雷喬杜里方程

在廣義相對論中，雷喬杜里方程（英語：Raychaudhuri equation），或朗道–雷喬杜里方程（英語：Landau–Raychaudhuri equation）是描述鄰近物質運動的基本方程。

它不僅是彭羅斯-霍金奇點定理和廣義相對論的精確解研究的基本引理，還具有獨特之處，即它指出引力應該是廣義相對論中任意質量-能量之間的普遍存在的吸引力，正如在牛頓引力理論中那樣。

這一方程由印度物理學家阿馬爾·庫馬爾·雷喬杜里和蘇聯物理學家列夫·朗道各自獨立發現。

數學表述

考慮一個類時的單位矢量場 ${\vec {X}}$（可理解為不相交的世界線的匯）, 雷喬杜里方程可寫為

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

式中

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

是剪切張量

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

和渦度張量

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

的二次不變量。這裡

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

是擴張張量，$\theta$是它的跡，稱為擴張純量。

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

是正交於${\vec {X}}$的超平面上的投影張量。另外，圓點表示對固有時的微分。潮汐張量$E[{\vec {X}}]_{ab}$的跡可寫為

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

這個量有時也稱為雷喬杜里純量。

參見

• 匯
• 引力奇點
• 彭羅斯-霍金奇點定理

參考資料

• 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.

外部連結

• The Meaning of Einstein's Field Equation （頁面存檔備份，存於網際網路檔案館） 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.