雷乔杜里方程

维基百科,自由的百科全书
跳到导航 跳到搜索

广义相对论中,雷乔杜里方程(英語:Raychaudhuri equation),或朗道–雷乔杜里方程(英語:Landau–Raychaudhuri equation[1]是描述邻近物质运动的基本方程。

它不仅是彭罗斯-霍金奇点定理广义相对论的精确解研究的基本引理,还具有独特之处,即它指出引力应该是广义相对论中任意质量-能量之间的普遍存在的吸引力,正如在牛顿引力理论中那样。

这一方程由印度物理学家阿马尔·库马尔·雷乔杜里英语Amal Kumar Raychaudhuri[2]和苏联物理学家列夫·朗道各自独立发现。[3]

数学表述[编辑]

考虑一个类时的单位矢量场 (可理解为不相交的世界线英语Congruence (general relativity)), 雷乔杜里方程可写为

式中

剪切张量

涡度张量

的二次不变量。这里

扩张张量是它的,称为扩张标量

是正交于的超平面上的投影张量。另外,圆点表示对固有时的微分。潮汐张量英语Electrogravitic tensor的迹可写为

+1

这个量有时也称为雷乔杜里标量


参见[编辑]

注释[编辑]

  1. ^ Spacetime as a deformable solid, M. O. Tahim, R. R. Landim, and C. A. S. Almeida, arXiv:0705.4120v1.
  2. ^ Dadhich, Naresh. Amal Kumar Raychaudhuri (1923–2005) (PDF). Current Science. August 2005, 89: 569–570. 
  3. ^ The large scale structure of space-time by Stephen W. Hawking and G. F. R. Ellis, Cambridge University Press, 1973, p. 84, ISBN 0-521-09906-4.

参考资料[编辑]

  • Poisson, Eric. A Relativist's Toolkit: The Mathematics of Black Hole Mechanics. Cambridge: Cambridge University Press. 2004. ISBN 0-521-83091-5.  See chapter 2 for an excellent discussion of Raychaudhuri's equation for both timelike and null geodesics, as well as the focusing theorem.
  • Carroll, Sean M. Spacetime and Geometry: An Introduction to General Relativity. San Francisco: Addison-Wesley. 2004. ISBN 0-8053-8732-3.  See appendix F.
  • Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius; Hertl, Eduard. Exact Solutions to Einstein's Field Equations (2nd ed.). Cambridge: Cambridge University Press. 2003. ISBN 0-521-46136-7.  See chapter 6 for a very detailed introduction to geodesic congruences, including the general form of Raychaudhuri's equation.
  • Hawking, Stephen & Ellis, G. F. R. The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press. 1973. ISBN 0-521-09906-4.  See section 4.1 for a discussion of the general form of Raychaudhuri's equation.
  • Raychaudhuri, A. K. Relativistic cosmology I.. Phys. Rev. 1955, 98 (4): 1123. Bibcode:1955PhRv...98.1123R. doi:10.1103/PhysRev.98.1123.  Raychaudhuri's paper introducing his equation.
  • Dasgupta, Anirvan; Nandan, Hemwati & Kar, Sayan. Kinematics of geodesic flows in stringy black hole backgrounds. Phys. Rev. D. 2009, 79 (12): 124004. Bibcode:2009PhRvD..79l4004D. arXiv:0809.3074. doi:10.1103/PhysRevD.79.124004.  See section IV for derivation of the general form of Raychaudhuri equations for three kinematical quantities (namely expansion scalar, shear and rotation).
  • Kar, Sayan & SenGupta, Soumitra. The Raychaudhuri equations: A Brief review. Pramana. 2007, 69: 49. Bibcode:2007Prama..69...49K. arXiv:gr-qc/0611123. doi:10.1007/s12043-007-0110-9.  See for a review on Raychaudhuri equations.

外部链接[编辑]

  • The Meaning of Einstein's Field Equation by John C. Baez and Emory F. Bunn. Raychaudhuri's equation takes center stage in this well known (and highly recommended) semi-technical exposition of what Einstein's equation says.