By Volker Perlick
This ebook is ready the mathematical thought of sunshine propagation in media on general-relativistic spacetimes. the 1st half discusses the transition from Maxwell's equations to ray optics. the second one half establishes a normal mathematical framework for treating ray optics as a thought in its personal correct, making huge use of the Hamiltonian formalism. This half additionally encompasses a distinctive dialogue of variational ideas (i.e., quite a few models of Fermat's precept) for gentle rays in general-relativistic media. a few functions, e.g. to gravitational lensing, are labored out. The reader is believed to have a few uncomplicated wisdom of common relativity and a few familiarity with differential geometry. many of the effects are released right here for the 1st time, e.g. a general-relativistic model of Fermat's precept for mild rays in a medium that has to fulfill a few regularity only.
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Additional resources for Ray Optics, Fermat’s Principle, and Applications to General Relatively
33) stays close to exact solutions, for 0 < a < ao, within some given error bounds. 40), formed with this family over a real interval [a,, a2l 9 [0, ao], deviates from all exact solutions by an arbitrarily large amount. 33) is of limited usefulness in a non-linear medium gives no information on non-monochromatic waves. These arguments show that it is somewhat problematic to apply the ray method to non-linear differential equations. As a matter of fact, the existing since it literature on this topic is much more "heuristic" than in the linear case.
6 Discussion of transport gab Xpa Xb t) -= a eigenvalue, 6pa) gab say el Xo- Ub a = E2. In this = equations 0 and Ea > 0 for the functions case o, 1, 2, 3, has = hi(x, p)2 and a double h2(X, p) 2 of bilinear with respect to the momentum coordinates. An example for medium is a uniaxial crystal. 91) Oa b X3 X3 - Ua Ub. Hence, either branch of the characteristic variety is the null cone bundle of a Lorentzian metric. 91) can be called the optical metrics of the medium. , it is of the second optical same kind as the optical metric in an isotropic medium.
Following this line of thought, the only possibility to treat dispersive meby violating the above-mentioned property (b). At first sight, the idea to smuggle the parameter a into the differential equations seems alien to optics. (This is a major difference to the JWKB method in wave mechanics. ) Nonetheless, there is a sound method of achieving this goal. Strictly speaking, this method comes in various different variants. The common feature is that one considers asymptotic behavior of approximate-plane-wave families on a one-parameter family of background geometries, rather than on a fixed background geometry.
Ray Optics, Fermat’s Principle, and Applications to General Relatively by Volker Perlick