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 :

$M:(u,v)\mapsto (u',v')$.

Argument

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

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

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 $n_{+}$ and the number of negative indices $n_{-}$. For the far field case, there is only one image, i.e., $\chi =n_{+}-n_{-}=1$. So the total number of images is $N=n_{+}+n_{-}=2n_{-}+1$, 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). 221 (20). Wiley: 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. 244. IOP Publishing: 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. 26 (7). AIP Publishing: 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. 32 (12). AIP Publishing: 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. 13 (2). World Scientific Pub Co Pte Lt: 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. {{cite journal}}: Cite journal requires |journal= (help)
• 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