Chronology, Causality, and Distinction

The lower causal ladder asks progressively stronger global questions. Chronology forbids returning to an event along a timelike curve; causality also forbids return along light; distinction requires the chronological future and past to identify each event uniquely. The last condition adds genuine resolving power: two events can no longer be invisible to the order structure even when no causal loop exists.

Required background. Lorentzian Metrics and Causal Cones supplies the causal-cone classification, and Smooth Manifolds and Tensors supplies the manifold topology used in the local characterizations. Helpful background. Geodesics and Jacobi Fields clarifies why a causal loop that is not everywhere null can be deformed into a timelike loop.

Closed curves and the first two rungs

Let $(M,g)$ be a connected, time-oriented $C^2$ spacetime. It is chronological when it contains no closed timelike curve. In terms of the chronological relation,

$$p\not\ll p \qquad\text{for every }p\in M.$$

It is causal when it contains no nonconstant closed causal curve. Since the reflexive relation $p\leq p$ also includes the constant curve, the useful order-theoretic form is antisymmetry:

$$p\leq q\ \text{and}\ q\leq p \quad\Longrightarrow\quad p=q.$$

Causality implies chronology because every timelike curve is causal. The converse fails. A closed null geodesic violates causality without violating chronology, provided no timelike loop exists. In a chronological spacetime any closed causal curve must in fact be an unbroken closed null geodesic: a corner, a timelike segment, or two nonparallel null segments can be pushed inside the cone to produce a closed timelike curve. This implication and its proof are stated in Minguzzi and Sánchez 2008, §§3.2–3.3, pp. 21–22.

These are global conditions despite their short formulas. Every point has arbitrarily small neighborhoods with excellent causal behavior, while a curve can travel far away and close only after probing the topology or large-scale cone geometry. Absence of curvature does not prevent the pathology: quotienting flat spacetime by a causal translation can create closed causal curves.

Future and past distinction

A spacetime is future distinguishing if

$$I^+(p)=I^+(q)\quad\Longrightarrow\quad p=q,$$

and past distinguishing if the corresponding statement holds for $I^-$. It is distinguishing when both properties hold. Thus the maps

$$p\longmapsto I^+(p), \qquad p\longmapsto I^-(p)$$

are injective. Different events then have different chronological futures and different chronological pasts.

This is more than the absence of loops. It says that the collection of all possible timelike influence experiments separates points. If two events had identical futures, no question of the form “can a timelike signal from here reach that later event?” would distinguish them. Past distinction supplies the time-reversed separation. Either future or past distinction already implies causality: events on a closed causal curve have the same chronological future and the same chronological past by push-up and concatenation.

Distinction also has a local-return characterization. Future distinction at $p$ means that, for every neighborhood $U$ of $p$, there is a smaller neighborhood $V\subset U$ such that a future-directed causal curve starting at $p$ meets $V$ in one connected parameter interval. Equivalently,

$$J^+(p,V)=J^+(p)\cap V.$$

The past version reverses the curves. The equivalence between injectivity and this anchored no-return property is proved in Minguzzi and Sánchez 2008, §3.4, pp. 22–24. The word “anchored” is important: strong causality controls causal curves with arbitrary nearby endpoints and is strictly stronger.

Separating the implications

The lower ladder can be summarized as

$$\text{distinguishing} \quad\Longrightarrow\quad \text{causal} \quad\Longrightarrow\quad \text{chronological}.$$

Neither arrow reverses in general. A chronological but noncausal example is the flat null cylinder considered in the exercise below. For the other separation, one can arrange a causal cylinder with a nearly closed null channel so that two distinct points along the channel have identical chronological futures; there is no closed causal curve, yet future distinction fails. An explicit construction and its one-sided variants appear in Minguzzi and Sánchez 2008, Figure 6 and §3.4, pp. 22–24.

Condition	Forbidden pathology	Information not yet guaranteed
Chronology	Closed timelike curves	Closed null curves
Causality	All nonconstant closed causal curves	Chronological futures need not separate events
Future distinction	Equal $I^+$ sets for distinct events	Past distinction and uniform neighborhood control
Distinction	Equal $I^+$ or equal $I^-$ sets for distinct events	Strong causality or stable behavior under cone widening

The table lists only one direction of implication. A spacetime satisfying a stronger row automatically satisfies the weaker rows above it, but an example at one row need not lie at the next.

What the order now remembers

In a distinguishing spacetime the causal order contains enough local information to recover the conformal metric. More precisely, a diffeomorphism between distinguishing spacetimes that preserves the causal relation in both directions preserves their conformal classes. The order determines which tangent directions are null; it does not determine the conformal factor, lengths, curvature scales, or the Einstein equation. The result and its hypotheses are given in Minguzzi and Sánchez 2008, Proposition 3.13, pp. 23–24.

Nor does distinction make chronological diamonds $I^+(p)\cap I^-(q)$ a basis for the manifold topology. Equality of the Alexandrov and manifold topologies characterizes strong causality, the next major rung. Distinction controls curves anchored at an event; strong causality makes that control uniform enough for arbitrary nearby passages.

Two terminology traps are worth avoiding. First, “chronological” does not mean that a global time coordinate exists; that belongs higher in the ladder. Second, “causal spacetime” is a technical global condition, not merely a spacetime equipped with light cones. Every time-oriented Lorentzian spacetime has causal vectors, including one with closed causal curves.

Exercise

Let $\widetilde M=\mathbb R^{1,1}$ with metric $-dt^2+dx^2$, and form the quotient by the discrete isometry

$$(t,x)\sim(t+L,x+L),\qquad L>0.$$

Show that the quotient is chronological but not causal.

Solution

The straight null line $s\mapsto(t+s,x+s)$ projects to a closed future-directed null curve after $s=L$. Hence causality fails.

Any closed curve in the quotient lifts to a curve in $\widetilde M$ whose endpoints differ by $n(L,L)$ for some integer $n$. For a future-directed closed causal curve, $n>0$ and the endpoint displacement is future null. In Minkowski space, two null-separated endpoints cannot be joined by a timelike curve: integrating the cone inequality gives $\Delta t\geq\lvert\Delta x\rvert$, with strict inequality for a curve containing a timelike segment. Here $\Delta t=\Delta x=nL$, so equality forces the lift to be null throughout. Therefore the quotient has no closed timelike curve and is chronological, although it has closed null curves and is not causal.

Continue to Strong Causality, Stable Causality, and Time Functions for neighborhood control and robustness under cone widening.

References

• Minguzzi, Ettore, “Lorentzian Causality Theory”, Living Reviews in Relativity 22, article 3, §§4.2–4.3, pp. 111–137, 2019.
• Minguzzi, Ettore, and Miguel Sánchez, “The Causal Hierarchy of Spacetimes”, in Recent Developments in Pseudo-Riemannian Geometry, §§3.2–3.4, pp. 21–24 in the arXiv version, European Mathematical Society, 2008.