Gravitational Lensing from a Spacetime Perspective Volker Perlick Institute of Theoretical Physics TU Berlin Sekr. PN 7-1 Hardenbergstrasse 36 10623 Berlin Germany email: [email protected] Accepted on 8 July 2004 Published on 17 September 2004 http://www.livingreviews.org/lrr-2004-9 Living Reviews in Relativity Published by the Max Planck Institute for Gravitational Physics Albert Einstein Institute, Germany Abstract The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations. More precisely, the review covers all aspects of gravita- tional 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 calcu- lating the position, the shape, and the brightness of images in an arbitrary general-relativistic 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. c Max Planck Society and the authors. Further information on copyright is given at http://relativity.livingreviews.org/Info/Copyright/ For permission to reproduce the article please contact [email protected]. Article Amendments On author request a Living Reviews article can be amended to include errata and small additions to ensure that the most accurate and up-to-date information possible is provided. If an article has been amended, a summary of the amendments will be listed on this page. 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Contents 1 Introduction 5 2 Lensing in Arbitrary Spacetimes 9 2.1 Light cone and exact lens map ............................. 9 2.2 Wave fronts ........................................ 12 2.3 Optical scalars and Sachs equations ........................... 16 2.4 Distance measures .................................... 20 2.5 Image distortion ..................................... 25 2.6 Brightness of images ................................... 28 2.7 Conjugate points and cut points ............................ 30 2.8 Criteria for multiple imaging .............................. 32 2.9 Fermat’s principle for light rays ............................. 33 3 Lensing in Globally Hyperbolic Spacetimes 36 3.1 Criteria for multiple imaging in globally hyperbolic spacetimes . 36 3.2 Wave fronts in globally hyperbolic spacetimes ..................... 37 3.3 Fermat’s principle and Morse theory in globally hyperbolic spacetimes . 38 3.4 Lensing in asymptotically simple and empty spacetimes . 41 4 Lensing in Spacetimes with Symmetry 43 4.1 Lensing in conformally flat spacetimes ......................... 43 4.2 Lensing in conformally stationary spacetimes ..................... 43 4.3 Lensing in spherically symmetric and static spacetimes . 46 4.4 Lensing in axisymmetric stationary spacetimes .................... 53 5 Examples 55 5.1 Schwarzschild spacetime ................................. 55 5.2 Kottler spacetime ..................................... 66 5.3 Reissner–Nordstr¨omspacetime ............................. 66 5.4 Morris–Thorne wormholes ................................ 67 5.5 Barriola–Vilenkin monopole ............................... 68 5.6 Janis–Newman–Winicour spacetime .......................... 71 5.7 Boson and fermion stars ................................. 71 5.8 Kerr spacetime ...................................... 73 5.9 Rotating disk of dust ................................... 79 5.10 Straight spinning string ................................. 79 5.11 Plane gravitational waves ................................ 89 6 Acknowledgements 93 References 94 Gravitational Lensing from a Spacetime Perspective 5 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 al- most 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 galax- ies. 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 ver- ified 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 evaluat- ing 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 approx- imations. 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 met- ric, 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 Living Reviews in Relativity http://www.livingreviews.org/lrr-2004-9 6 Volker Perlick space, Newtonian time, Newtonian gravitational field). The quasi-Newtonian approximation for- malism 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 as- sumes 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 topo- logical or causal structure