Causal Boundary Constructions

A causal boundary tries to recover ideal endpoints from the causal relation itself, without assuming that the physical metric can be smoothly rescaled across a manifold boundary. The guiding idea is operational: two inextendible observers approach the same ideal endpoint when their accumulated chronological pasts and futures carry the same limiting information. Turning that idea into a well-behaved topology is subtle, so a causal completion is a construction with hypotheses and choices—not a universally unique replacement for conformal infinity.

Required background. Lorentzian Metrics and Causal Cones supplies chronological relations. Helpful background. Geodesics and Jacobi Fields distinguishes geodesic from general curve endpoints; Causality and Global Structure supplies the causal ladder; Exact Solutions of Einstein’s Equations provides comparison examples.

Past and future sets as endpoint data

Work in a smooth, time-oriented, strongly causal spacetime $(M,g)$, the setting of the standard construction reviewed here. A subset $P\subset M$ is a past set when

$$P=I^-(P),$$

and a future set $F$ satisfies $F=I^+(F)$. A nonempty past set is indecomposable if it cannot be written as the union of two proper past subsets. Indecomposable future sets are defined by reversing time.

Two classes of indecomposable past sets matter. A proper indecomposable past set is the chronological past of a physical event,

$$P=I^-(p).$$

A terminal indecomposable past set is not $I^-(p)$ for any event. Under the stated hypotheses, every such set is generated by a future-inextendible timelike curve $\gamma$:

$$P=I^-(\gamma)=\bigcup_s I^-(\gamma(s)).$$

It represents a candidate future ideal endpoint. Dually, a past-inextendible timelike curve produces a terminal indecomposable future set. This construction depends only on chronology, so it can still be applied when curvature blows up or no differentiable conformal factor is known.

The causal assumption also protects the original events. Strong causality implies distinction, so each of the maps $p\mapsto I^-(p)$ and $p\mapsto I^+(p)$ is injective. For the pair $p\mapsto(I^-(p),I^+(p))$, injectivity needs only weak distinction: equality of both pasts and futures must imply equality of points. Equality of the pasts alone does not make the pair identical. These are different separation requirements Minguzzi 2019, Definition 4.47 and Theorem 4.58. The strongly causal construction and its static examples are reviewed in Sánchez 2022, §§4.3–4.4, pp. 25–30.

Pairing future and past information

A future endpoint cannot always be represented faithfully by a past set alone. At a timelike ideal endpoint, there may also be a compatible terminal future set. Modern completions therefore represent an ideal point by a pair $(P,F)$, allowing one component to be empty. Informally, $P$ is the past visible from the endpoint and $F$ the future visible from it.

The standard compatibility rule requires $P$ to be a maximal indecomposable past set in the common past of $F$, and $F$ to be a maximal indecomposable future set in the common future of $P$. “Common future” means events chronologically after every point of $P$; “maximal” means that no larger indecomposable future set has that property. Both directions are imposed. An empty component is admitted when no nonempty partner exists, rather than chosen at will Sánchez 2022, §4.3.1, p. 26. This says more than merely attaching a past and future that look compatible in a sketch.

The pairing is necessary near boundaries where several candidate terminal sets would otherwise be overcounted or incorrectly identified. It also exposes a genuine difficulty: which pairs should be identified, how sequences converge to them, and whether the completed space is Hausdorff are not determined by the bare set of terminal pasts and futures alone. Different historical definitions of the causal boundary made different choices here.

A useful construction must specify at least:

• the causal regularity assumed of the spacetime;
• which indecomposable sets are included;
• the compatibility or equivalence relation used to form endpoints;
• the chronology assigned on the completion;
• the topology on physical and ideal points; and
• the sense in which the physical spacetime embeds densely.

Without these items, “the causal boundary” is underspecified. A good completion should reproduce familiar endpoints in standard examples and remain stable under the class of spacetime perturbations relevant to the application, but no single list of desiderata resolves every pathological case.

Minkowski and punctured-spacetime checks

In $1+1$ Minkowski spacetime, consider the future-complete inertial observer $\gamma(s)=(s,0)$ as $s\to+\infty$. Every fixed event $(t,x)$ is in $I^-(\gamma)$: choose an observer time $s\gt t+\lvert x\rvert$. Thus this terminal past set is all of $M$, representing $i^+$.

An accelerated observer gives a different terminal past. With $a\gt0$, take

$$t(\tau)=a^{-1}\sinh(a\tau),\qquad x(\tau)=a^{-1}\cosh(a\tau).$$

Its null coordinates obey $u(\tau)=t-x=-a^{-1}e^{-a\tau}\to0$ from below and $v(\tau)=t+x=a^{-1}e^{a\tau}\to+\infty$. Chronological precedence in this chart requires both null coordinates to increase. Therefore

$$I^-(\gamma)=\{(t,x):t-x\lt0\}.$$

Every event with $u\lt0$ can signal to a sufficiently late point of the observer; no event with $u\geq0$ can. This terminal past represents a point of the right-hand component of $\mathscr I^+$, not $i^+$. It is a proper subset of $M$ but a terminal, not proper, indecomposable past set: the two uses of “proper” must not be confused. Reversing time supplies the past-endpoint examples.

Now delete one event $q$ from two-dimensional Minkowski space. Timelike curves can terminate at the missing location in finite parameter even though the remaining metric is flat. Curves approaching $q$ from its chronological past generate terminal past sets associated with the hole. A conformal compactification designed only for large radius would not automatically represent this finite missing endpoint; causal sets detect it through the failure of continuation.

The example also warns against equating a causal-boundary point with a curvature singularity. The punctured spacetime has vanishing curvature, and the missing event can be restored by a smooth extension. The boundary records causal incompleteness of the chosen manifold. Curvature blowup, inextendibility, and physical singular behavior require separate arguments.

Comparison with conformal completion

When a regular conformal completion exists and satisfies suitable causality assumptions, its boundary points often correspond naturally to terminal indecomposable sets. The conformal construction adds differentiable and metric data: one can compute the boundary normal, induced conformal geometry, and regular limits of rescaled fields. The causal construction uses less structure and can reach examples with nonsmooth or absent conformal boundaries.

Neither method automatically dominates. A conformal boundary can distinguish directions that a coarse causal quotient identifies, while a causal completion can attach endpoints hidden by a failed smooth rescaling. Topologies can be non-Hausdorff, ideal points can split or merge, and conformal embeddings themselves are nonunique. The correct question is therefore not “which boundary is real?” but “which completion, under which assumptions, preserves the causal or asymptotic information needed for this theorem?”

Penrose diagrams often borrow boundary labels from a conformal completion. If only a causal boundary has been constructed, smoothness, causal character from $\nabla\Omega$, and metric claims cannot be read into the picture. Conversely, a smooth diagram does not by itself establish that its endpoint identifications match a chosen causal completion.

Exercise

Let $M$ be two-dimensional Minkowski spacetime with the origin removed, and let $\gamma(s)=(-e^{-s},0)$ for $s\geq0$. Explain why $I^-(\gamma)$ is terminal in $M$ but would be proper after restoring the origin.

Solution

The curve is future timelike and approaches the missing origin as $s\to\infty$. Its accumulated chronological past is the set of events in $M$ that can send a timelike signal to some point of $\gamma$. No physical point $p\in M$ has exactly this past: the only candidate in the smooth extension is the deleted origin. Thus $I^-(\gamma)$ is a terminal indecomposable past set in $M$. After adjoining the origin $q$, the same set becomes $I^-(q)$, a proper indecomposable past set. Terminality therefore describes the chosen spacetime, not an invariant curvature diagnosis.

What Penrose Diagrams Suppress now gathers the information that neither a two-dimensional conformal picture nor endpoint labels can carry by itself.

References

• Minguzzi, Ettore, “Lorentzian Causality Theory”, Living Reviews in Relativity 22, article 3, §§2–4, 2019.
• Sánchez, Miguel, “Globally Hyperbolic Spacetimes: Slicings, Boundaries and Counterexamples”, General Relativity and Gravitation 54, article 124, §§4.3–4.4, pp. 25–30, 2022.