在广义相对论中，雷乔杜里方程（），或朗道–雷乔杜里方程（）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 \vec{X}（可理解为不相交的世界线的）, 雷乔杜里方程可写为
: \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\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。E[\vec{X}]_{ab}的迹可写为
: {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
*
- 引力奇点
- 彭罗斯-霍金奇点定理
注释
参考资料
- See chapter 2 for an excellent discussion of Raychaudhuri's equation for both timelike and null geodesics, as well as the focusing theorem.
- See appendix F.
- See chapter 6 for a very detailed introduction to geodesic congruences, including the general form of Raychaudhuri's equation.
- See section 4.1 for a discussion of the general form of Raychaudhuri's equation.
- Raychaudhuri's paper introducing his equation.
- See section IV for derivation of the general form of Raychaudhuri equations for three kinematical quantities (namely expansion scalar, shear and rotation).
- See for a review on Raychaudhuri equations.
外部链接
- [http://math.ucr.edu/home/baez/einstein/ 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.
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