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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 Cauchy surface ： ウィキペディア英語版 Cauchy surface Intuitively, a Cauchy surface is a plane in space-time which is like an ''instant'' of time; its significance is that giving the initial conditions on this plane determines the future (and the past) uniquely. More precisely, a Cauchy surface is any subset of space-time which is intersected by every inextensible, non-spacelike (i.e. causal) curve exactly once. A partial Cauchy surface is a hypersurface which is intersected by any causal curve ''at most'' once. It is named for French mathematician Augustin Louis Cauchy. == Discussion == If $\backslash mathcal$ is a space-like surface (i.e., a collection of points such that every pair is space-like separated), then $D^(\backslash mathcal)$ is the future of $\backslash mathcal$, i. e.: $D^+(\backslash mathcal):=\backslash p\backslash text\backslash mathcal)$ Similarly $D^(\backslash mathcal)$, the past of $\backslash mathcal$, is the same thing going forward in time. When there are no closed timelike curves, $D^$ and $D^$ are two different regions. When the time dimension closes up on itself everywhere so that it makes a circle, the future and the past of $\backslash mathcal$ are the same and both include $\backslash mathcal$. The Cauchy surface is defined rigorously in terms of intersections with inextensible curves in order to deal with this case of circular time. An inextensible curve is a curve with no ends: either it goes on forever, remaining timelike or null, or it closes in on itself to make a circle, a closed non-spacelike curve. When there are closed timelike curves, or even when there are closed non-spacelike curves, a Cauchy surface still determines the future, but the future includes the surface itself. This means that the initial conditions obey a constraint, and the Cauchy surface is not of the same character as when the future and the past are disjoint. If there are no closed timelike curves, then given $\backslash mathcal$ a partial Cauchy surface and if $D^(\backslash mathcal)\backslash cup\; \backslash mathcal\backslash cup\; D^(\backslash mathcal)\; =\; \backslash mathcal$, the entire manifold, then $\backslash mathcal$ is a Cauchy surface. Any surface of constant $t$ in Minkowski space-time is a Cauchy surface. 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「Cauchy surface」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.