Penrose's singularity theorem (1965)

Step 1 (trapped surface, in symbols). Let $S$ be a closed spacelike 2-surface, and let $\theta$ denote the expansion of a bundle of null geodesics leaving $S$ orthogonally — the fractional rate at which the bundle's cross-sectional area grows. Far from any mass, the outgoing bundle expands ($\theta_{\text{out}}>0$) and the ingoing one contracts ($\theta_{\text{in}}<0$). $S$ is trapped if instead $\theta_{\text{out}}<0, \qquad \theta_{\text{in}}<0$ on all of $S$: even the outgoing light is being dragged inward.

Step 2 (the Raychaudhuri equation). Differentiating the definition $\theta=\nabla_a k^a$ of the expansion along the null congruence with tangent $k^a$, and using the definition of the Riemann tensor to commute derivatives, gives the exact kinematic identity $$\frac{d\theta}{d\lambda} = -\frac{\theta^2}{2} - \sigma_{ab}\sigma^{ab} - R_{ab}k^ak^b$$ where $\sigma_{ab}$ is the shear tensor of the congruence. This equation is purely geometric — a statement about how any family of light rays must bend, following from the definition of curvature, before any physics is assumed.

Step 3 (energy condition forces further focusing). Physically reasonable matter satisfies the null energy condition $$R_{ab}k^ak^b \ge 0$$ and the shear term $\sigma_{ab}\sigma^{ab}\ge0$ always (it is a sum of squares); dropping both non-negative terms from the right of Step 2's identity gives the inequality $$\frac{d\theta}{d\lambda} \le -\frac{\theta^2}{2}$$

Step 4 (the Riccati inequality forces a finite-time blow-up). On a trapped surface $\theta_0<0$ at $\lambda=0$. As long as $\theta$ stays negative, dividing the inequality by $\theta^2>0$ and rearranging gives $$\frac{d}{d\lambda}\left(\frac{1}{\theta}\right) \ge \frac12 \;\Longrightarrow\; \frac{1}{\theta(\lambda)} \ge \frac{1}{\theta_0}+\frac{\lambda}{2}$$ Since the right side grows without bound as $\lambda$ increases while $1/\theta$ (negative) must satisfy this lower bound, $1/\theta$ is forced up toward $0^-$, i.e. $\theta\to-\infty$, at some finite affine parameter $\lambda_*\le 2/|\theta_0|$: the null congruence orthogonal to $S$ must develop a conjugate point (a caustic, where neighboring light rays cross) within finite affine time.

Step 5 (from local focusing to global incompleteness). A standard result of causal structure theory (used but not re-derived here) shows that a null geodesic in a globally hyperbolic spacetime cannot remain achronal — cannot stay on the boundary of the future of a set — past a conjugate point. Combined with the non-compact Cauchy surface, this rules out the geodesic looping back or terminating in an ordinary way, and forces at least one null geodesic orthogonal to $S$ to be future-incomplete: it cannot be extended to arbitrarily large affine parameter. By definition, this incompleteness is what Penrose calls a singularity — the theorem shows it must occur without any assumption of symmetry, whenever a trapped surface forms.