The Raychaudhuri equation for a geodesic congruence and finite-time focusing

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold, let $U\subseteq M$ be open and let $n$ be a smooth future unit timelike geodesic congruence on $U$ with kinematic quantities $B$, $\theta$, $\sigma$, $\omega$ and rest-space metric $h$ (Geodesic congruences and their expansion, shear and vorticity). Let $q\in U$ and let ${\gamma }_q:I_q\to U$ be the observer curve through $q$ with affine parameter $s$, $s\in I_q$ (Through each point there is a unique maximal integral curve).

(a) Raychaudhuri equation. For every $s\in I_q$, $\frac{d}{ds}\theta ({\gamma }_q(s))=-\frac{1}{3}\theta ({\gamma }_q(s){)}^2-{\sigma }_{ab}{\sigma }^{ab}({\gamma }_q(s))+{\omega }_{ab}{\omega }^{ab}({\gamma }_q(s))-{\mathrm{Ric}}_{ab}({\gamma }_q(s))n^a({\gamma }_q(s))n^b({\gamma }_q(s)),$ where all contractions use $g$ and $\mathrm{Ric}$ is the Ricci tensor of the metric-selected connection (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors).

(b) Focusing. Suppose $\omega =0$ and ${\mathrm{Ric}}_{ab}{n}^a{n}^b\ge 0$ on $U$, and suppose there is $s_0\in I_q$ with ${\theta }_0:=\theta ({\gamma }_q(s_0))<0$. Then $\theta ({\gamma }_q(s))<0$ for every $s\in I_q$ with $s>s_0$, and $\frac{1}{-\theta ({\gamma }_q(s))}\le \frac{1}{-{\theta }_0}-\frac{s-s_0}{3}\mathrm{.}$ Consequently $s<s_0+3\mathrm{/}(-{\theta }_0)$ for every such $s$, so $\sup I_q\le s_0+3\mathrm{/}(-{\theta }_0)$ for the maximal integral curve in $U$. Any continuation beyond this bound that remains a smooth geodesic congruence and continues to satisfy $\omega =0$ and ${\mathrm{Ric}}_{ab}{n}^a{n}^b\ge 0$ along the continued flow line is impossible. A continuation outside $U$ that does not preserve these hypotheses is not ruled out by this result.

• What focusing does and does not say. The bound concerns the maximal flow line in the region where the smooth congruence and the stated sign hypotheses hold. It does not assert that the spacetime or the individual geodesic ends at $s_0+3\mathrm{/}(-{\theta }_0)$, and it does not constrain continuations on which the hypotheses fail. In flat spacetime the radial congruence from a point reaches infinite expansion in finite affine time at its focal point while the spacetime and the geodesics continue, as the companion examples page shows.
• Hypotheses. Irrotationality ($\omega =0$) and the sign of the Ricci term are both essential: the vorticity term enters with the opposite sign, and without the Ricci sign assumption the expansion need not decrease. No energy condition is asserted: the inequality ${\mathrm{Ric}}_{ab}{n}^a{n}^b\ge 0$ is a hypothesis on the geometry, and whether a matter model satisfies it is a separate question.