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On Fermat’s principle in general relativity. I: The general case. (English) Zbl 0707.53054
[For part II, cf. the review below.]
The paper provides a rigorous proof of a generic formulation of Fermat’s principle (I. Kovner 1990). The spacetime is arbitrary with no symmetries, all that is needed is a Lorentzian conformal structure. The principle states that among all null curves connecting some point (event) with some timelike curve the geodesic lines are those for which the arrival time is either minimal or is given by a saddle point. It is shown that the arrival time is minimal if the geodesic line is free of conjugate points and it is a saddle point if the line contains a conjugate point. E.g. the latter case occurs for a light signal travelling once around the Einstein’s static universe. The principle is formulated both in the modern global language and in the coordinate notation.
Reviewer: L.M.Sokolowski

53C80 Applications of global differential geometry to the sciences
83C50 Electromagnetic fields in general relativity and gravitational theory
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