Caustics, Focal Points, and Volume Evolution

A caustic occurs where the map from curve labels to spacetime loses rank, so neighboring members cease to provide regular local coordinates. For a geodesic family this infinitesimal degeneracy is detected by Jacobi fields. Distinct, widely separated labels can also reach the same event without local rank loss: such a crossing is a failure of global injectivity and need not be a caustic. Vanishing cross-sectional volume and divergent expansion are related symptoms of rank loss, but neither implies that spacetime curvature is singular.

Required background. The Riemann Curvature Tensor supplies the curvature map in geodesic deviation; Affine and Variational Geodesics supplies affinely parametrized geodesics and the endpoint-variation setup; Jacobi Fields and Geodesic Deviation supplies the Jacobi equation, its initial data, and the exponential-map rank test used to identify conjugate and focal behavior. Helpful background. Calculus and Linear Algebra for Relativity supplies rank, determinants, and linearized maps; Frames, Coframes, and Anholonomy supplies transverse components.

Rank loss in the separation map

Choose a reference curve and a set of connecting fields spanning its transverse directions. Their components form a Jacobi matrix $D(\lambda)$ that maps initial labels to current separations. Before a focal point, $D$ is invertible and the deformation map is

$$B=\dot D D^{-1}.$$

The transverse volume density is proportional to

$$\mathcal V(\lambda)=\left|\det D(\lambda)\right|.$$

Jacobi’s determinant identity gives

$$\frac{\dot{\mathcal V}}{\mathcal V} =\operatorname{tr}B =\theta.$$

Integrating on an interval free of zeros,

$$\mathcal V(\lambda) =\mathcal V(\lambda_0) \exp\left(\int_{\lambda_0}^{\lambda}\theta(\lambda')\,d\lambda'\right).$$

If $\det D$ reaches zero, at least one nonzero initial transverse label maps to zero separation. The congruence map loses rank and $D^{-1}$ fails. The Jacobi fields and geodesics themselves may remain smooth, so the second-order deviation equation is the appropriate continuation through the event.

Caustic, focal point, and conjugate point

A caustic is the image of critical points of the map that launches or labels a family of curves. It is a property of that family and can involve folds, cusps, or higher degeneracies.

A point along a geodesic is conjugate to the initial point when a nonzero Jacobi field vanishes at both endpoints. More generally, it is focal to an initial submanifold when a nontrivial Jacobi field satisfies the boundary condition determined by variations orthogonal to that submanifold and vanishes later. These notions use specific endpoint data; a zero of an arbitrarily chosen separation component is not enough.

Every such event signals degeneracy of a relevant exponential or endpoint map, but the words are not interchangeable without saying what family and boundary conditions are being used. Witten treats transverse area and null focal behavior in Witten 2020, §5.4, pp. 49–53 and develops the Jacobi-field and endpoint-variation relation in Witten 2020, §§8.3–8.5, pp. 79–87.

Expansion near a simple focus

Suppose $m$ transverse directions shrink linearly as $\lambda$ approaches $\lambda_c$, while the remaining directions stay finite. Then

$$\mathcal V\sim C\left|\lambda-\lambda_c\right|^m,$$

and hence

$$\theta =\frac{d}{d\lambda}\ln\mathcal V \sim\frac{m}{\lambda-\lambda_c}.$$

The pole in $\theta$ records the multiplicity of the rank loss. Its sign depends on which side of the caustic and which parameter orientation is used. The absolute-value definition keeps the unsigned volume positive, while an oriented determinant changes sign when the map crosses a simple caustic.

For a two-dimensional null screen, $m$ can be one or two. One vanishing principal axis gives a line focus of the infinitesimal beam; two give a point focus. The optical tensor cannot be evaluated exactly at either because it contains $D^{-1}$.

Flat-spacetime counterexample

Consider ingoing radial null geodesics in Minkowski spacetime, affinely parametrized so that $r=-\lambda$ for $\lambda<0$ and the rays meet at $r=0$. Their Jacobi map on angular labels is

$$D(\lambda)=(-\lambda)I,$$

so

$$\mathcal A\propto\lambda^2, \qquad \theta=\frac2\lambda.$$

The expansion tends to negative infinity as the rays approach the origin. Nevertheless $R^\rho{}_{\sigma\mu\nu}=0$ everywhere, including at the meeting event. Cartesian coordinates and each radial null geodesic continue smoothly; only the angularly labeled congruence and its spherical screen coordinates degenerate.

The same lesson holds for inertial timelike rays emanating from one event: the Milne volume scales as $\tau^3$ and $\theta=3/\tau$, although the ambient spacetime is flat. These examples are decisive counterexamples to the claim that a caustic is automatically a curvature singularity.

Interpretation and limitations

In geometric optics, a caustic can produce large idealized magnification and a change in image parity. Finite wavelength, finite source size, and detector resolution regularize what a ray model predicts. Those observational refinements do not alter the geometric rank condition. The relation among the Jacobi determinant, angular-diameter distance, and image distortion is developed in Perlick 2004, §§2.3–2.5.

A caustic also does not establish geodesic incompleteness: the affine parameter may extend beyond it. Conversely, incompleteness need not be preceded by a caustic. Curvature blow-up, failure of a coordinate chart, failure of a congruence map, and failure to extend a geodesic are logically distinct diagnoses.

Common pitfalls

A divergent trace is not the primary definition. The invariant event is rank loss of the relevant separation or endpoint map. The trace is a useful symptom only where the inverse map exists.

The initial set matters. Conjugacy to a point and focality to a surface impose different Jacobi boundary data. State which endpoint problem is under discussion.

Ray optics has a regime. Infinitesimal beams neglect diffraction and finite-source structure. Near an ideal caustic, those omitted scales may become physically decisive.

Exercises

For $0<\lambda<\pi$, consider the regular Jacobi maps

$$D_1(\lambda)=\operatorname{diag}(\sin\lambda,1), \qquad D_2(\lambda)=\sin\lambda\,I.$$

Verify that each solves $\ddot D=-\mathcal T D$ for a smooth tidal matrix $\mathcal T$. Compute the volume density and expansion, then compare the pole residue at $\lambda=\pi$ with the dimension of $\ker D(\pi)$.

Solution

The first map satisfies

$$D_1''=-\operatorname{diag}(1,0)D_1, \qquad \mathcal V_1=\sin\lambda, \qquad \theta_1=\cot\lambda.$$

Near $\lambda=\pi$, $\sin\lambda\sim-(\lambda-\pi)$, so $\theta_1\sim1/(\lambda-\pi)$. Exactly one transverse direction lies in $\ker D_1(\pi)$, matching residue one. The second map satisfies

$$D_2''=-I D_2, \qquad \mathcal V_2=\sin^2\lambda, \qquad \theta_2=2\cot\lambda \sim\frac{2}{\lambda-\pi}.$$

Now both transverse directions vanish, so the kernel dimension and residue are both two. By contrast, an expression such as $\operatorname{diag}(\lambda,\lambda^2)$ cannot be a nontrivial regular Jacobi map through $\lambda=0$ for a smooth tidal matrix: its second component would have both value and first derivative zero there, and uniqueness for the Jacobi equation would force that component to vanish identically.

Continue to Optical Tensors and Sachs Evolution for the matrix equation away from the rank-loss point.

References

• Perlick, Volker. “Gravitational Lensing from a Spacetime Perspective.” Living Reviews in Relativity 7, 9 (2004). doi:10.12942/lrr-2004-9.
• Witten, Edward. “Light Rays, Singularities, and All That.” Reviews of Modern Physics 92, 045004 (2020). DOI; Open PDF.