Physics:Odd number theorem

From HandWiki

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 :

[math]\displaystyle{ M: (u,v) \mapsto (u',v') }[/math].

Argument

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

However, only in some specific directions [math]\displaystyle{ V_0:(s_0,w_0) }[/math], will the bent light rays reach the observer, i.e., the images only form where [math]\displaystyle{ D=\delta V=0|_{(s_0,w_0)} }[/math]. Then we can directly apply the Poincaré–Hopf theorem [math]\displaystyle{ \chi=\sum \text{index}_D = \text{constant} }[/math].

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

References




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