Ray Optics, Fermat’s Principle and Applications to General Relativity (2000)(en)(236s)

Volker Perlick9783540668985, 3540668985

This book is about the mathematical theory of light propagation in media on general-relativistic spacetimes. The first part discusses the transition from Maxwell’s equations to ray optics. The second part establishes a general mathematical framework for treating ray optics as a theory in its own right, making extensive use of the Hamiltonian formalism. This part also includes a detailed discussion of variational principles (i.e., various versions of Fermat’s principle) for light rays in general-relativistic media. Some applications, e.g. to gravitational lensing, are worked out. The reader is assumed to have some basic knowledge of general relativity and some familiarity with differential geometry. Some of the results are published here for the first time, e.g. a general-relativistic version of Fermat’s principle for light rays in a medium that has to satisfy some regularity condition only.

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Table of contents :
front-matter.pdf……Page 1
Ray Optics, Fermat’s Principle, and Applications to General Relativity……Page 3
Preface……Page 5
Contents……Page 9
1.1 A brief guide to the literature……Page 11
1.2 Assumptions and notations……Page 13
2.1 Maxwell’s equations in linear dielectric and permeable media……Page 15
2.2 Approximate-plane-wave families……Page 22
2.3 Asymptotic solutions of Maxwell’s equations……Page 25
2.4 Derivation of the eikonal equation and transport equations……Page 27
2.5 Discussion of the eikonal equation……Page 32
2.6 Discussion of transport equations and the introduction of rays……Page 39
2.7 Ray optics as an approximation scheme……Page 44
03.pdf……Page 50
3.1 Methodological remarks on dispersive media……Page 51
3.2 Light propagation in a non-magnetized plasma……Page 53
4.1 A brief guide to the literature……Page 66
4.2 Assumptions and notations……Page 68
5.1 Definition and basic properties of ray-optical structures……Page 71
5.2 Regularity notions for ray-optical structures……Page 80
5.3 Symmetries of ray-optical structures……Page 86
5.4 Dilation-invariant ray-optical structures……Page 91
5.5 Eikonal equation……Page 96
5.6 Caustics……Page 104
6.1 The vacuum ray-optical structure……Page 114
6.2 Observer fields, frequency, and redshift……Page 116
6.3 Isotropic ray-optical structures……Page 123
6.4 Light bundles in isotropic media……Page 126
6.5 Stationary ray-optical structures……Page 134
6.6 Stationary ray optics in vacuum and in simple media……Page 144
7.1 The principle of stationary action: The general case……Page 151
7.2 The principle of stationary action: The strongly regular case……Page 156
7.3 Fermat’s principle……Page 158
7.4 A Hilbert manifold setting for variational problems……Page 167
7.5 A Morse theory for strongly hyperregular ray-optical structures……Page 170
8.1 Doppler effect, aberration, and drag effect in isotropic media……Page 184
8.2 Light rays in a uniformly accelerated medium on Minkowski space……Page 191
8.3 Light propagation in a plasma on Kerr spacetime……Page 194
8.4 Gravitational lensing……Page 200
References
……Page 212
Index
……Page 218