The theory, observations, and applications ofgravitational lensingconstitute one ofthe most rapidly growing branches ofextragalactic astrophysics. The deflection of light from very distant sources by intervening masses provides a unique possibility for the investigation of both background sources and lens mass distributions. Gravitational lensing manifestsitselfmost distinctly through multiply imaged QSOs and the formation of highly elongated im­ ages of distant galaxies ('arcs') and spectacular ring-like images of extra­ galactic radio sources. But the effects of gravitational light deflection are not limited to these prominent image configurations; more subtle, since not directly observable, consequences of lensing are the, possibly strong, mag­ nification of sources, which may permit observation of intrinsically fainter, or more distant, sources than would be visible without these natural tele­ scopes. Such light deflection can also affect the source counts of QSOs and of other compact extragalactic sources, and can lead to flux variability of sources owing to propagation effects. Trying to summarizethe theory and observationalstatus ofgravitational lensing in a monograph turned out to be a bigger problem than any of the authors anticipated when we started this project at the end of 1987, encour­ aged by Martin Harwit, who originally approached us. The development in the field has been very rapid during the last four years, both through the­ ory and through observation, and many sections have been rewritten several times, as the previous versions became out of date.

Inhalt

1. Introduction.- 1.1 Historical remarks.- 1.1.1 Before 1919.- 1.1.2 The period 1919-1937.- 1.1.3 The period 1963-1979.- 1.1.4 Post-1979.- 1.2 Outline of the book.- 1.3 Remarks about notation.- 2. Basic facts and the observational situation.- 2.1 The Schwarzschild lens.- 2.2 The general lens.- 2.3 The magnification factor.- 2.4 Observing gravitational lens systems.- 2.4.1 Expectations for point sources..- 2.4.2 Expectations for extended sources.- 2.5 Known gravitational lens systems.- 2.5.1 Doubles.- 2.5.2 Triples.- 2.5.3 Quadruples.- 2.5.4 Additional candidates.- 2.5.5 Arcs.- 2.5.6 Rings.- 2.5.7 A rapidly growing list of candidates.- 2.5.8 Speculations on other gravitational lens systems.- 2.5.9 Gravitational lenses and cosmology.- 3. Optics in curved spacetime.- 3.1 The vacuum Maxwell equations.- 3.2 Locally approximately plane waves.- 3.3 Fermat's principle.- 3.4 Geometry of ray bundles.- 3.4.1 Ray systems and their connection vectors.- 3.4.2 Optical scalars and their transport equations.- 3.5 Distances based on light rays. Caustics.- 3.6 Luminosity, flux and intensity.- 4. Derivation of the lens equation.- 4.1 Einstein's gravitational field equation.- 4.2 Approximate metrics of isolated, slowly moving, non-compact matter distributions.- 4.3 Light deflection by quasistationary, isolated mass distributions.- 4.4 Summary of Friedmann-Lemaître cosmological models.- 4.5 Light propagation and redshift-distance relations in homogeneous and inhomogeneous model universes.- 4.5.1 Flux conservation and the focusing theorem.- 4.5.2 Redshift-distance relations.- 4.5.3 The Dyer-Roeder equation.- 4.6 The lens mapping in cosmology.- 4.7 Wave optics in lens theory.- 5. Properties of the lens mapping.- 5.1 Basic equations of the lens theory.- 5.2 Magnification and critical curves.- 5.3 Time delay and Fermat's principle.- 5.4 Two general theorems about gravitational lensing.- 5.4.1 The case of a single lens plane.- 5.4.2 Generalizations.- 5.4.3 Necessary and sufficient conditions for multiple imaging.- 5.5 The topography of time delay (Fermat) surfaces.- 6. Lensing near critical points.- 6.1 The lens mapping near ordinary images.- 6.2 Stable singularities of lens mappings.- 6.2.1 Folds. Rules for truncating Taylor expansions.- 6.2.2 Cusps.- 6.2.3 Whitney's theorem. Singularities of generic lens maps.- 6.3 Stable singularities of one-parameter families of lens mappings; metamorphoses.- 6.3.1 Umbilics.- 6.3.2 Swallowtails.- 6.3.3 Lips and beak-to-beaks.- 6.3.4 Concluding remarks about singularities.- 6.4 Magnification of extended sources near folds.- 7. Wave optics in gravitational lensing.- 7.1 Preliminaries; magnification of ordinary images.- 7.2 Magnification near isolated caustic points.- 7.3 Magnification near fold catastrophes.- 8. Simple lens models.- 8.1 Axially symmetric lenses.- 8.1.1 General properties.- 8.1.2 The Schwarzschild lens.- 8.1.3 Disks as lenses.- 8.1.4 The singular isothermal sphere.- 8.1.5 A family of lens models for galaxies.- 8.1.6 A uniform ring.- 8.2 Lenses with perturbed symmetry (Quadrupole lenses).- 8.2.1 The perturbed Plummer model.- 8.2.2 The perturbed Schwarzschild lens ('Chang-Refsdallens').- 8.3 The two point-mass lens.- 8.3.1 Two equal point masses.- 8.3.2 Two point masses with arbitrary mass ratio.- 8.3.3 Two point masses with external shear.- 8.3.4 Generalization to N point masses.- 8.4 Lenses with elliptical symmetry.- 8.4.1 Elliptical isodensity curves.- 8.4.2 Elliptical isopotentials.- 8.4.3 A practical approach to (nearly) elliptical lenses.- 8.5 Marginal lenses.- 8.6 Generic properties of "elliptical lenses".- 8.6.1 Evolution of the caustic structure.- 8.6.2 Imaging properties.- 9. Multiple light deflection.- 9.1 The multiple lens-plane theory.- 9.1.1 The lens equation.- 9.1.2 The magnification matrix.- 9.1.3 Particular cases.- 9.2 Time delay and Fermat's principle.- 9.3 The generalized quadrupole lens.- 10. Numerical methods.- 10.1 Roots of one-dimensional equations.- 10.2 Images of extended sources.- 10.3 Interactive methods for model fitting.- 10.4 Grid search methods.- 10.5 Transport of images.- 10.6 Ray shooting.- 10.7 Constructing lens and source models from resolved images.- 11. Statistical gravitational lensing: General considerations.- 11.1 Cross-sections.- 11.1.1 Multiple image cross-sections.- 11.1.2 Magnification cross-sections.- 11.2 The random star field.- 11.2.1 Probability distribution for the deflection.- 11.2.2 Shear and magnification.- 11.2.3 Inclusion of external shear and smooth matter density.- 11.2.4 Correlated deflection probability.- 11.2.5 Spatial distribution of magnifications.- 11.3 Probabilities in a clumpy universe.- 11.4 Light propagation in inhomogeneous universes.- 11.4.1 Statistics for light rays.- 11.4.2 Statistics over sources.- 11.5 Maximum probabilities.- 12. Statistical gravitational lensing: Applications.- 12.1 Amplification bias and the luminosity function of QSOs.- 12.1.1 Amplification bias: Preliminary discussion.- 12.1.2 QSO source counts and their luminosity function.- 12.2 Statistics of multiply imaged sources.- 12.2.1 Statistics for point-mass lenses.- 12.2.2 Statistics for isothermal spheres.- 12.2.3 Modifications of the lens model: Symmetric lenses.- 12.2.4 Modification of the lens model: Asymmetric lenses.- 12.2.5 Lens surveys.- 12.3 QSO-galaxy associations.- 12.3.1 Observational challenges.- 12.3.2 Mathematical formulation of the lensing problem.- 12.3.3 Maximal overdensity.- 12.3.4 Lens models.- 12.3.5 Relation to observations.- 12.4 Microlensing: Astrophysical discussion.- 12.4.1 Lens-induced variability.- 12.4.2 Microlensing in 2237 + 0305.- 12.4.3 Microlensing and broad emission lines of QSOs.- 12.4.4 Microlensing and the classification of AGNs.- 12.5 The amplification bias: Detailed discussion.- 12.5.1 Theoretical analysis.- 12.5.2 Observational hints of amplification bias.- 12.5.3 QSO-galaxy associations revisited.- 12.6 Distortion of images.- 12.7 Lensing of supernovae.- 12.8 Further applications of statistical lensing.- 12.8.1 Gravitational microlensing by the galactic halo.- 12.8.2 Recurrence of ?-ray bursters.- 12.8.3 Multiple imaging from an ensemble of galaxies, and the 'missing lens' problem.- 13. Gravitational lenses as astrophysical tools.- 13.1 Estimation of model parameters.- 13.1.1 Invariance transformations.- 13.1.2 Determination of lens mass and Hubble constant.- 1…