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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