Ray optics, Fermat’s principle, and applications to general relativity

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.

---

Table of contents :
Cover ……Page 1
Title ……Page 2
Date-line ……Page 3
Preface ……Page 4
Contents……Page 8
Part I. From Maxwell’s equations to ray optics……Page 10
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
3. Light propagation in other kinds of media ……Page 50
3.1 Methodological remarks on dispersive media ……Page 51
3.2 Light propagation in a non-magnetized plasma ……Page 53
Part II. A mathematical framework for ray optics……Page 66
4.1 A brief guide to the literature ……Page 67
4.2 Assumptions and notations ……Page 69
5.1 Definition and basic properties of ray-optical structures ……Page 72
5.2 Regularity notions for ray-optical structures ……Page 81
5.3 Symmetries of ray-optical structures ……Page 87
5.4 Dilation-invariant ray-optical structures ……Page 92
5.5 Eikonal equation ……Page 97
5.6 Caustics ……Page 105
6.1 The vacuum ray-optical structure ……Page 115
6.2 Observer fields, frequency, and redshift ……Page 117
6.3 Isotropic ray-optical structures ……Page 124
6.4 Light bundles in isotropic media ……Page 127
6.5 Stationary ray-optical structures ……Page 135
6.6 Stationary ray optics in vacuum and in simple media ……Page 145
7.1 The principle of stationary action: The general case ……Page 152
7.2 The principle of stationary action: The strongly regular case ……Page 157
7.3 Fermat’s principle ……Page 159
7.4 A Hubert manifold setting for variational problems ……Page 168
7.5 A Morse theory for strongly hyperregular ray-optical structures ……Page 171
8.1 Doppler effect, aberration, and drag effect in isotropic media ……Page 185
8.2 Light rays in a uniformly accelerated medium on Minkowski space ……Page 192
8.3 Light propagation in a plasma on Kerr spacetime ……Page 195
8.4 Gravitational lensing ……Page 201
References ……Page 213
Index……Page 219
Back cover……Page 223