The odd number theorem is a theorem in strong gravitational lensing which comes directly from differential topology.

The theorem states that the number of multiple images produced by a bounded transparent lens must be odd.

Formulation

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The gravitational lensing is a thought to mapped from what's known as image plane to source plane following the formula :

 .

Argument

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If we use direction cosines describing the bent light rays, we can write a vector field on   plane  .

However, only in some specific directions  , will the bent light rays reach the observer, i.e., the images only form where  . Then we can directly apply the Poincaré–Hopf theorem  .

The index of sources and sinks is +1, and that of saddle points is −1. So the Euler characteristic equals the difference between the number of positive indices   and the number of negative indices  . For the far field case, there is only one image, i.e.,  . So the total number of images is  , i.e., odd. The strict proof needs Uhlenbeck's Morse theory of null geodesics.

References

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  • Chwolson, O. (1924). "Über eine mögliche Form fiktiver Doppelsterne". Astronomische Nachrichten (in German). 221 (20). Wiley: 329–330. Bibcode:1924AN....221..329C. doi:10.1002/asna.19242212003. ISSN 0004-6337.
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  • McKenzie, Ross H. (1985). "A gravitational lens produces an odd number of images". Journal of Mathematical Physics. 26 (7). AIP Publishing: 1592–1596. Bibcode:1985JMP....26.1592M. doi:10.1063/1.526923. ISSN 0022-2488.
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  • Frittelli, Simonetta; Newman, Ezra T. (1999-04-28). "Exact universal gravitational lensing equation". Physical Review D. 59 (12): 124001. arXiv:gr-qc/9810017. Bibcode:1999PhRvD..59l4001F. doi:10.1103/physrevd.59.124001. ISSN 0556-2821. S2CID 248125.
  • Perlick, Volker (1999). "Gravitational Lensing from a Geometric Viewpoint". Einstein's Field Equations and Their Physical Implications: Selected Essays in Honour of Jürgen Ehlers. Lecture Notes in Physics. Vol. 540. pp. 373–425. doi:10.1007/3-540-46580-4_6. ISBN 978-3-540-67073-5.
  • Perlick, Volker (September 2004). "Gravitational lensing from a spacetime perspective". Living Reviews in Relativity. 7 (1): 9. arXiv:1010.3416. Bibcode:2004LRR.....7....9P. doi:10.12942/lrr-2004-9. PMC 5255571. PMID 28179867.