Singularity Theory and Gravitational Lensing

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Springer Science & Business Media, Jul 1, 2001 - Mathematics - 603 pages
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This monograph, unique in the literature, is the first to develop a mathematical theory of gravitational lensing. The theory applies to any finite number of deflector planes and highlights the distinctions between single and multiple plane lensing.

Introductory material in Parts I and II present historical highlights and the astrophysical aspects of the subject. Among the lensing topics discussed are multiple quasars, giant luminous arcs, Einstein rings, the detection of dark matter and planets with lensing, time delays and the age of the universe (Hubble’s constant), microlensing of stars and quasars.

The main part of the book---Part III---employs the ideas and results of singularity theory to put gravitational lensing on a rigorous mathematical foundation and solve certain key lensing problems. Results are published here for the first time.

Mathematical topics discussed: Morse theory, Whitney singularity theory, Thom catastrophe theory, Mather stability theory, Arnold singularity theory, and the Euler characteristic via projectivized rotation numbers. These tools are applied to the study of stable lens systems, local and global geometry of caustics, caustic metamorphoses, multiple lens images, lensed image magnification, magnification cross sections, and lensing by singular and nonsingular deflectors.

Examples, illustrations, bibliography and index make this a suitable text for an undergraduate/graduate course, seminar, or independent these project on gravitational lensing. The book is also an excellent reference text for professional mathematicians, mathematical physicists, astrophysicists, and physicists.

  

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Fajna książka:) Jest mi bardzo potrzebna więc jeśli ktoś by ją udostępnił to byłoby super:):)

Contents

Historical Highlights
3
12 Detecting Gravitational Lens Effects
10
Central Problems
15
21 Two Basic Problems
16
22 Mathematicians versus Physicists
18
ASTROPHYSICAL ASPECTS
23
Basic Physical Concepts
25
31 Ingredients of Gravitational Lens Systems
26
772 1Parameter Families of Lagrangian Caustics
285
Classification and Genericity of Stable Lens Systems
287
81 Some Cautionary Remarks
288
82 Stability of Time Delay Functions and Lensing Maps
291
83 Generalization of Thorns Theorem for Two Parameters
295
832 Proof of Generalization
298
84 Most Time Delay Families have Stable Lensing Maps
310
842 Proof of Main Theorem
311

311 General Relativity and Friedmann Cosmology
27
312 Cosmic Distances
37
313 Spacetime Geometry for Gravitational Lensing
43
Density Perturbations
46
315 Deflected Light Rays and Bending Angles
58
316 Cosmic Light Sources
64
32 Gravitational Lens Optics
65
Time Delays
66
322 Computing the Time Delay Function
67
Ray Tracing SinglePlane Lens Equation
77
324 Dimensionless Variables
79
325 Magnification and Flux Conservation Definition of Magnification
82
326 Critical Curves Caustics and Light Curves Critical Curves and Caustics
88
327 Shear Convergence and Extended Sources
95
33 Two Important Families of Lens Models
101
332 Elliptical Lens Models
105
333 Point Mass versus Elliptical Lens Models
108
35 Weak Lensing by Large Scale Structures
112
Physical Applications
119
42 Dark Matter
121
421 Detecting Compact Dark Objects and Planets
122
422 Dark Matter in Galaxy Clusters
126
423 Cluster Mass Reconstruction via Weak Lensing
128
43 Structure of Quasars via Microlensing
131
44 Time Delay and Hubbles Constant
134
45 Statistical Lensing and Limits on Cosmology
137
Cosmic Strings
139
47 Outlook
140
Observations of Gravitational Lensing
143
51 Multiple Quasars
146
The Double Quasar Q0957+561
147
Giant Luminous Arcs
150
521 Fantastic Arcs in Galaxy Cluster CL0024+1654
152
531 The Einstein Ring 1938+666
155
54 Quasar Microlensing
156
541 Microlensing in Quadruple Quasar Q2237+0305
157
From MACHOS to Planets
159
56 Weak Lensing
166
MATHEMATICAL ASPECTS
169
Time Delay and Lensing Maps
171
61 Gravitational Lens Potentials
172
62 SinglePlane Lensing
176
622 Time Delay Functions and Light Rays SinglePlane Time Delay Functions
177
623 Local Geometry of Time Delay Surfaces
179
624 Lensing Maps Lensed Images and Magnification Lens Equation Lensed Images and Lensing Maps
180
63 Simple Models
185
632 Singular Isothermal Sphere
186
633 Point Mass
187
634 A Gallery of Gravitational Lens Models
192
64 Multiple Plane Lensing
193
641 Multiplane Time Delay Functions
195
642 Multiplane Lensing Maps
198
65 Relating Time Delay and Lensing Maps
203
Critical Points and Stability
209
71 Jets and Transversality
210
712 Trans versality and Multijet Transversality
216
72 Mather Stability Theory
226
722 Transverse and Global Stability Transverse Stability
229
73 Singularity Manifolds
231
732 Singularities Srs XY and Srs f
235
74 Morse Theory
238
742 Stability of Functions
240
75 Whitney Singularity Theory
244
752 Characterizations of Folds and Cusps Whitneys Forms for Folds and Cusps
247
753 Stability of Maps between 2Manifolds
260
76 Thorn Catastrophe Theory
266
762 Elementary Catastrophes and Thoms Theorem
272
77 Arnold Singularity Theory
276
771 Lagrangian Maps and Generating Families
277
Local Lensing Geometry
327
91 Qualitative Features of Multiplane Lensing Near Caustics
328
911 Lensing Near Folds
329
912 Lensing Near Positive and Negative Cusps
332
913 Magnification Lensed Caustics and Orientation
338
92 Folds and Cusps via Taylor Coefficients of the Potential
341
93 Local Convexity of Fold Caustics
353
932 Local Convexity in the SinglePlane Case
359
933 DoublePlane Lensing Maps Displaying Folds
360
934 Violation of Convexity in DoublePlane Case
364
94 Folds and Cusps via Directional Derivatives
367
95 Caustic Metamorphoses in Lensing
375
952 Equations for Caustics and Their Metamorphoses
378
953 Illustrations of SinglePlane Caustic Metamorphoses
383
954 Single versus DoublePlane Caustic Metamorphoses
388
Morse Inequalities
393
101 Betti Numbers
394
1012 Precise Treatment of Betti Numbers
400
102 Morse Inequalities A and B
403
103 Proof of the Morse Inequalities
407
1032 Relative Betti Numbers
411
1033 Derivation of Morse Inequalities A and B
414
Counting Lensed Images SinglePlane Case
419
111 General SinglePlane Gravitational Lens
421
112 Isolated Gravitational Lenses
424
113 Nonsingular Isolated Lenses
426
114 Point Masses with Continuous Matter and Shear
429
1141 Subcritical Case
431
1142 Strong Shear Case
432
1143 Supercritical Case
434
115 Upper Bounds on Number of Lensed Images
436
1151 Resultants
437
1152 Application to Point Masses
439
116 Location of Lensed Images due to Point Masses
441
Counting Lensed Images Multiplane Case
445
121 General Multiplane Gravitational Lens
446
1212 Genericity of Morse Boundary Conditions B
451
122 Nonsingular Isolated Multiplane Lenses
455
123 Point Masses in ThreeDimensional Arrays
456
1231 LensedImage Counting Formulas and Lower Bounds
457
1232 Upper Bound on Number of Lensed Images
459
Total Magnification
467
SinglePlane Case
468
1311 General SinglePlane Gravitational Lens
469
1312 Nonsingular Isolated Lenses
470
1313 Point Masses with Continuous Matter and Shear
471
Multiplane Case
473
133 Magnification Cross Section for Multiplane Lensing
475
1331 Coarea Formula for Magnification Cross Sections
476
1332 Magnification Cross Section Near Folds and Cusps
479
Computing the Euler Characteristic
487
141 Locally Stable Maps from Surfaces into the Plane
488
142 Projectivized Rotation Numbers
489
143 Orientation of Critical Curves
495
144 A Formula for the Euler Characteristic
496
Global Geometry of Caustics
503
151 Critical Points and Lens Equation in Complex Form
504
152 Caustics of One and Two Point Masses
509
1522 Binary Point Masses
514
1523 Binaries as a Point Mass Plus ChangRefsdal Lens
524
153 Caustics of Finitely Many Point Masses
531
1532 Bounds on Number of Caustics and Metamorphoses
535
1533 Bounds on Number of Cusps
536
154 Curvature of Caustics
544
1542 Linking Curvature and Obstruction Points
549
Bibliography
561
Index of Notation
589
Index
593
Copyright

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