雷乔杜里方程

在广义相对论中，雷乔杜里方程（Template:Lang-en），或朗道–雷乔杜里方程（Template:Lang-en）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Template:Le和苏联物理学家列夫·朗道各自独立发现。

数学表述

考虑一个类时的单位矢量场 $\overrightarrow{X}$（可理解为不相交的世界线的Template:Le）, 雷乔杜里方程可写为

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

式中

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

是剪切张量

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

和涡度张量

${\omega }_{ab}={h^m}_a{h^n}_b{X}_{[m;n]}$

的二次不变量。这里

${\theta }_{ab}={h^m}_a{h^n}_b{X}_{(m;n)}$

是扩张张量，$\theta$是它的迹，称为扩张标量。

$h_{ab}=g_{ab}+X_a{X}_b$

是正交于$\overrightarrow{X}$的超平面上的投影张量。另外，圆点表示对固有时的微分。Template:Le$E[\overrightarrow{X}{]}_{ab}$的迹可写为

${E[\overrightarrow{X}{]}^a}_a=R_{mn}{X}^m{X}^n$ +1

这个量有时也称为雷乔杜里标量。

参见

• Template:Le
• 引力奇点
• 彭罗斯-霍金奇点定理

参考资料

• Template:Cite book See chapter 2 for an excellent discussion of Raychaudhuri's equation for both timelike and null geodesics, as well as the focusing theorem.
• Template:Cite book See appendix F.
• Template:Cite book See chapter 6 for a very detailed introduction to geodesic congruences, including the general form of Raychaudhuri's equation.
• Template:Cite book See section 4.1 for a discussion of the general form of Raychaudhuri's equation.
• Template:Cite journal Raychaudhuri's paper introducing his equation.
• Template:Cite journal See section IV for derivation of the general form of Raychaudhuri equations for three kinematical quantities (namely expansion scalar, shear and rotation).
• Template:Cite journal See for a review on Raychaudhuri equations.

外部链接

• The Meaning of Einstein's Field Equation Template:Wayback 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.