Odd number theorem

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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[edit]

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

.

Argument[edit]

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[edit]

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  • Schneider, P.; Ehlers, J.; Falco, E. E. (1999). Gravitational Lenses". Astronomy and Astrophysics Library. Springer. ISBN 9783540665069.
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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.
  • 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". arXiv:1010.3416. 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