Odd number theorem

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 :

[math]\displaystyle{ M: (u,v) \mapsto (u',v') }[/math].

## Argument

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

However, only in some specific directions [math]\displaystyle{ V_0:(s_0,w_0) }[/math], will the bent light rays reach the observer, i.e., the images only form where [math]\displaystyle{ D=\delta V=0|_{(s_0,w_0)} }[/math]. Then we can directly apply the Poincaré–Hopf theorem [math]\displaystyle{ \chi=\sum \text{index}_D = \text{constant} }[/math].

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

## References

* Chwolson, O. (1924). "Über eine mögliche Form fiktiver Doppelsterne" (in de). Astronomische Nachrichten (Wiley) 221 (20): 329–330. doi:10.1002/asna.19242212003. ISSN 0004-6337. Bibcode: 1924AN....221..329C.
* Burke, W. L. (1981). "Multiple Gravitational Imaging by Distributed Masses". The Astrophysical Journal (IOP Publishing) 244: L1. doi:10.1086/183466. ISSN 0004-637X. Bibcode: 1981ApJ...244L...1B.
* McKenzie, Ross H. (1985). "A gravitational lens produces an odd number of images". Journal of Mathematical Physics (AIP Publishing) 26 (7): 1592–1596. doi:10.1063/1.526923. ISSN 0022-2488. Bibcode: 1985JMP....26.1592M.
* Kozameh, Carlos; Lamberti, Pedro W.; Reula, Oscar (1991). "Global aspects of light cone cuts". Journal of Mathematical Physics (AIP Publishing) 32 (12): 3423–3426. doi:10.1063/1.529456. ISSN 0022-2488. Bibcode: 1991JMP....32.3423K.
* 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. doi:10.1142/s0217732398000115. ISSN 0217-7323. Bibcode: 1998MPLA...13...83L.
* Wambsganss, Joachim (1998). "Gravitational Lensing in Astronomy". Living Reviews in Relativity 1 (1): 12. doi:10.12942/lrr-1998-12. PMID 28937183. Bibcode: 1998LRR.....1...12W.
* 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. doi:10.1088/0264-9381/16/6/303. Bibcode: 1999CQGra..16.1689G.
* Frittelli, Simonetta; Newman, Ezra T. (1999-04-28). "Exact universal gravitational lensing equation". Physical Review D 59 (12): 124001. doi:10.1103/physrevd.59.124001. ISSN 0556-2821. Bibcode: 1999PhRvD..59l4001F.
* Perlick, Volker (1999). "Gravitational Lensing from a Geometric Viewpoint". Einstein's Field Equations and Their Physical Implications. Lecture Notes in Physics. 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.
* 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

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