Odd number theorem

The odd number theorem is a theorem in strong gravitational lensing which comes directly from differential topology. It says that the number of multiple images produced by a bounded transparent lens must be odd.

In fact, the gravitational lensing is a mapping from image plane to source plane . 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.

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It is also a theorem that the sum of the first odd numbers is a square number[1]

References


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