在广义相对论中，雷乔杜里方程（英语：Raychaudhuri equation），或朗道–雷乔杜里方程（英语：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）^[2]和苏联物理学家列夫·朗道各自独立发现。^[3]

数学表述

考虑一个类时的单位矢量场 ${\displaystyle {\vec {X))}$（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

${\displaystyle {\dot {\theta ))=-{\frac {\theta ^{2)){3))-2\sigma ^{2}+2\omega ^{2}-{E[{\vec {X))]^{a))_{a}+((\dot {X))^{a))_{;a))$

式中

${\displaystyle \sigma ^{2}={\frac {1}{2))\sigma _{mn}\,\sigma ^{mn},\;\omega ^{2}={\frac {1}{2))\omega _{mn}\,\omega ^{mn))$

是剪切张量

${\displaystyle \sigma _{ab}=\theta _{ab}-{\frac {1}{3))\,\theta \,h_{ab))$

和涡度张量

${\displaystyle \omega _{ab}={h^{m))_{a}\,{h^{n))_{b}X_{[m;n]))$

的二次不变量。这里

${\displaystyle \theta _{ab}={h^{m))_{a}\,{h^{n))_{b}X_{(m;n)))$

是扩张张量，${\displaystyle \theta }$是它的迹，称为扩张标量。

${\displaystyle h_{ab}=g_{ab}+X_{a}\,X_{b))$

是正交于${\displaystyle {\vec {X))}$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）${\displaystyle E[{\vec {X))]_{ab))$的迹可写为

${\displaystyle {E[{\vec {X))]^{a))_{a}=R_{mn}\,X^{m}\,X^{n))$ +1

这个量有时也称为雷乔杜里标量。

参见

• 汇（英语：Congruence (general relativity)）
• 引力奇点
• 彭罗斯-霍金奇点定理

注释

参考资料

• 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

• 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.