Horizon Regularity, Caustics, and Crease Sets

A dynamical event horizon is always a closed achronal topological hypersurface, but it need not be smooth or even continuously differentiable on any prescribed open set. Its null generators can enter through past endpoints; several can enter at the same point, producing a crease. Caustics instead record focal behavior of a generator family, and the two mechanisms can overlap. These nonsmooth sets control how horizon cross-sections appear, merge, and change topology without being curvature singularities of spacetime.

Required background. We use timelike congruences and spatial projection for optical behavior and hypersurface normals and tangential projectors for smooth cross-sections and normal bundles.

Helpful background. Pseudo-Riemannian metrics and signature supply the local cones, affine and variational geodesics supply Jacobi and focal points, Causality and Global Structure gives the global context, and Exact Solutions of Einstein’s Equations supplies regular horizon-crossing examples.

Regularity guaranteed by causal structure

Let $\mathcal H$ be a future horizon: a closed, achronal, future-null-geodesically ruled topological hypersurface in a smooth, time-oriented spacetime. Its generators have no future endpoints on $\mathcal H$, but they may have past endpoints where they enter. A future black-hole event horizon is the main example; a past Cauchy horizon has the same orientation.

The general regularity guarantee is local Lipschitz continuity, not $C^1$ smoothness. Locally, $\mathcal H$ is the graph of a Lipschitz function over a spacelike slice. Rademacher’s theorem then gives differentiability almost everywhere. In globally hyperbolic settings, future horizons are also semiconvex, which supplies second Alexandrov derivatives almost everywhere and makes weak optical quantities possible. Chruściel reviews these levels of regularity in Chruściel 2002, §1.5, pp. 17–19 and Appendix A, pp. 32–35.

Almost-everywhere differentiability must not be upgraded to piecewise smoothness. There are mathematically valid horizons nondifferentiable on dense sets. Conversely, a stationary, suitably regular event horizon can be smooth, but stationarity and the field-equation hypotheses are extra inputs. The causal definition alone permits far rougher behavior.

At a differentiability point there is one null tangent hyperplane and one generator direction. Let $N(p)$ be the number of generators through a point $p\in\mathcal H$. For the standard smooth-spacetime horizon theory,

$$\mathcal H\text{ is differentiable at }p \quad\Longleftrightarrow\quad N(p)=1.$$

An endpoint can nevertheless have $N(p)=1$ and be differentiable at that single point; every neighborhood of it contains more complicated endpoint behavior. The endpoint set and the nondifferentiable set are therefore not identical.

Endpoint, crease, and caustic sets

Denote by $\mathcal H_{\rm end}$ the set of past endpoints of future-horizon generators. On this page the crease set means

$$\mathcal C=\{p\in\mathcal H_{\rm end}:N(p)\gt1\}.$$

At a crease point, at least two null wavefront sheets reach the same cut point. Continuing both sheets farther to the past would destroy achronality; the event horizon retains only their future portions. A three-sheet transverse meeting is often called a corner. Different authors sometimes use “crease set” for the entire endpoint set, so the multiplicity convention must accompany the term.

A caustic point is a focal point of a family of generators: the null normal exponential map from a smooth cross-section loses transverse rank. Equivalently, a Jacobi field induced by varying the starting point on that cross-section, initially nonzero in its tangent space, has vanishing transverse separation there. The transverse, or screen, projection removes a displacement along the null generator itself; an arbitrary Jacobi field with unrelated initial data is not a focal-point test for this cross-section. This definition is clean when the horizon is smooth on a late cross-section and its generators are extended backward as a wavefront. A caustic may have one generator or may also lie in the crease set. A crease need not be a caustic; it can arise when distinct smooth wavefront sheets cross before either sheet focuses.

This is the same distinction as in gravitational lensing: a conjugate point is rank loss within one family, whereas a cut point can occur because two competing null routes arrive. Both tell a generator where it ceases to lie on an achronal boundary. Neither is a spacetime singularity; the metric and curvature can remain smooth through the event.

The measure-theoretic structure is stronger than a sketch suggests. For an $n$-dimensional future horizon in an $(n+1)$-dimensional spacetime, the full endpoint set has vanishing $n$-dimensional Hausdorff measure, and sets with increasing generator multiplicity have progressively smaller rectifiable dimension. Exact hypotheses and statements appear in Chruściel, Fu, Galloway, and Howard 2002, Theorems 1–2, pp. 2–3. Zero horizon measure does not mean the endpoint set is finite, closed in every setup, or physically negligible.

Generic late-smooth horizons

Sharper dimensions require sharper assumptions. In four spacetime dimensions, suppose the spacetime is globally hyperbolic and a connected future horizon has a smooth, compact, oriented cross-section lying to the future of every generator endpoint, with the horizon smooth near that late cross-section. Under these conditions, a normal two-generator noncaustic crease set is a smooth spacelike two-dimensional submanifold of spacetime. Normal three-generator corners form a spacelike one-dimensional submanifold. The caustic set has Hausdorff dimension at most one.

These statements are not universal properties of arbitrary horizons. They use global hyperbolicity, smoothness at late time, and—in more detailed singularity classifications—a specified mathematical notion of genericity. Gadioux and Reall state the assumptions separately from the conclusions in Gadioux and Reall 2023, §§2.1–2.2, pp. 8–18. Symmetry can lower dimensions: spherical collapse can compress the relevant endpoint structure to a point, while axisymmetry can produce a line.

The distinction also prevents a common overstatement. A caustic of the extended null wavefront need not lie on the horizon. Once a generator has passed its cut point, it no longer belongs to the achronal boundary. One must intersect the wavefront singularity structure with the retained horizon portion before labeling a feature of $\mathcal H$.

Cross-section topology and slicing

Choose a time function $t$ with spacelike level sets $\Sigma_t$. The visible horizon at that time is the two-dimensional cut $S_t=\mathcal H\cap\Sigma_t$. Away from generator endpoints and tangencies, nearby cuts are diffeomorphic. A topology change occurs when the foliation passes through the endpoint or crease structure. New generators enter toward the future, allowing disconnected components to merge or a handle to close.

Consequently a dynamical event horizon can admit a transient toroidal cut even when sufficiently late regular cuts are spherical. This does not mean that the three-dimensional horizon hypersurface instantaneously changes its own topology, and the torus is not foliation invariant. A different time function can move the critical cut, alter the duration of the toroidal phase, or avoid that particular presentation. The local restructuring of normal creases under a time foliation is analyzed, with explicit dependence on the chosen time function, in Gadioux and Reall 2023, §3, pp. 18–35.

These possibilities do not contradict horizon-topology or topological-censorship theorems. Such theorems impose their own energy, asymptotic, stability, outermost, or “good cut” hypotheses. Applying a stationary spherical-section conclusion to an arbitrary dynamical slicing drops precisely the assumptions that control the result.

Smooth-horizon intuition and its limits

Raychaudhuri’s equation and the null second fundamental form are pointwise smooth tools. On a nonsmooth event horizon they are defined classically only on regular portions, or almost everywhere in an Alexandrov framework under additional semiconvexity. An area theorem for such a horizon also requires null convergence plus completeness or asymptotic predictability conditions; Lipschitz regularity by itself is not an area law.

Numerical event-horizon finders often trace null rays backward from a late-time surface. The method works because the event horizon is global and late information selects the relevant generators. A numerical crossover or caustic must still be distinguished from grid error, interpolation nonsmoothness, or a coordinate pathology. The present page supplies the causal classification; algorithms and convergence studies belong to Black-Hole Dynamics.

Exercises

In Minkowski spacetime with coordinates $(t,x,y,z)$, consider the graph

$$\mathcal H=\{t=\max(x,y)\}.$$

Show that it is locally an achronal Lipschitz hypersurface with two null sheets meeting on a spacelike two-dimensional crease. Identify the two future generator directions.

Solution

The function $f(x,y,z)=\max(x,y)$ is 1-Lipschitz because each branch has Euclidean gradient of norm one and the maximum preserves that bound. Therefore its graph is achronal. Where $x\gt y$, the sheet is $t=x$ and has future null generator $k_1=\partial_t+\partial_x$. Where $y\gt x$, the sheet is $t=y$ and has generator $k_2=\partial_t+\partial_y$. They meet where $t=x=y$, with $z$ free. Tangents along increasing common value and along $z$ are $(1,1,1,0)$ and $(0,0,0,1)$; both are spacelike and mutually orthogonal. The meeting set is therefore a spacelike two-surface. The graph is not differentiable there because its two null tangent hyperplanes differ, and the two generators enter the future portions of their respective sheets from the crease.

References

• Chruściel, Piotr T. “Black Holes.” Lecture Notes in Physics 604, 61–102 (2002). arXiv.
• Chruściel, Piotr T., Joseph H. G. Fu, Gregory J. Galloway, and Ralph Howard. “On Fine Differentiability Properties of Horizons and Applications to Riemannian Geometry.” Journal of Geometry and Physics 41, 1–12 (2002). DOI.
• Gadioux, Maxime, and Harvey S. Reall. “Creases, Corners, and Caustics: Properties of Nonsmooth Structures on Black Hole Horizons.” Physical Review D 108, 084021 (2023). DOI.