# 雷乔杜里方程

## 数学表述

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

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

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

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

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

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

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

## 注释

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

• 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 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.