Causal Relations and the Causal Ladder

Causal theory converts the pointwise light cones of a Lorentzian metric into a global order on events. This chapter builds that conversion in six steps: admissible curves, chronological and causal futures, achronal sets, the lower causal conditions, robust time ordering, and the continuity or closure properties just below global hyperbolicity. Each stronger rung removes a specific pathology; none licenses an unstated converse.

Helpful background. Lorentzian Metrics and Causal Cones reviews the local cones, Smooth Manifolds and Tensors supplies the topology behind closures and neighborhoods, and Geodesics and Jacobi Fields helps interpret null generators and limit curves.

Enter causal order

The metric tells whether one tangent vector is timelike, null, or spacelike. Global causality asks a different question: can those allowed tangents be integrated into a curve connecting two specified events, and how does that reachability change under limits or perturbations? A flat metric can have poor global causality after a quotient or an excision, while every spacetime has arbitrarily small neighborhoods with good causal behavior. Curvature alone therefore cannot read off the causal ladder.

Throughout the chapter, $(M,g)$ is connected and time oriented, normally with a $C^2$ Lorentzian metric. The regularity assumption supports the standard push-up, corner-rounding, and limit-curve results. Lower-regularity cone structures require separate definitions because causal bubbling can split relations that coincide here.

The central discipline is to name both the set and the operation. $I^+(p)$ is an open chronological future; $J^+(p)$ is a causal future that need not be closed; $\overline{J^+}$ is a closure, not automatically a causal relation realized by curves. Likewise, a time function, a temporal function, and a Cauchy temporal function impose progressively different conditions.

Six steps through the chapter

1. Causal Curves and Time Orientation separates the cone choice from the time orientation and distinguishes general causal curves from geodesics. The Minkowski cone inequality supplies the first independent check.
2. Chronological and Causal Futures defines $I^\pm$ and $J^\pm$, proves openness, transitivity, and push-up, and uses punctured Minkowski spacetime to show why $J^+(p)$ may not be closed.
3. Achronal, Acausal, and Edgeless Sets compares three conditions often conflated in discussions of “instants.” A null line, a bounded spacelike disk, and a non-Cauchy spacelike graph discriminate them.
4. Chronology, Causality, and Distinction excludes first closed timelike curves, then closed causal curves, then failures of event separation by chronological futures and pasts.
5. Strong Causality, Stable Causality, and Time Functions moves from neighborhood control to robustness under cone widening. It states the equivalence among stable causality, a continuous time function, and a smooth temporal function.
6. Causal Continuity and Causal Simplicity adds continuous variation of $I^\pm$ and then closedness of $J^+$. A half-plane example exposes the remaining gap: closed causal reachability does not make diamonds compact.

The sequence follows the standard modern hierarchy in Minguzzi 2019, §§4.1–4.5, pp. 104–149. It deliberately stops before domains of dependence, Cauchy surfaces, and global hyperbolicity.

The abbreviated causal ladder

Reading downward, each row implies every row below it:

Condition	Added requirement	Immediate interpretive gain
Causal simplicity	Causality and closed $J^+$	Limiting causal endpoint pairs remain causally related
Causal continuity	Distinction and reflectivity	Chronological futures and pasts vary continuously
Stable causality	Some wider cone field remains causal	A global time or temporal function exists
Strong causality	Arbitrarily small causally convex neighborhoods	No arbitrarily close causal return
Distinction	$I^+$ and $I^-$ each separate events	Chronological order identifies event labels
Causality	No nonconstant closed causal curve	Neither light nor matter returns causally to its start
Chronology	No closed timelike curve	Matter cannot return timelike to its start

This is an abbreviated teaching ladder. Refined conditions—non-imprisonment, one-sided distinction, causal easiness, and related subladders—can be inserted between rows for specialized arguments. The definitions and strict separating examples for the displayed rungs are collected in Minguzzi and Sánchez 2008, §§3.2–3.10, pp. 21–42.

The direction of reasoning matters. A temporal function proves stable causality and therefore every weaker condition. It does not prove causal continuity. Closed $J^+$ proves causal simplicity only when causality is also known. Compact causal diamonds are not part of causal simplicity and must be checked separately.

A reusable method for causal claims

When evaluating a spacetime or a proposed theorem hypothesis, proceed in four stages.

First, fix the spacetime domain and time orientation. Removing a point, taking a quotient, or changing a boundary from included to excluded can alter global reachability without changing the local metric formula.

Second, compute or characterize $I^\pm$ and $J^\pm$. In symmetric examples, lift curves to a covering spacetime or use an endpoint cone inequality. In general, distinguish existence of a curve from membership in the closure of a relation.

Third, identify the exact failure mode. A closed timelike curve violates chronology; a closed null curve can violate only causality; equal chronological futures violate distinction; a curve that leaves and returns to every small neighborhood violates strong causality; a causal loop after every cone widening violates stable causality; jumping futures violate causal continuity; and a disappearing limiting connection violates causal simplicity.

Finally, test the nearest converse with a standard example. The flat null cylinder separates chronology from causality. A nearly closed causal channel separates causality from distinction. Punctured Minkowski space separates causal continuity from simplicity. The Minkowski half-plane separates causal simplicity from global hyperbolicity. These checks prevent the ladder from becoming a list of names detached from its logical content.

What this chapter does not establish

No condition developed here, by itself, states that initial data determine an entire spacetime. That requires a Cauchy surface or an equivalent global-hyperbolicity criterion. Nor does any rung imply geodesic completeness, absence of curvature blowup, an event horizon, or the Einstein field equations. Causal structure is invariant under conformal rescaling, so it cannot determine lengths or curvature scales without additional metric information.

The next chapter, Domains of Dependence and Global Hyperbolicity, adds the compactness and coverage properties needed for prediction and hyperbolic propagation.

Chapter diagnostic

For each statement below, identify the strongest conclusion guaranteed by the information given. Do not infer an unstated converse.

1. A spacetime admits a continuous function strictly increasing along every future causal curve.
2. A spacetime is causal and $J^+$ is closed in $M\times M$.
3. A spacetime is causal and every diamond $J^+(p)\cap J^-(q)$ is compact.
4. In Minkowski space, $S$ is the complete null line $t=x$.

Solution

1. The function is a time function, so the stable-causality theorem guarantees stable causality, hence strong causality, distinction, causality, and chronology. Causal continuity is not guaranteed.
2. This is causal simplicity by definition. It implies causal continuity and all weaker displayed rungs, but compact diamonds are not guaranteed.
3. Causality plus compact causal diamonds is a standard smooth criterion for global hyperbolicity. That conclusion belongs to the next chapter and is stronger than causal simplicity.
4. The line is achronal and edgeless but not acausal, because distinct points on it are joined by null segments. These are properties of a subset, not rungs assigned to the whole Minkowski spacetime.

References

• Minguzzi, Ettore, “Lorentzian Causality Theory”, Living Reviews in Relativity 22, article 3, §§1.11–1.14, §§2.14–2.18, and §§4.1–4.5, 2019.
• Minguzzi, Ettore, and Miguel Sánchez, “The Causal Hierarchy of Spacetimes”, in Recent Developments in Pseudo-Riemannian Geometry, §§2–3.10, pp. 3–42 in the arXiv version, European Mathematical Society, 2008.