]. He also discussed the matching to an interior solution, without, however, associating it with strings (which were no issue at that time). The same metric was investigated by Sokolov and Starobinsky [308] as an example for a conic singularity. Later Vilenkin [332, 333] showed that within the linearized Einstein theory the metric (133) with a = 0 describes the spacetime outside a straight nonspinning string. Hiscock [159], Gott [144], and Linet [207] realized that the same is true in the full (nonlinear) Einstein theory. Basic features of lensing by a nonspinning string were found by Vilenkin [333] and Gott [144]. The matching to an interior solution for a spinning string, a ≠ 0, was worked out by Jensen and Soleng [170]. Already earlier, the restriction of the metric (133) with a ≠ 0 to the hyperplane z = 0 was studied as the spacetime of a spinning particle in 2 + 1 dimensions by Deser, Jackiw, and’ t Hooft [77]. The geodesics in this (2+ 1)dimensional metric were first investigated by Clément [60] (cf. Krori, Goswami, and Das [192] for the (3 + 1)dimensional case). For geodesics in string metrics one may also consult Galtsov and Masar [131]. The metric (133) can be generalized to the case of several parallel strings (see Letelier [205] for the nonspinning case, and Krori, Goswami, and Das [192] for the spinning case). Clarke, Ellis and Vickers [58] found obstructions against embedding a string model close to metric (133) into an almostRobertsonWalker spacetime. This is a caveat, indicating that the lensing properties of “real” cosmic strings might be significantly different from the lensing properties of the metric (133).
5.10.2 Redshift and Fermat geometry
5.10.3 Light cone
5.10.4 Lensing by a nontransparent string
5.10.5 Lensing by a transparent string
In comparison to a nontransparent string, a transparent string produces additional images. These additional images correspond to light rays that pass through the string. We consider the case a = 0 and 1 < 1/k < 2, which is illustrated by Figures 24 and 25. The general features do not depend on the form of the interior metric, as long as it monotonously interpolates between a regular axis and the boundary of the string. In the nontransparent case, there is a singleimaging region and a doubleimaging region. In the transparent case, the doubleimaging region becomes a tripleimaging region. The additional image corresponds to a light ray that passes through the interior of the string and then smoothly slips over one of the cusp ridges. The point where this light ray meets the worldline of the light source is on the sheet of the light cone between the two cusp ridges in Figure 25, i.e., on the sheet that does not exist in the nontransparent case of Figure 24. From the picture it is obvious that the additional image shows the light source at a younger age than the other two images (so it is a “tertiary image”). A light source whose worldline meets the caustic of the observer’s past light cone is on the borderline between singleimaging and tripleimaging. In this case the tertiary image coincides with the secondary image and it is particularly bright (even infinitely bright according to the rayoptical treatment; recall Section 2.6). Under a small perturbation of the worldline the bright image either splits into two or vanishes, so one is left either with three images or with one image.
5.11 Plane gravitational waves
In spite of their high idealization, plane gravitational waves are interesting mathematical models for studying the lensing effect of gravitational waves. In particular, the focusing effect of plane gravitational waves on light rays can be studied quite explicitly, without any weakfield or smallangle approximations. This focusing effect is reflected by an interesting light cone structure.
The geodesic and causal structure of plane gravitational waves and, more generally, of ppwaves is also studied in [163, 51].
One often considers profile functions f and g with Diracdeltalike singularities (“impulsive gravitational waves”). Then a mathematically rigorous treatment of the geodesic equation, and of the geodesic deviation equation, is delicate because it involves operations on distributions which are not obviously welldefined. For a detailed mathematical study of this situation see [310, 193].
Garfinkle [132] discovered an interesting example for a ppwave which is singular on a 2dimensional worldsheet. This exact solution of Einstein’s vacuum field equation can be interpreted as a wave that travels along a cosmic string. Lensing in this spacetime was numerically discussed by Vollick and Unruh [340].
The vast majority of work on lensing by gravitational waves is done in the weakfield approximation. For the exact treatment and in the weakfield approximation one may use Kovner’s version of Fermat’s principle (see Section 2.9), which has the advantage that it allows for timedependent situations. Applications of this principle to gravitational waves have been worked out in the original article by Kovner [187] and by Faraoni [110, 111].
Notes
Acknowledgements
I have profited very much from many suggestions and comments by Jürgen Ehlers. Also, I wish to thank an anonymous referee for his detailed and very helpful report.
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