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

2

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

References

↑ http://mathworld.wolfram.com/OddNumberTheorem.html

Chwolson O., 1924. "Über eine mögliche Form fiktiver Doppelsterne", "Astronomische Nachrichten" 221, 329-330.
Burke W.L., 1981. "Multiple gravitational imaging by distributed masses", Astrophysical Journal 244, L1.
McKenzie R.H., 1985. "A gravitational lens produced an odd number of images", Journal of Mathematical Physics 26, 1592.
Kozameh C, Lamberti P. W., Reula O. Global aspects of light cone cuts. J. Math. Phys. 32, 3423-3426 (1991).
Lombardi M., An application of the topological degree to gravitational lenses. Modern Phys. Lett. A 13, 83-86 (1998).
Wambsganss J., 1998. "Gravitational lensing in astronomy" http://relativity.livingreviews.org/Articles/lrr-1998-12/download/lrr-1998-12BW.pdf
Schneider P., Ehlers J., Falco E. E. 1999. "Gravitational Lenses" Astronomy and Astrophysics Library. Springer
Giannoni F., Lombardi M, 1999. "Gravitational lenses: Odd or even images?" "Class. Quantum Grav." 16, 375-415.
Fritelli S., Newman E. T., 1999. "Exact universal gravitational lens equations" "Phys. Rev." D 59, 124001
Perlick V., Gravitational lensing in asymptotically simple and empty spacetimes, Annalen der Physik 9, SI139-SI142 (2000)
Perlick V., Gravitational lensing from a geometric viewpoint, in B. Schmidt (ed.) "Einstein's field equations and their physical interpretations" Selected Essays in Honour of Jürgen Ehlers, Springer, Heidelberg (2000) pp. 373–425
Perlick V., 2010. "Gravitational Lensing from a Spacetime Perspective" http://arxiv.org/abs/1010.3416

This article is issued from Wikipedia - version of the 10/29/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.