×

zbMATH — the first resource for mathematics

A gravitational lens produces an odd number of images. (English) Zbl 0569.53043
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1086/158365 · doi:10.1086/158365
[2] DOI: 10.1086/158750 · doi:10.1086/158750
[3] DOI: 10.1086/183759 · doi:10.1086/183759
[4] DOI: 10.1126/science.223.4631.46 · doi:10.1126/science.223.4631.46
[5] DOI: 10.1086/183260 · doi:10.1086/183260
[6] DOI: 10.1086/183466 · doi:10.1086/183466
[7] DOI: 10.2307/3968435 · doi:10.2307/3968435
[8] DOI: 10.1016/0040-9383(75)90037-3 · Zbl 0323.58010 · doi:10.1016/0040-9383(75)90037-3
[9] DOI: 10.2307/1969485 · Zbl 0045.26003 · doi:10.2307/1969485
[10] DOI: 10.1016/0040-9383(63)90013-2 · Zbl 0122.10702 · doi:10.1016/0040-9383(63)90013-2
[11] DOI: 10.1086/153300 · doi:10.1086/153300
[12] DOI: 10.1086/158751 · doi:10.1086/158751
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.