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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Odd number theorem ： ウィキペディア英語版 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 $M:\; (u,v)\; \backslash mapsto\; (u\text{'},v\text{'})\backslash ,$. If we use direction cosines describing the bent light rays, we can write a vector field on $(u,v)\backslash ,$ plane $V:(s,w)\backslash ,$. However, only in some specific directions $V\_0:(s\_0,w\_0)\backslash ,$, will the bent light rays reach the observer, i.e., the images only form where $D=\backslash delta\; V=0|\_$. Then we can directly apply the Poincaré–Hopf theorem $\backslash chi=\backslash sum\; \backslash text\_D\; =\; \backslash text\backslash ,$. 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 $n\_\backslash ,$ and the number of negative indices $n\_\backslash ,$. For the far field case, there is only one image, i.e., $\backslash chi=n\_-n\_=1\backslash ,$. So the total number of images is $N=n\_+n\_=2n\_+1\; \backslash ,$, i.e., odd. The strict proof needs Uhlenbeck’s Morse theory of null geodesics. == References == * 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. "Gravtational lensing in astronomy" http://mathnet.preprints.org/EMIS/journals/LRG/Articles/lrr-1998-12/ * 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 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Odd number theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.