Focusing and Conjugate-Point Lemmas

Focusing becomes useful in a singularity theorem only after two translations. Raychaudhuri turns negative expansion and Ricci convergence into loss of rank of a Jacobi map within finite affine or proper parameter. Lorentzian variational theory then turns that focal or conjugate point into loss of a maximizing or achronal-boundary property. Neither translation alone proves incompleteness: the geodesic may continue after focusing, and a global causal argument is still required.

Required background. Congruences and Optical Scalars supplies the optical decomposition; Einstein Equations relates matter inequalities to Ricci convergence. Helpful background. Lorentzian Metrics and Causal Cones fixes causal signs; Geodesics and Jacobi Fields supplies variations and the exponential map; Causality and Global Structure supplies maximizing and boundary arguments; Curvature fixes the tidal operator.

Jacobi maps and optical scalars

Let $\gamma$ be an affinely parametrized causal geodesic in an $n$-dimensional $C^2$ spacetime. For a unit timelike tangent $u=\dot\gamma$, transverse Jacobi fields live in the positive-definite $(n-1)$-dimensional space $u^\perp$. For a null tangent $k$, the relevant positive-definite screen is the quotient

$$[k]^\perp=k^\perp/\operatorname{span}\{k\},$$

of dimension $n-2$. Write $d$ for the appropriate transverse dimension.

A Jacobi map $A(\lambda)$ collects a basis of transverse Jacobi fields. Before it becomes singular,

$$B=\dot A A^{-1}, \qquad \theta=\operatorname{tr}B =\frac{d}{d\lambda}\log\lvert\det A\rvert.$$

For a twist-free, or equivalently Lagrange, Jacobi family, $B$ is self-adjoint and decomposes as $B=(\theta/d)I+\sigma$. Its matrix Riccati equation gives

$$\dot\theta =-\frac{\theta^2}{d} -\operatorname{tr}(\sigma^2) -R_{ab}\dot\gamma^a\dot\gamma^b.$$

The vorticity term is absent because point-emitted and surface-orthogonal congruences are hypersurface orthogonal until their first caustic. For a general twisting congruence it would enter with the opposite sign and could obstruct the simple inequality. The Jacobi-map construction, including why the null tangent itself must be quotiented out, is given in Steinbauer 2023, §3.2, pp. 10–12.

Finite-parameter focusing lemma

Assume along the geodesic segment that

$$R_{ab}\dot\gamma^a\dot\gamma^b\geq0,$$

and that the expansion at some regular parameter value $\lambda_0$ is $\theta_0<0$. Since the screen metric is positive definite, $\operatorname{tr}(\sigma^2)\geq0$, so

$$\dot\theta\leq-\frac{\theta^2}{d}.$$

Divide by $\theta^2$ while $\theta$ remains finite and negative:

$$\frac{d}{d\lambda}\left(\frac1\theta\right) =-\frac{\dot\theta}{\theta^2} \geq\frac1d.$$

Integration gives

$$\frac1{\theta(\lambda)} \geq \frac1{\theta_0}+\frac{\lambda-\lambda_0}{d}.$$

The right-hand side reaches zero at $\lambda_0+d/\lvert\theta_0\rvert$. A regular negative $\theta$ cannot cross through $-\infty$ to satisfy the inequality. Instead $\det A$ vanishes no later than that bound. For a Jacobi map based at a point, this is a conjugate point to the base point. For a family initially normal to a submanifold, it is a focal point to that submanifold. The precise proposition and its “provided the geodesic exists that long” qualification appear in Steinbauer 2023, Propositions 3.3–3.4, pp. 12–14.

That qualification is logically decisive. If an inextendible geodesic ends before the bound at finite affine or proper parameter, it is already incomplete. If it exists through the bound, focusing forces a focal or conjugate point. The lemma is therefore often used as a dichotomy rather than as an unconditional promise that a conjugate point lies in the spacetime.

In four dimensions, $d=3$ for timelike geodesics and $d=2$ for null geodesics. Thus a twist-free null bundle with $\theta_0<0$ focuses within affine parameter at most $2/\lvert\theta_0\rvert$. The number has meaning only after the null tangent’s affine normalization has been fixed: $k\mapsto ck$ rescales both affine parameter and expansion.

Focal data from hypersurfaces

For a spacelike hypersurface $S$ with chosen future unit normal $n$, consider future timelike geodesics leaving $S$ orthogonally. The initial derivative of their Jacobi map is set by the shape operator. With a declared extrinsic-curvature convention, the trace fixes their initial expansion. A uniform inequality $\theta(0)\leq-\theta_*$ with $\theta_*>0$ makes every such normal geodesic focal to $S$ by proper time at most $(n-1)/\theta_*$, provided it lasts that long. This is the analytic core of a future-directed Hawking theorem. Reversing time converts present expansion into past-directed convergence.

For a compact spacelike codimension-two surface $P$, there are two independent future null-normal families. If both expansions are negative everywhere, $P$ is future trapped. Compactness makes the largest focusing bound finite: after fixing the null normalization continuously, the maximum of $2/\lvert\theta_\pm\rvert$ over $P$ exists in four dimensions. Under null convergence, every generator that lasts sufficiently long becomes focal to $P$. The surface definitions and normalization dependence are separated on Trapped and Marginal Surfaces.

Negative expansion need not be caused by curvature. In flat Minkowski spacetime, ingoing radial null geodesics orthogonal to a round sphere have $\theta=-2/r$ and meet at the center. Ricci convergence holds with equality, yet the focal point reflects the initial geometry. A trapped surface requires both future null families to converge and, in the Penrose theorem, compactness and additional global hypotheses.

Worked example: a caustic in flat spacetime

Take a round sphere of radius $R$ at $t=0$ in four-dimensional Minkowski spacetime. Normalize each future null tangent by $dt/d\lambda=1$. The outgoing and ingoing sphere radii, before the ingoing rays reach the center, are

$$r_+(\lambda)=R+\lambda, \qquad r_-(\lambda)=R-\lambda, \qquad 0\leq\lambda\lt R.$$

Their areas are $A_\pm=4\pi r_\pm^2$. Thus

$$\theta_+=\frac{d\log A_+}{d\lambda} =\frac{2}{R+\lambda}, \qquad \theta_-=-\frac{2}{R-\lambda}.$$

The ingoing family saturates $d\theta_-/d\lambda=-\theta_-^2/2$ and focuses exactly at $\lambda=R=2/\lvert\theta_-(0)\rvert$. There is no curvature: $R_{abcd}=0$. In Cartesian coordinates a ray labeled by a unit vector $\boldsymbol n$ is simply

$$t=\lambda, \qquad \boldsymbol x=(R-\lambda)\boldsymbol n.$$

It continues smoothly through the center for every $\lambda\gt R$. What fails at $R$ is the one-to-one labeling of a congruence by its initial sphere, not the geodesic equation. After the crossing the rays lie inside the chronological future of the initial sphere and cannot continue generating its achronal future boundary. This explicit calculation separates a focal point from a singular endpoint. The sphere is not trapped because $\theta_+\gt0$; compare the two-expansion sign diagram.

Conjugate points destroy maximizing behavior

For timelike-separated points $p$ and $q$, a maximizing causal curve is a timelike geodesic. If a timelike geodesic contains a point conjugate to $p$ before $q$, its index form has a variation that increases proper time. It therefore cannot maximize from $p$ to $q$. Similarly, a timelike geodesic normal to $S$ ceases to maximize the Lorentzian distance $d(S,\cdot)$ after its first focal point to $S$. These variational statements are reviewed in Steinbauer 2023, Proposition 3.2 and §3.3, pp. 11–14.

For null geodesics the relevant property is achronality. A null segment generating $\partial J^+(p)$ or $E^+(P)$ cannot contain a conjugate or focal point before its endpoint: beyond such a point, a variation gives a timelike curve between the relevant endpoints, moving the endpoint into a chronological future. Thus focal points force null generators to leave the boundary even though the null geodesics themselves can continue smoothly inside it. This is the bridge used by the Penrose proof.

The converse is false. A geodesic can stop maximizing before any conjugate point because a distinct geodesic of equal or greater length reaches the same endpoint—the cut-locus phenomenon. Likewise, generators can meet at a crossover or caustic. Conjugate points are valuable because curvature inequalities bound their occurrence; they are not the only mechanism for lost maximality.

Genericity and limits of the elementary estimate

The elementary lemma requires a negative expansion somewhere. The Hawking–Penrose theorem instead assumes that every causal geodesic encounters nontrivial transverse tidal curvature and applies a stronger matrix-Riccati argument. If a complete causal geodesic never acquires negative expansion, Raychaudhuri under convergence forces both expansion and shear to remain zero; the Riccati equation then makes the transverse tidal operator vanish, contradicting genericity. One obtains a pair of conjugate points on every complete causal geodesic. This is stronger than the one-sided bound above and should not be smuggled into it.

The classical derivation uses pointwise Ricci curvature, uniquely defined geodesics, Jacobi fields, and a differentiable exponential map. At $C^{1,1}$ or $C^1$ regularity, generalized theorems use smooth approximants and distributional curvature inequalities, while some $C^1$ arguments must address branching of maximizing geodesics. These results are surveyed separately in Steinbauer 2023, §§4.2–4.10, pp. 22–40. The smooth lemma stated here is not a distributional assertion.

Exercises

In four dimensions, a twist-free affinely parametrized null congruence has $\theta(0)=-4/L$ and obeys null convergence. Give the latest possible affine parameter for its first focal point. If a generator is known to remain on an achronal boundary until parameter $L$, what follows?

Solution

The null screen has dimension $d=2$, so the focusing bound is

$$\lambda_{\mathrm{foc}} \leq\frac{2}{\lvert\theta(0)\rvert} =\frac{L}{2}.$$

If the generator exists through $L/2$, it has a focal point by then. A null generator cannot remain on an achronal boundary beyond a focal point, because a timelike variation then connects the relevant endpoints. Therefore it cannot remain a generator until parameter $L$. If it fails to exist before $L/2$ and is inextendible with finite affine length, the alternative is null incompleteness.

For the flat-space sphere above, consider the ingoing ray at $\lambda=R+\epsilon$, where $0\lt\epsilon\lt R$. Show directly that its endpoint is timelike related to another point of the initial sphere. Explain why this proves loss of the boundary-generating property without proving incompleteness.

Solution

The endpoint is $(t,\boldsymbol x)=(R+\epsilon,-\epsilon\boldsymbol n)$. Choose the antipodal initial point $(0,-R\boldsymbol n)$. The spatial separation is $R-\epsilon$, smaller than the elapsed time $R+\epsilon$, and their squared interval is

$$-(R+\epsilon)^2+(R-\epsilon)^2=-4R\epsilon\lt0.$$

Hence the endpoint lies in $I^+(P)$, not in $E^+(P)$. The original ray nevertheless remains a straight, affinely parametrized null geodesic. Boundary generation ends; the geodesic continues.

References

• Kontou, Eleni-Alexandra, and Ko Sanders, “Energy Conditions in General Relativity and Quantum Field Theory”, Classical and Quantum Gravity 37, §2, pp. 6–9, 2020.
• Steinbauer, Roland, “The Singularity Theorems of General Relativity and Their Low Regularity Extensions”, Jahresbericht der Deutschen Mathematiker-Vereinigung 125, §§3.2–3.3, pp. 10–15, 2023.