Doubt regarding the existence of conjugate points on timelike geodesics

Question

The expansion of a timelike geodesic congruence with (normalized) tangent vector field $\xi^a$ is defined as $\theta=\nabla_a\xi^a$. Assuming the strong energy condition, $R_{ab}\xi^a\xi^b\geq 0$, and that the congruence is hypersurface-orthogonal (i.e. $\nabla_b\xi_a=\nabla_a\xi_b$), then the time evolution of $\theta$ obeys the following equation (Wald 1984, Sec. 9.2): $$\theta^{-1}(\tau)\geq\theta_0^{-1}+\frac{1}{3}\tau\tag*{(1)}$$ From this it follows that for negative initial expansion, $\theta_0<0$, $\theta^{-1}(\tau)$ approaches zero on the negative side as $\tau$ increases, meaning the expansion $\theta$ diverges to $-\infty$. Gravity is attractive!

So far so good... But what if the initial expansion were positive? In that case $\theta(\tau)$ remains positive for all $\tau$, approaching zero for $\tau\rightarrow\infty$. From how I understand it, this seems to be saying that there is no matter distribution (satisfying the strong energy condition) that can re-converge an initially expanding geodesic congruence. But that seems obviously wrong. The initially-expanding geodesics of particles passing sufficiently close to, say, a planet can obviously reconverge due to deflection by gravity right? Indeed, the point of this whole section is essentially to show that geodesics possess conjugate points (pairs of points on a geodesic that each have $\theta=-\infty$). Wald claims that

If $R_{ab}\xi^a\xi^b\geq 0$ everywhere along the geodesic $\gamma$ and $R_{ab}\xi^a\xi^b > 0$ at point $r\in\gamma$, then one can show that for $p$ sufficiently far from $r$, the expansion of the timelike geodesic congruence emanating from $p$ must be negative at $r$. Hence $p$ will have a conjugate point $q$ on $\gamma$.

The last statement regarding $q$ follows from the same reasoning I presented at the beginning for negative initial expansion at $r$. However he also says that the expansion emanating from $p$ can become negative at a later point $r$ assuming that $R_{ab}\xi^a\xi^b>0$ at $r$ (as I would expect intuitively). Presumably the expansion emanating from $p$ is positive (somebody correct me if not), so this seems to be in direct contradiction with (1). What gives?

Answer 1

For a simple example, consider empty spacetime, curved as though a planet were centered on the origin. Consider a spherical shell of dust that is also centered on the origin, initially expanding. The worldlines of the motes of dust are the timelike geodesics in the congruence. The expansion is initially positive, but if the initial outward velocity is not too great, then the shell's expansion will eventually slow to zero, after which it will being to shrink (negative expansion).

This does not contradict (1), because (1) is only an inequality, not an equation. The inequality (1) is derived from Raychaudhuri's equation, which is $$ \frac{d\theta}{d\tau}=-\frac{1}{3}\theta^2-\sigma_{ab}\sigma^{ab} +\omega_{ab}\omega^{ab}-R_{ab}\xi^a\xi^b. \tag{2} $$ This is equation (9.2.11) in Wald's book. For the simple example that I just described, the rotation $\omega_{ab}$ and shear $\sigma_{ab}$ are both zero because of the scenario's spherical symmetry, so equation (2) reduces to $$ \frac{d\theta}{d\tau}=-\frac{1}{3}\theta^2-R_{ab}\xi^a\xi^b. \tag{3} $$ Initially, our shell's expansion is positive: $\theta>0$. Thanks to the strong energy condition, both terms on the right-hand side of (3) are negative, so equation (3) says that the magnitude of the expansion decreases. Eventually the rate of expansion goes to zero ($\theta=0$), so that the first term on the right-hand side is zero. But the second term on the right-hand side is still negative, so the equation says that $\theta$ continues to decrease, thus becoming negative, exactly as our physical intuition says it should.

The inequality (1) is derived from the fact that equation (3) implies the inequality $$ \frac{d\theta}{d\tau}\leq -\frac{1}{3}\theta^2. \tag{4} $$ The key is that this is an inequality, not an equation. It comes from equation (3), which is perfectly consistent with our physical intuition that an initially-positive expansion can eventually become negative.