雷乔杜里方程

在广义相对论中，雷乔杜里方程（英語：Raychaudhuri equation），或朗道–雷乔杜里方程（英語：Landau–Raychaudhuri equation）是描述邻近物质运动的基本方程。 它不仅是彭罗斯-霍金奇点定理和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）和苏联物理学家列夫·朗道各自独立发现。 考虑一个类时的单位矢量场 X → {\displaystyle {\vec {X}}} （可理解为不相交的世界线的汇（英语：Congruence (general relativity)））, 雷乔杜里方程可写为 θ ˙ = − θ 2 3 − 2 σ 2 + 2 ω 2 − E [ X → ] a a + X ˙ a ; a {\displaystyle {\dot {\theta }}=-{\frac {\theta ^{2}}{3}}-2\sigma ^{2}+2\omega ^{2}-{E[{\vec {X}}]^{a}}_{a}+{{\dot {X}}^{a}}_{;a}} 式中 σ 2 = 1 2 σ m n σ m n , ω 2 = 1 2 ω m n ω m n {\displaystyle \sigma ^{2}={\frac {1}{2}}\sigma _{mn}\,\sigma ^{mn},\;\omega ^{2}={\frac {1}{2}}\omega _{mn}\,\omega ^{mn}} 是剪切张量 σ a b = θ a b − 1 3 θ h a b {\displaystyle \sigma _{ab}=\theta _{ab}-{\frac {1}{3}}\,\theta \,h_{ab}} 和涡度张量 ω a b = h m a h n b X [ m ; n ] {\displaystyle \omega _{ab}={h^{m}}_{a}\,{h^{n}}_{b}X_{[m;n]}} 的二次不变量。这里 θ a b = h m a h n b X ( m ; n ) {\displaystyle \theta _{ab}={h^{m}}_{a}\,{h^{n}}_{b}X_{(m;n)}} 是扩张张量， θ {\displaystyle \theta } 是它的迹，称为扩张标量。 h a b = g a b + X a X b {\displaystyle h_{ab}=g_{ab}+X_{a}\,X_{b}} 是正交于 X → {\displaystyle {\vec {X}}} 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） E [ X → ] a b {\displaystyle E[{\vec {X}}]_{ab}} 的迹可写为 E [ X → ] a a = R m n X m X n {\displaystyle {E[{\vec {X}}]^{a}}_{a}=R_{mn}\,X^{m}\,X^{n}} +1 这个量有时也称为雷乔杜里标量。 汇（英语：Congruence (general relativity)） 引力奇点 彭罗斯-霍金奇点定理 Spacetime as a deformable solid, M. O. Tahim, R. R. Landim, and C. A. S. Almeida, arXiv:0705.4120v1 . Dadhich, Naresh. Amal Kumar Raychaudhuri (1923–2005) (PDF). Current Science. August 2005, 89: 569–570 [2018-10-29]. （原始内容存档 (PDF)于2020-01-03）.  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. 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. The Meaning of Einstein's Field Equation （页面存档备份，存于互联网档案馆） 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.