Raychaudhuri equation

Remarks

The proof differentiates $B_{ab} = \nabla_b u_a$ along $u$: using the geodesic equation and the definition of the Riemann tensor, $u^c \nabla_c B_{ab} = -B_{ac} B^c{}_b + R_{cba}{}^d u^c u_d$ in a standard convention. Taking the trace and inserting the decomposition $B_{ab} = \frac13 \theta h_{ab} + \sigma_{ab} + \omega_{ab}$ gives the equation.

The most important consequence is focusing. Assume $\omega_{ab} = 0$, for example for a congruence orthogonal to a spacelike hypersurface, and $R_{ab} u^a u^b \ge 0$. By the Einstein field equations, the latter holds when matter satisfies the strong energy condition. Since $\sigma_{ab}\sigma^{ab} \ge 0$, one gets $d\theta/d\tau \le -\theta^2/3$, hence $\theta^{-1} \ge \theta_0^{-1} + \tau/3$. So if $\theta_0 = \theta(0) < 0$, then $\theta \to -\infty$ within proper time $\tau \le 3/\abs{\theta_0}$: nearby geodesics cross. Without the sign condition on $R_{ab} u^a u^b$, focusing need not happen, and the rotation term $+\omega_{ab}\omega^{ab}$ also works against focusing.

Example: for the radially expanding congruence $u^\mu = x^\mu / \tau$ in Minkowski space, $\theta = 3/\tau$, $\sigma = 0$, $\omega = 0$ and $R_{ab} = 0$; indeed $d\theta/d\tau = -3/\tau^2 = -\theta^2/3$.

Going deeper

For a student

The equation is the engine of the Penrose–Hawking singularity theorems: focusing of geodesics plus a global causality argument forces some geodesic to be incomplete. For null geodesic congruences there is an analogous equation, with $\frac12 \theta^2$ in place of $\frac13 \theta^2$. Together with the null energy condition, it is used in the area theorem for black holes and in the theory of trapped surfaces. In cosmology, applied to the matter flow of a homogeneous universe, it reduces to the acceleration equation of Friedmann.