Odd number theorem

Odd number theorem This article may need to be rewritten to comply with Wikipedia's quality standards. You can help. The talk page may contain suggestions. (January 2020) This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (August 2009) (Learn how and when to remove this template message) 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 The gravitational lensing is a thought to mapped from what's known as image plane to source plane following the formula : {displaystyle M:(u,v)mapsto (u',v')} .

Argument If we use direction cosines describing the bent light rays, we can write a vector field on {displaystyle (u,v)} plane {displaystyle V:(s,w)} .

However, only in some specific directions {displaystyle V_{0}:(s_{0},w_{0})} , will the bent light rays reach the observer, i.e., the images only form where {displaystyle D=delta V=0|_{(s_{0},w_{0})}} . Then we can directly apply the Poincaré–Hopf theorem {displaystyle chi =sum {text{index}}_{D}={text{constant}}} .

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 {displaystyle n_{+}} and the number of negative indices {displaystyle n_{-}} . For the far field case, there is only one image, i.e., {displaystyle chi =n_{+}-n_{-}=1} . So the total number of images is {displaystyle N=n_{+}+n_{-}=2n_{-}+1} , i.e., odd. The strict proof needs Uhlenbeck's Morse theory of null geodesics.

References Chwolson, O. (1924). "Über eine mögliche Form fiktiver Doppelsterne". Astronomische Nachrichten (in German). Wiley. 221 (20): 329–330. Bibcode:1924AN....221..329C. doi:10.1002/asna.19242212003. ISSN 0004-6337. Burke, W. L. (1981). "Multiple Gravitational Imaging by Distributed Masses". The Astrophysical Journal. IOP Publishing. 244: L1. Bibcode:1981ApJ...244L...1B. doi:10.1086/183466. ISSN 0004-637X. McKenzie, Ross H. (1985). "A gravitational lens produces an odd number of images". Journal of Mathematical Physics. AIP Publishing. 26 (7): 1592–1596. Bibcode:1985JMP....26.1592M. doi:10.1063/1.526923. ISSN 0022-2488. Kozameh, Carlos; Lamberti, Pedro W.; Reula, Oscar (1991). "Global aspects of light cone cuts". Journal of Mathematical Physics. AIP Publishing. 32 (12): 3423–3426. Bibcode:1991JMP....32.3423K. doi:10.1063/1.529456. ISSN 0022-2488. Lombardi, Marco (1998-01-20). "An application of the topological degree to gravitational lenses". Modern Physics Letters A. World Scientific Pub Co Pte Lt. 13 (2): 83–86. Bibcode:1998MPLA...13...83L. doi:10.1142/s0217732398000115. ISSN 0217-7323. Wambsganss, Joachim (1998). "Gravitational Lensing in Astronomy". Living Reviews in Relativity. 1 (1): 12. arXiv:astro-ph/9812021. Bibcode:1998LRR.....1...12W. doi:10.12942/lrr-1998-12. PMC 5567250. PMID 28937183. Schneider, P.; Ehlers, J.; Falco, E. E. (1999). Gravitational Lenses". Astronomy and Astrophysics Library. Springer. ISBN 9783540665069. Giannoni, Fabio; Lombardi, Marco (1999). "Gravitational lenses: Odd or even images?". Classical and Quantum Gravity. 16 (6): 1689–1694. Bibcode:1999CQGra..16.1689G. doi:10.1088/0264-9381/16/6/303. S2CID 250827307. 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. 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 (2010). "Gravitational Lensing from a Spacetime Perspective". arXiv:1010.3416. Perlick V., Gravitational lensing from a geometric viewpoint, in B. Schmidt (ed.) "Einstein's field equations and their physical interpretations" Selected Essays in Honour of Jürgen Ehlers, Springer, Heidelberg (2000) pp. 373–425 This astronomy-related article is a stub. You can help Wikipedia by expanding it.

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