Petters, Arlie O., Levine, Harold, Wambsganss, Joachim
2001, XXV, 603 p.
A product of Birkhäuser Basel
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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.
Foreword / D. Spergel * Preface * I. Introduction * 1. Historical Highlights * 2. The Central Problems * II. Astrophysical Aspects * 3. Basic Physical Concepts * 4. Physical Applications * 5. Observations of Gravitational Lensing * III. Mathematical Aspects * 6. Time Delay and Lensing Maps * 7. Critical Points and Stability * 8. Classification and Genericity of Stable Lens Systems * 9. Local Geometry of Caustics * 10. Morse Inequalities * 11. Counting Lensed Images: Single Plane Case * 12. Counting Lensed Images: Multiplane Case * 13. Magnification of Lensed Images * 14. Computing the Euler Characteristic * 15. Global Geometry of Caustics * Bibliography * Index of Notation * Index