翻訳と辞書 Words near each other ・ Van Speijk-class frigate ・ Van Spence ・ Van Sprang ・ Van Springel ・ Van Stadens Bridge ・ Van Stadens Pass ・ Van Stadens River ・ Van Stadens Wind Farm ・ Van Stadensriviermond ・ Van Stadensrus ・ Van Staveren ・ Van Steeden ・ Van Steen ・ Van Steenwinckel ・ Van Stephenson ・ Van Stockum dust ・ Van Straaten ・ Van Straelen ・ Van Straten ・ Van Subregion ・ Van Sung's shrew ・ Van Susteren ・ Van Swearingen-Shepherd House ・ Van Sweringen brothers ・ Van Sweringen railroad holdings ・ Van Swinden ・ Van Syckel, New Jersey ・ Van T. Barfoot ・ Van Tard ・ Van Tassel Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Van Stockum dust ： ウィキペディア英語版 Van Stockum dust In general relativity, the van Stockum dust is an exact solution of the Einstein field equation in which the gravitational field is generated by dust rotating about an axis of cylindrical symmetry. Since the density of the dust is ''increasing'' with distance from this axis, the solution is rather artificial, but as one of the simplest known solutions in general relativity, it stands as a pedagogically important example. This solution is named for Willem Jacob van Stockum, who rediscovered it in 1937, independently of an earlier discovery by Cornelius Lanczos in 1924. ==Derivation== One way of obtaining this solution is to look for a cylindrically symmetric perfect fluid solution in which the fluid exhibits ''rigid rotation''. That is, we demand that the world lines of the fluid particles form a timelike congruence having nonzero vorticity but vanishing expansion and shear. (In fact, since dust particles feel no forces, this will turn out to be a timelike ''geodesic'' congruence, but we won't need to assume this in advance.) A simple Ansatz corresponding to this demand is expressed by the following frame field, which contains two undetermined functions of $r$: :$\backslash vec\_0\; =\; \backslash partial\_t,\; \backslash ;\; \backslash vec\_1\; =\; f(r)\; \backslash ,\; \backslash partial\_z,\; \backslash ;\; \backslash vec\_2\; =\; f(r)\; \backslash ,\; \backslash partial\_r,\; \backslash ;\; \backslash vec\_3\; =\; \backslash frac\; \backslash ,\; \backslash partial\_\backslash phi\; -\; h(r)\; \backslash ,\; \backslash partial\_t$ To prevent misunderstanding, we should emphasize that taking the ''dual coframe'' :$\backslash sigma^0\; =\; -dt\; +\; h(r)\; r\; \backslash ,\; d\backslash phi,\; \backslash ;\; \backslash sigma^1\; =\; \backslash frac\; \backslash ,\; dz,\; \backslash ;\; \backslash sigma^2\; =\; \backslash frac\; \backslash ,\; dr,\; \backslash ;\; \backslash sigma^3\; =\; r\; d\backslash phi$ gives the metric tensor in terms of the same two undetermined functions: :$g\; =\; -\backslash sigma^0\; \backslash otimes\; \backslash sigma^0\; +\; \backslash sigma^1\; \backslash otimes\; \backslash sigma^1\; +\; \backslash sigma^2\; \backslash otimes\; \backslash sigma^2\; +\; \backslash sigma^3\; \backslash otimes\; \backslash sigma^3$ Multiplying out gives :$ds^2\; =\; -dt^2\; -\; 2\; h(r)\; r\; \backslash ,\; dt\; \backslash ,\; d\backslash phi\; +\; (1-h(r)^2)\; r^2\; \backslash ,\; d\backslash phi^2\; +\; \backslash frac$ : We compute the Einstein tensor with respect to this frame, in terms of the two undetermined functions, and demand that the result have the form appropriate for a perfect fluid solution with the timelike unit vector $\backslash vec\_0$ everywhere tangent to the world line of a fluid particle. That is, we demand that :$G^\}\; =\; 8\; \backslash pi\; \backslash mu\; \backslash ,\; \backslash operatorname(1,0,0,0)\; +\; 8\; \backslash pi\; p\; \backslash ,\; \backslash operatorname(0,1,1,1)$ This gives the conditions :$f^\; =\; \backslash frac\; +\; \backslash frac,\; \backslash ;\; (h^\backslash prime)^2\; +\; \backslash frac\; +\; \backslash frac\; =\; \backslash frac$ Solving for $f$ and then for $h$ gives the desired frame defining the van Stockum solution: :$\backslash vec\_0\; =\; \backslash partial\_t,\; \backslash ;\; \backslash vec\_1\; =\; \backslash exp(a^2\; r^2/2)\; \backslash ,\; \backslash partial\_z,\; \backslash ;\; \backslash vec\_2\; =\; \backslash exp(a^2\; r^2/2)\; \backslash ,\; \backslash partial\_r,\; \backslash ;\; \backslash vec\_3\; =\; \backslash frac\; \backslash ,\; \backslash partial\_\backslash phi\; -\; a\; r\; \backslash ,\; \backslash partial\_t$ Note that this frame is only defined on $r>0$. 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「Van Stockum dust」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.