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 :

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

Argument

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

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

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

References

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