Chronological and Causal Futures

The chronological and causal futures of an event turn the local light cone into two global reachability sets. The chronological future records destinations reachable with timelike velocity, whereas the causal future also admits lightlike propagation. Their difference is subtle but essential: $I^+(p)$ is always open for a smooth spacetime, while $J^+(p)$ can fail to be closed because a limiting causal route may pass through a point missing from the spacetime.

Required background. Lorentzian Metrics and Causal Cones supplies the pointwise cone classification, and Smooth Manifolds and Tensors supplies the topology and differentiable curves used below. Helpful background. Geodesics and Jacobi Fields explains why the boundary of causal reachability is locally generated by null geodesics.

Four endpoint relations

Let $(M,g)$ be a connected, time-oriented spacetime with a $C^2$ Lorentzian metric. For events $p,q\in M$, write

$$\begin{aligned} p\ll q &\quad\Longleftrightarrow\quad \text{a future-directed timelike curve runs from }p\text{ to }q,\\ p\leq q &\quad\Longleftrightarrow\quad p=q\text{ or a future-directed causal curve runs from }p\text{ to }q. \end{aligned}$$

Thus $\ll$ is the chronological relation, while $\leq$ is the reflexive causal relation. Their pointwise future and past sets are

$$I^+(p)=\{q:p\ll q\},\qquad J^+(p)=\{q:p\leq q\},$$ $$I^-(p)=\{q:q\ll p\},\qquad J^-(p)=\{q:q\leq p\}.$$

For a subset $A\subset M$, the convention is $I^+(A)=\bigcup_{p\in A}I^+(p)$ and likewise for the other three operations. One must keep the pointwise sets separate from the relations $I^+,J^+\subset M\times M$: closing a relation in the product topology is not automatically the same operation as closing every fiber. The definitions and their local normal-neighborhood interpretation are developed in Minguzzi and Sánchez 2008, §§2.3–2.4, pp. 7–12.

In two-dimensional Minkowski spacetime, with $p=(t_p,x_p)$,

$$I^+(p)=\{(t,x):t-t_p>\lvert x-x_p\rvert\},$$

whereas $J^+(p)$ replaces $>$ by $\geq$. The diagonal point $p$ belongs to $J^+(p)$ but not to $I^+(p)$. This flat calculation is a useful check, not a general coordinate formula.

Openness, transitivity, and push-up

The chronological relation is open in $M\times M$; consequently every $I^+(p)$ and $I^-(p)$ is open. To see the mechanism, suppose $p\ll q$. Choose short portions of the timelike curve near its endpoints inside convex normal neighborhoods. Timelikeness is an open condition on tangent vectors, so sufficiently small changes of both endpoints can be joined to those interior portions by timelike segments. The perturbed endpoints remain chronologically related. This is a local stability argument along a global curve.

Both $\ll$ and $\leq$ are transitive because compatible curves can be concatenated and then reparametrized. More strongly, the push-up property says

$$p\leq q\ll r\quad\Longrightarrow\quad p\ll r, \qquad p\ll q\leq r\quad\Longrightarrow\quad p\ll r.$$

At the junction, a timelike piece allows a small deformation that pushes the adjoining causal piece strictly inside the cone. The smooth result and its corner-rounding argument appear in Minguzzi 2019, §§1.11–1.14 and §2.8, pp. 15–25 and 38–39. It follows immediately that $I^+(J^+(A))=I^+(A)$ and $J^+(I^+(A))=I^+(A)$.

These statements have regularity hypotheses. For merely continuous metrics, causal bubbles can obstruct familiar push-up equivalences, so a low-regularity theorem must specify which causal relation it uses. This page uses the classical $C^2$ setting.

The same rules give useful nesting tests. If $p\leq q$, every event chronologically after $q$ is chronologically after $p$, so $I^+(q)\subseteq I^+(p)$. If $p\ll q$, the stronger inclusion $J^+(q)\subseteq I^+(p)$ follows by push-up. Reversing time gives the corresponding inclusions for pasts. These implications are reliable; their converses require additional rungs of the causal ladder. In particular, recovering causal order from inclusion of chronological futures is guaranteed only under stronger closure and separation hypotheses.

For any subset $A$, $I^+(A)$ is open because it is a union of open point futures. It is also a future set: once an event lies chronologically after $A$, every event to its timelike future does as well. Thus

$$I^+(I^+(A))=I^+(A).$$

The boundary of a future set is achronal. If two boundary points were timelike related, openness and the future-set property would force the later one into the interior, a contradiction. This elementary observation is the common source of many null boundaries developed later in the volume. It does not prove smoothness, null geodesic completeness, or any horizon interpretation; those conclusions need separate hypotheses.

Finally, all four operations are computed in the chosen ambient spacetime. For an open subspacetime $U$, a point may belong to $J^+(p)$ in $M$ but not to $J^+(p,U)$ because every connecting curve leaves $U$. Causal convexity is precisely the condition that blocks this discrepancy for endpoints in $U$.

Closure and the punctured-spacetime test

The causal relation need not be closed. Remove the origin from two-dimensional Minkowski spacetime and choose

$$p=(-1,-1),\qquad q=(1,1).$$

In the unpunctured plane the displacement from $p$ to $q$ is future null, and the only causal curve joining the endpoints traces the straight null segment. That segment passes through the removed origin, so $q\notin J^+(p)$ in the punctured spacetime. Nevertheless, points

$$q_n=(1,1-1/n)$$

are timelike separated from $p$. For large $n$ they can be joined to $p$ by timelike curves detouring around the missing point, and $q_n\to q$. Hence $q\in\overline{J^+(p)}$ although $q\notin J^+(p)$.

This example isolates the global issue: the cones remain exactly Minkowskian at every surviving point, yet a limiting route has no curve in the manifold. Closedness of $J^+$ is therefore not a local curvature property. It is one of the additional conditions reached later at causal simplicity.

In a smooth spacetime one still has

$$\operatorname{int}J^+(p)=I^+(p), \qquad \overline{J^+(p)}=\overline{I^+(p)}.$$

The first equality uses openness plus push-up. For the second, take a point causally reachable from $p$ and move an arbitrarily small distance to its timelike future; those nearby points lie in $I^+(p)$. What can fail is the removal of the closure bar.

Boundaries and horismos

The common boundary

$$\partial I^+(p)=\partial J^+(p)$$

is an achronal boundary: no two of its points can be timelike related. Locally its regular pieces are ruled by null geodesic segments. Globally it can have corners, caustics, and limited differentiability, and some boundary points need not be reachable from $p$ when $J^+(p)$ is not closed.

This qualification distinguishes the boundary from future horismos,

$$E^+(p)=J^+(p)\setminus I^+(p).$$

Always $E^+(p)\subseteq\partial I^+(p)$, but equality requires the relevant causal reachability points not to disappear in a limit. Calling the entire boundary “the light cone from $p$” can therefore conceal a global existence claim. The next page separates achronality from the stronger conditions needed for a set to behave like a time slice.

Exercise

In two-dimensional Minkowski spacetime, let $p=(0,0)$ and $q=(2,0)$. Determine $I^+(p)\cap I^-(q)$ and its closure. Is the closure equal to $J^+(p)\cap J^-(q)$?

Solution

An event $(t,x)$ lies chronologically after $p$ when $t>\lvert x\rvert$, and chronologically before $q$ when $2-t>\lvert x\rvert$. Hence

$$I^+(p)\cap I^-(q) =\{(t,x):\lvert x\rvert<t<2-\lvert x\rvert\}.$$

Its closure is obtained by replacing both strict inequalities by weak ones:

$$\{(t,x):\lvert x\rvert\leq t\leq2-\lvert x\rvert\} =J^+(p)\cap J^-(q).$$

The equality holds here because Minkowski causal futures and pasts are closed. In a spacetime with an excised point, closing the intersection and intersecting the causal sets can give a different result; compactness and closedness must be checked rather than inferred from the flat diagram.

Continue to Achronal, Acausal, and Edgeless Sets to study subsets whose points cannot influence one another in specified causal senses.

References

• Minguzzi, Ettore, “Lorentzian Causality Theory”, Living Reviews in Relativity 22, article 3, §§1.11–1.14 and §§2.8, 2.14, 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.