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 4-dimensional 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 4-dimensional manifold with a Lorentzian metric.

The first observation of a ‘gravitational lensing’ effect was made when the deflection of star light by our Sun was verified during a Solar eclipse in 1919. Today, the list of observed phenomena includes the following:

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 large-scale 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 quasi-Newtonian 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 general-relativistic spacetime metric, without further approximations. (The term “non-perturbative lensing” is sometimes used in the same sense.) “Quasi-Newtonian approximation” means that the general-relativistic spacetime formalism is reduced by approximative assumptions to essentially Newtonian terms (Newtonian space, Newtonian time, Newtonian gravitational field). The quasi-Newtonian approximation formalism of lensing comes in several variants, and the relation to the exact formalism is not always evident because sometimes plausibility and ad-hoc assumptions are implicitly made. A common feature of all variants is that they are “weak-field 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 Robertson–Walker spacetime (lens embedded in a cosmological model). The background then defines a Euclidean 3-space, similar to Newtonian space, and the gravitational field of the lens is similar to a Newtonian gravitational field on this Euclidean 3-space. 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 3-space. 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 3-space, but there are also variants for “thick” and moving lenses. Also, modifications for a spatially curved Robertson–Walker background exist, but in all variants a non-trivial topological or causal structure of spacetime is (explicitly or implicitly) excluded. At the center of the quasi-Newtonian 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 3-space (“lens plane”). The light rays are taken to be straight lines in Euclidean 3-space 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 quasi-Newtonian approximation formalism can be found not only in the above-mentioned Living Review [343], but also in the monographs of Schneider, Ehlers, and Falco [298] and Petters, Levine, and Wambsganss [275].

The quasi-Newtonian approximation formalism has proven very successful for using gravitational lensing as a tool in astrophysics. This is impressively demonstrated by the work reviewed in [343]. On the other hand, studying lensing from a spacetime perspective is of relevance under three aspects:

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 non-trivial causal or topological structure demonstrates that, in principle, lensing situations can be much more complicated than suggested by the quasi-Newtonian 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 quasi-Newtonian 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 quasi-Newtonian formalism which would treat these light rays as small perturbations of straight lines.

The present review tries to elucidate all three aspects. More precisely, the following subjects will be covered:

• The basic equations and all relevant techniques that are needed for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime are reviewed. Part of this material is well-established 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 self-contained. 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 (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 quasi-Newtonian approximation. (It is, of course, true that this number is still small in comparison to the huge number of all articles on lensing; see [297, 204] for extensive lensing bibliographies.)

This introduction ends with some notes on subjects not covered in this review:

Wave optics.
In the electromagnetic theory, light is described by wavelike solutions to Maxwell’s equations. The ray-optical treatment used throughout this review is the standard high-frequency approximation (geometric optics approximation) of the electromagnetic theory for light propagation in vacuum on a general-relativistic spacetime (see, e.g., [226], § 22.5 or [298], Section 3.2). (Other notions of vacuum light rays, based on a different approximation procedure, have been occasionally suggested [218], 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 ray-optical treatment is valid and appropriate. An exception, where wave-optical 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 non-dispersive medium, they are the lightlike geodesics of another metric which is again of Lorentzian signature. (This “optical metric” was introduced by Gordon [142]. For a rigourous derivation, starting from Maxwell’s equation in an isotropic non-dispersive medium, see Ehlers [88].) Hence, the formalism used throughout this review still applies to this situation after an appropriate re-interpretation 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 general-relativistic 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 general-relativistic kinetic theory see, e.g., [89]. Apart from some occasional remarks, kinetic theory will not be considered in this review.

Derivation of the quasi-Newtonian formalism.
It is not satisfacory if the quasi-Newtonian formalism of lensing is set up with the help of ad-hoc 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 quasi-Newtonian lens equation by a well-defined approximation procedure. In comparison to earlier such derivations [298, 293, 302] more recent effort has led to considerable improvements. For lenses embedded in a cosmological model, see Pyne and Birkinshaw [284] who consider lenses that need not be thin and may be moving on a Robertson–Walker background (with positive, negative, or zero spatial curvature). For the non-cosmological situation, a Lorentz covariant approximation formalism was derived by Kopeikin and Schäfer [184]. Here Minkowski spacetime is taken as the background, and again the lenses need not be thin and may be moving.