Cauchy Horizons and Their Generators

A Cauchy horizon is the boundary beyond which an achronal data set no longer intercepts every relevant causal history. It is therefore a boundary of prediction relative to that set, not automatically a black-hole event horizon or a curvature singularity. Its null generators record how causal curves can just fail to enter the domain of dependence, while the edge of the data set or a global defect explains where those generators come from.

Required background. Geodesics and Jacobi Fields supplies null geodesics, inextendibility, and conjugate-point language. Helpful background. Lorentzian Metrics and Causal Cones fixes causal character; Causality and Global Structure locates horizons in the wider theory; Partial Differential Equations and Green Functions supplies the evolution problem whose geometric boundary is being diagnosed.

The future Cauchy horizon

Let $S$ be an achronal subset of a smooth, time-oriented spacetime $(M,g)$. Write $D^+(S)$ for its future domain of dependence. The future Cauchy horizon is

$$H^+(S)=\overline{D^+(S)}\setminus I^-(D^+(S)).$$

Equivalently, a point $p$ in the closure of $D^+(S)$ lies on $H^+(S)$ when no point chronologically to its future remains in $D^+(S)$:

$$p\in H^+(S) \quad\Longleftrightarrow\quad p\in\overline{D^+(S)} \ \text{and}\ I^+(p)\cap D^+(S)=\varnothing.$$

The past Cauchy horizon $H^-(S)$ is obtained by reversing every time orientation, and $H(S)=H^+(S)\cup H^-(S)$. These definitions, including the closure that captures limiting points, are given in Minguzzi and Sánchez 2008, Definition 3.73, pp. 43–44.

For a closed achronal set, the topological boundary of the future development obeys

$$\partial D^+(S)=S\cup H^+(S).$$

The union need not be disjoint: points on the edge of $S$ may already lie on its future horizon. If $S$ is not closed, accumulation points missing from $S$ complicate this compact formula, which is why closedness must be stated rather than inferred from a sketch.

The definition is asymmetric for a reason. $H^+(S)$ is the last boundary of prediction toward the future. Moving slightly to its timelike future cannot leave one in $D^+(S)$; a causal history can now avoid $S$. The horizon itself is still a subset of the given spacetime. A missing endpoint approached by it is not silently added to $M$.

Why the horizon is null-ruled

In the standard smooth theory, $H^+(S)$ is achronal. If two horizon points were timelike related, openness of the chronological relation and approximation by points of $D^+(S)$ would place the earlier point in $I^-(D^+(S))$, contradicting the definition. An achronal boundary cannot have a timelike tangent direction.

The stronger statement is that it is ruled by null geodesics. Through each horizon point away from $\operatorname{edge}(S)$ passes a null geodesic segment contained in the horizon. Extend such a generator to the past while it remains on $H^+(S)$. It either has a past endpoint on $\operatorname{edge}(S)$ or is past-inextendible in $M$. The proof architecture is a limit-curve argument: approach the boundary by points in the development, take causal curves that nearly escape $S$, extract a limiting causal curve, and use achronality to show that the limit must be a null pregeodesic rather than a timelike curve. The smooth causal-boundary results and their regularity assumptions are organized in Minguzzi 2019, §§2–3.

A generator cannot remain on an achronal horizon beyond a point conjugate to an earlier point of that generator. More precisely, if $r$ is conjugate to $p$ along a null geodesic and $q$ lies strictly beyond $r$, the segment from $p$ to $q$ can be deformed to a timelike curve. The two horizon points $p,q$ would then violate achronality. The strict placement of $r$ between the endpoints is part of the theorem Hawking and Ellis 1973, Proposition 4.5.12, pp. 115–116.

A first conjugate point can itself be the last point of an achronal segment. For a direct check, in the Einstein static universe $g=-dt^2+L^2d\omega_3^2$, a null ray from the north pole reaches the antipode at $t=\pi L$. Nearby great circles focus there, but no timelike curve reaches the antipode in time $\pi L$: its spatial distance is exactly $\pi L$. Continuing the ray any farther loses achronality. Caustics and intersections can similarly mark endpoints of horizon generators. “Generated by null geodesics” therefore does not imply everywhere smoothness, a Killing symmetry, or a common affine parameter for all generators.

Edge-generated and past-inextendible generators

The distinction between the two past behaviors is physically informative. Take the finite interval

$$S=\{(0,x):-a\leq x\leq a\}$$

in two-dimensional Minkowski spacetime. Its future domain is the upper diamond $\lvert x\rvert+t\leq a$, $t\geq0$. The two null sides of the diamond are $H^+(S)$, and their generators have past endpoints at $x=\pm a$, precisely the edge of $S$. Prediction ends because the data patch ends.

Now delete a point $q$ from otherwise flat spacetime and choose an edgeless spacelike set to its past. Null rays emanating from the missing location can bound the region whose past curves still reach the set. Within the punctured manifold, those rays have no past endpoint: they approach $q$, but $q\notin M$. They are past-inextendible horizon generators. Prediction ends because the spacetime has a causal hole, even though its curvature vanishes everywhere.

In more elaborate exact solutions, a Cauchy horizon can separate a maximal globally hyperbolic development from an extension. The generator theorem identifies its causal geometry; it does not assert that a smooth extension exists, that an extension is unique, or that perturbations leave the horizon regular. Those are equations-and-regularity questions, taken up later in Spacetime Extensions and Maximality and Strong Cosmic Censorship.

Cauchy, event, and observer horizons

A Cauchy horizon depends on a proposed data set $S$. Change $S$, and $D(S)$ and $H(S)$ can change. An event horizon instead depends on the causal past of a specified future infinity. An observer horizon depends on the causal past or future of an observer’s worldline. These boundaries can coincide in special geometries, but their definitions do not.

The flat finite-interval example is decisive: it has a nonempty Cauchy horizon but no black hole and no curvature. Conversely, a globally hyperbolic spacetime can contain an event horizon: Cauchy slices can cross the event horizon in the maximally extended Schwarzschild solution. The event horizon is a boundary of escape to infinity, not a failure of those slices to determine the spacetime. One should therefore ask “horizon of what set or observer?” before importing physical language.

Crossing $H^+(S)$ also has a limited logical meaning. It says that data on $S$ no longer causally screen the event from every alternative history. A chosen equation and additional boundary or extension data may still produce solutions beyond the horizon. Such a continuation is not determined by the original data set alone; that is the precise loss being recorded.

Exercise

For the interval $S=\{(0,x):-a\leq x\leq a\}$ in two-dimensional Minkowski spacetime, determine $H^+(S)$ from its definition rather than by naming the null sides. Which points of $S$ itself belong to the horizon?

Solution

The future domain is

$$D^+(S)=S\cup\{(t,x):t>0,\ \lvert x\rvert+t\leq a\}.$$

An interior point of this diamond has a timelike future neighbor still in $D^+(S)$, so it lies in $I^-(D^+(S))$ and not on the horizon. A point satisfying $\lvert x\rvert+t=a$ with $0\leq t\leq a$ has no timelike future point in the diamond, although it lies in its closure. Hence

$$H^+(S)=\{(t,x):0\leq t\leq a,\ \lvert x\rvert+t=a\}.$$

At $t=0$, this set contains exactly the endpoints $(0,\pm a)=\operatorname{edge}(S)$. The interior of $S$ is not on $H^+(S)$ because a short timelike step toward the interior of the diamond stays in $D^+(S)$.

References

• Hawking, Stephen W., and George F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, Proposition 4.5.12, pp. 115–116, and §6.5, 1973.
• Minguzzi, Ettore, “Lorentzian Causality Theory”, Living Reviews in Relativity 22, article 3, §§2–3, 2019.
• Minguzzi, Ettore, and Miguel Sánchez, “The Causal Hierarchy of Spacetimes”, in Recent Developments in Pseudo-Riemannian Geometry, §3.11.2, pp. 43–45 in the arXiv version, European Mathematical Society, 2008.