## Gravitational LensesLight observed from distant objects is found to be deflected by the gravitational field of massive objects near the line of sight - an effect predicted by Einstein in his first paper setting forth the general theory of relativity, and confirmed by Eddington soon afterwards. If the source of the light is sufficiently distant and bright, and if the intervening object is massive enough and near enough to the line of sight, the gravitational field acts like a lens, focusing the light and producing one or more bright images of the source. This book, by renowned researchers in the field, begins by discussing the basic physics behind gravitational lenses: the optics of curved space-time. It then derives the appropriate equations for predicting the properties of these lenses. In addition, it presents up-to-date observational evidence for gravitational lenses and describes the particular properties of the observed cases. The authors also discuss applications of the results to problems in cosmology. |

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### Contents

Introduction | 1 |

112 The period 19191937 | 3 |

113 The period 19631979 | 6 |

114 Post1979 | 9 |

12 Outline of the book | 11 |

13 Remarks about notation | 21 |

Basic facts and the observational situation | 25 |

22 The general lens | 29 |

Multiple light deflection | 281 |

91 The multiple lensplane theory | 282 |

912 The magnification matrix | 285 |

913 Particular cases | 287 |

92 Time delay and Fermats principle | 288 |

93 The generalized quadrupole lens | 291 |

Numerical methods | 295 |

101 Roots of onedimensional equations | 296 |

23 The magnification factor | 33 |

24 Observing gravitational lens systems | 41 |

241 Expectations for point sources | 42 |

242 Expectations for extended sources | 46 |

25 Known gravitational lens systems | 47 |

251 Doubles | 48 |

252 Triples | 60 |

253 Quadruples | 64 |

254 Additional candidates | 71 |

255 Arcs | 72 |

256 Rings | 77 |

257 A rapidly growing list of candidates | 84 |

259 Gravitational lenses and cosmology | 89 |

Optics in curved spacetime | 91 |

32 Locally approximately plane waves | 93 |

33 Fermats principle | 100 |

34 Geometry of ray bundles | 104 |

342 Optical scalars and their transport equations | 106 |

35 Distances based on light rays Caustics | 110 |

36 Luminosity flux and intensity | 115 |

Derivation of the lens equation | 119 |

42 Approximate metrics of isolated slowly moving noncompact matter distributions | 121 |

43 Light deflection by quasistationary isolated mass distributions | 123 |

44 Summary of FriedmannLemaitre cosmological models | 127 |

45 Light propagation and redshiftdistance relations in homogeneous and inhomogeneous model universes | 132 |

452 Redshiftdistance relations | 134 |

453 The DyerRoeder equation | 137 |

46 The lens mapping in cosmology | 143 |

47 Wave optics in lens theory | 150 |

Properties of the lens mapping | 157 |

52 Magnification and critical curves | 161 |

53 Time delay and Fermats principle | 166 |

54 Two general theorems about gravitational lensing | 172 |

542 Generalizations | 176 |

543 Necessary and sufficient conditions for multiple imaging | 177 |

Lensing near critical points | 183 |

61 The lens mapping near ordinary images | 184 |

62 Stable singularities of lens mappings | 185 |

621 Folds Rules for truncating Taylor expansions | 186 |

622 Cusps | 192 |

623 Whitneys theorem Singularities of generic lens maps | 197 |

63 Stable singularities of oneparameter families of lens mappings metamorphoses | 198 |

631 Umbilics | 199 |

632 Swallowtails | 203 |

633 Lips and beaktobeaks | 207 |

634 Concluding remarks about singularities | 211 |

64 Magnification of extended sources near folds | 215 |

Wave optics in gravitational lensing | 217 |

72 Magnification near isolated caustic points | 220 |

73 Magnification near fold catastrophes | 222 |

Simple lens models | 229 |

81 Axially symmetric lenses | 230 |

812 The Schwarzschild lens | 239 |

813 Disks as lenses | 240 |

814 The singular isothermal sphere | 243 |

815 A family of lens models for galaxies | 244 |

816 A uniform ring | 247 |

82 Lenses with perturbed symmetry Quadrupole lenses | 249 |

821 The perturbed Plummer model | 252 |

822 The perturbed Schwarzschild lens ChangRefsdal lens | 255 |

83 The two pointmass lens | 261 |

832 Two point masses with arbitrary mass ratio | 264 |

834 Generalization to N point masses | 265 |

84 Lenses with elliptical symmetry | 266 |

841 Elliptical isodensity curves | 267 |

842 Elliptical isopotentials | 268 |

843 A practical approach to nearly elliptical lenses | 271 |

85 Marginal lenses | 274 |

86 Generic properties of elliptical lenses | 277 |

862 Imaging properties | 278 |

102 Images of extended sources | 298 |

103 Interactive methods for model fitting | 299 |

104 Grid search methods | 300 |

105 Transport of images | 302 |

106 Ray shooting | 303 |

107 Constructing lens and source models from resolved images | 307 |

Statistical gravitational lensing General considerations | 309 |

111 Crosssections | 310 |

1111 Multiple image crosssections | 311 |

1112 Magnification crosssections | 313 |

112 The random star field | 320 |

1121 Probability distribution for the deflection | 322 |

1122 Shear and magnification | 328 |

1123 Inclusion of external shear and smooth matter density | 330 |

1124 Correlated deflection probability | 334 |

1125 Spatial distribution of magnifications | 337 |

113 Probabilities in a clumpy universe | 344 |

114 Light propagation in inhomogeneous universes | 348 |

1141 Statistics for light rays | 350 |

1142 Statistics over sources | 364 |

115 Maximum probabilities | 366 |

Statistical gravitational lensing Applications | 371 |

121 Amplification bias and the luminosity function of QSOs | 373 |

1212 QSO source counts and their luminosity function | 378 |

122 Statistics of multiply imaged sources | 380 |

1221 Statistics for pointmass lenses | 381 |

1222 Statistics for isothermal spheres | 385 |

Symmetric lenses | 395 |

Asymmetric lenses | 399 |

1225 Lens surveys | 401 |

1231 Observational challenges | 404 |

1232 Mathematical formulation of the leasing problem | 407 |

1233 Maximal overdensity | 408 |

1234 Lens models | 411 |

1235 Relation to observations | 415 |

Astrophysical discussion | 419 |

1241 Lensinduced variability | 421 |

1242 Microlensing in 2237+0305 | 425 |

1243 Microlensing and broad emission lines of QSOs | 429 |

1244 Microlensing and the classification of AGNs | 433 |

Detailed discussion | 435 |

1252 Observational hints of amplification bias | 444 |

1253 QSOgalaxy associations revisited | 447 |

126 Distortion of images | 448 |

127 Lensing of supernovae | 453 |

128 Further applications of statistical lensing | 456 |

1282 Recurrence of 7ray bursters | 460 |

1283 Multiple imaging from an ensemble of galaxies and the missing lens problem | 461 |

Gravitational lenses as astrophysical tools | 467 |

131 Estimation of model parameters | 468 |

1311 Invariance transformations | 471 |

determination of Ha is not possible unless | 473 |

1313 Application to the 0957+561 system | 476 |

132 Arcs in clusters of galaxies | 483 |

1322 The nearly spherical lens | 485 |

1323 Analysis of the observations arcs as astronomical tools | 492 |

1324 Statistics of arcs and arclets | 498 |

133 Additional applications | 501 |

1332 Scanning of the source by caustics | 504 |

1333 The parallax effect | 508 |

1334 Cosmic strings | 509 |

1335 Upper limits to the mass of some QSOs | 511 |

1336 Gravitational lensing and superluminal motion | 512 |

134 Miscellaneous topics | 513 |

1342 Light deflection in the Solar System | 514 |

517 | |

545 | |

547 | |

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### Common terms and phrases

amplification bias angular separation approximation arcs arcseconds assumed BL Lac objects brightness caustic Chap clumpy universe compact objects components consider corresponding cosmological critical curve cross-section cusps dark matter defined deflection angle deflector delay denotes derived described determined dimensionless discussed distance elliptical extended sources factor Fermat potential Fermat's principle flux ratio geometrical gravitational field gravitational lensing Hence high-redshift integral intrinsic isothermal sphere Jacobian matrix larger lens equation lens mapping lens model lens plane light bundles light curves light deflection light rays luminosity function magnification magnification probability mass distribution matter distribution method metric multiple images number of images observed obtained optical depth overdensity parameters perturbed point masses point source probability distribution properties QSOs radial radio galaxies radius redshift sample Schwarzschild lens Sect shear singularities smooth source counts source plane source position spacetime statistical surface mass density symmetric tangent theorem values variability vector velocity VLBI yields

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