A gravitational lens produces an odd number of images. (English) Zbl 0569.53043

McKenzie, Ross H. A.

J. Math. Phys. 26, 1592-1596 (1985).

Rigorous results are given to the effect that a transparent gravitational lens produces an odd number of images. Suppose that \(p\) is an event and \(T\) the history of a light source in a globally hyperbolic space-time \((M,g)\). Uhlenbeck’s Morse theory of null geodesics is used to show under quite general conditions that if there are at most a finite number \(n\) of future-directed null geodesics from \(T\) to \(p\), then \(M\) is contractible to a point. Moreover, \(n\) is odd and \({1/2}(n-1)\) of the images of the source seen by an observer at \(p\) have the opposite orientation to the source. An analogous result is noted for Riemannian manifolds with positive definite metric.

MSC:

53Z05	Applications of differential geometry to physics
53C80	Applications of global differential geometry to the sciences
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
83C99	General relativity

Keywords:

gravitational lens; globally hyperbolic space-time; null geodesics

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