Achronal, Acausal, and Edgeless Sets

An achronal set contains no pair of events that can be joined timelike; an acausal set also forbids null communication; and an edgeless achronal set has no local gap through which a timelike curve can slip without meeting it. These conditions answer different questions. In particular, achronality alone does not make a set an admissible “instant of time”: a Minkowski null hyperplane is achronal, yet light can travel within it, and a bounded spacelike disk has an edge.

Required background. Lorentzian Metrics and Causal Cones provides timelike and causal tangent cones; Smooth Manifolds and Tensors provides subspaces, closures, and hypersurfaces. Helpful background. Geodesics and Jacobi Fields helps interpret the null generators that can lie in an achronal set.

Achronality and acausality

Let $S\subset M$ in a time-oriented $C^2$ spacetime. The set is achronal when

$$I^+(S)\cap S=\varnothing.$$

Equivalently, no future-directed timelike curve begins and ends on $S$. It is acausal when no nonconstant future-directed causal curve begins and ends on $S$; in a causal spacetime this can be written

$$J^+(p)\cap S=\{p\} \qquad\text{for every }p\in S.$$

Acausality implies achronality, but not conversely. In two-dimensional Minkowski spacetime the null line $S=\{t=x\}$ is achronal: two points on it are never timelike separated. They are, however, joined by a null segment, so the line is not acausal. By contrast, $S=\{t=0\}$ is acausal.

The distinction is conformal and global. A set may be locally achronal in every sufficiently small neighborhood yet fail to be globally achronal because a timelike curve leaves the neighborhood and returns to another point of the set. Conversely, no differentiability is built into the word “achronal.” An achronal set may be rough, disconnected, lower-dimensional, or have boundary. The structural results and the precise separation between achronal and acausal sets are given in Minguzzi 2019, §§2.14–2.15, pp. 59–65.

For a graph $t=f(x)$ over an interval in two-dimensional Minkowski space, the familiar slope test illustrates the definitions. If $f$ is differentiable and $\lvert f'(x)\rvert\lt1$, the mean-value theorem gives $\lvert f(x_2)-f(x_1)\rvert\lt\lvert x_2-x_1\rvert$ for distinct endpoints, so the graph is acausal. The weaker bound $\lvert f'\rvert\leq1$ gives achronality; a segment of slope $+1$ or $-1$ introduces null-related pairs. A local tangent test controls this entire graph because the interval contains the segment between any two base points. It is not a universal replacement for checking $I^+(S)\cap S$: a general null hypersurface, for example, need not be globally achronal.

The edge of an achronal set

The edge detects failure to separate a neighborhood into a timelike past side and a timelike future side. A point $q\in\overline S$ lies in $\operatorname{edge}(S)$ if every neighborhood $U$ of $q$ contains points

$$p\in I^-(q,U),\qquad r\in I^+(q,U),$$

that can be joined by a timelike curve lying in $U$ and avoiding $S$. Here $I^\pm(q,U)$ are computed in the spacetime $U$ rather than in all of $M$. The use of every neighborhood makes the definition local at $q$, while $q\in\overline S$ prevents unrelated holes elsewhere from counting.

Take the closed interval

$$S=\{(0,x):-1\leq x\leq1\}$$

in two-dimensional Minkowski spacetime. It is acausal, but its endpoints form its edge. Near $(0,1)$ a timelike curve can pass from the local past to the local future just to the right of the interval. At an interior point, every sufficiently small such crossing must meet $t=0$ inside the interval. If the endpoints are deleted, they remain edge points because the edge is allowed to lie in $\overline S\setminus S$.

This last observation gives a useful theorem: for achronal $S$,

$$\overline S\setminus S\subseteq\operatorname{edge}(S).$$

Consequently, an edgeless achronal set is automatically closed. The local bypass definition, closedness of the edge, and this inclusion are proved in Minguzzi 2019, §2.18, pp. 75–78.

Edgeless sets and hypersurface behavior

An achronal set is edgeless when $\operatorname{edge}(S)=\varnothing$. Around each of its points, nearby events fall on the set, to its timelike future, or to its timelike past. More precisely, a closed achronal set is edgeless exactly when

$$S\cup I^+(S)\cup I^-(S)$$

is open. This is the local separation property expected of a hypersurface, even when the hypersurface is only Lipschitz. Every achronal boundary—such as the boundary of a future set—has empty edge, although it can contain null generators and need not be smooth.

The three conditions now have distinct roles:

Condition	What it excludes	What it still allows
Achronal	Timelike communication between points of $S$	Null-related points and local edges
Acausal	All nonconstant causal communication between points of $S$	A boundary or gap in $S$
Edgeless and achronal	Local timelike bypasses around $S$	Null generators unless acausality is added

The point is logical: none of the stronger properties in the last two rows can be reconstructed from achronality alone.

Why this is not yet a Cauchy surface

A partial Cauchy hypersurface is conventionally a closed, acausal, edgeless set. Even that local package does not guarantee that every inextendible causal curve meets the set. The latter is the genuinely global requirement for a Cauchy surface.

For example, in two-dimensional Minkowski spacetime the graph

$$t=\sqrt{1+x^2}$$

is a smooth, closed, spacelike—and hence acausal—edgeless hypersurface. Yet the future-directed null line $t=x$ never intersects it, since $x=\sqrt{1+x^2}$ has no solution. Thus the graph is not a Cauchy surface. Achronality controls repeated intersections, edgelessness controls local gaps, and the Cauchy property controls global coverage. Domains of Dependence develops that additional condition.

Two common inferences therefore fail. “Spacelike” is a tangent-space property and does not by itself prove that a disconnected or globally folded set is acausal. Conversely, “achronal” does not mean spacelike: null hypersurfaces provide the standard counterexample. The relevant condition must be matched to the claim being made.

Exercise

In two-dimensional Minkowski spacetime, classify the complete straight line $S_a=\{(t,x):t=ax\}$ as achronal or acausal for each real $a$, and determine when it is edgeless.

Solution

A tangent to $S_a$ is $v=(a,1)$, with

$$g(v,v)=1-a^2.$$

If $\lvert a\rvert<1$, the tangent is spacelike. Every displacement between two distinct points has the same spacelike direction, so the line is acausal. If $\lvert a\rvert=1$, the line is null: it is achronal but distinct points are null-related, so it is not acausal. If $\lvert a\rvert>1$, distinct points are timelike related and the line is not achronal.

For $\lvert a\rvert\leq1$, the complete line separates the plane locally and has no endpoint through which a timelike curve can bypass it; it is edgeless. The word “edgeless” is defined only here in conjunction with achronality, so the timelike cases are not promoted to edgeless achronal sets.

Next, Chronology, Causality, and Distinction applies these relations to the spacetime as a whole.

References

• Minguzzi, Ettore, “Lorentzian Causality Theory”, Living Reviews in Relativity 22, article 3, §§2.14–2.15 and §2.18, pp. 59–65 and 75–82, 2019.
• Minguzzi, Ettore, and Miguel Sánchez, “The Causal Hierarchy of Spacetimes”, in Recent Developments in Pseudo-Riemannian Geometry, §§2.3–2.4, pp. 7–12 in the arXiv version, European Mathematical Society, 2008.