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

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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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• 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)
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