Odd number theorem

The gravitational lensing is a thought to mapped from what's known as image plane to source plane following the formula :

$M:\; (u,v)\; \backslash mapsto\; (u\text{'},v\text{'})$.

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 $D=\backslash delta\; V=0|\_\{(s\_0,w\_0)\}$. Then we can directly apply the Poincaré–Hopf theorem $\backslash chi=\backslash sum\; \backslash text\{index\}\_D\; =\; \backslash text\{constant\}$.

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., $\backslash chi=n\_\{+\}-n\_\{-\}=1$. 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.

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• {{cite journal

| last = Perlick | first = Volker

| arxiv = 1010.3416

| date = September 2004

| doi = 10.12942/lrr-2004-9

| issue = 1

| journal = Living Reviews in Relativity

| title = Gravitational lensing from a spacetime perspective

| volume = 7| page = 9

| doi-access = free

| pmid = 28179867

| pmc = 5255571

| bibcode = 2004LRR.....7....9P

}}

Category:Gravitational lensing

Category:Physics theorems

Category:Equations of astronomy

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