Abstract Boundaries and Ideal Singular Endpoints

An ideal-boundary construction represents ways of leaving a spacetime without pretending that the limiting object already belongs to the physical manifold. The abstract-boundary approach compares boundary sets produced by all admissible embeddings and groups mutually covering representations. The resulting a-boundary is defined from the smooth manifold alone. A metric, differentiability threshold, and admissible curve family then classify representatives; only equivalence-invariant properties give well-defined classifications of the a-boundary points themselves.

Required background. Geodesics and Jacobi Fields supplies incomplete curve families; Curvature supplies regularity tests. Helpful background. Lorentzian Metrics and Causal Cones supplies metric extensions; Causality and Global Structure supplies causal boundaries; Exact Solutions of Einstein’s Equations supplies embedding examples.

Envelopments and boundary sets

Let $M$ be an $n$-dimensional spacetime. An envelopment is a smooth embedding

$$\phi:M\longrightarrow \widehat M$$

into another $n$-manifold. The topological boundary $\partial\phi(M)$ in $\widehat M$ supplies candidate boundary points. Different envelopments can make the same physical limiting process appear as one point, several points, or an extended set. No single embedding is privileged in advance.

For example, ordinary polar coordinates can make the regular origin look like a missing boundary circle if angular information is not identified correctly. A conformal compactification can put null infinity on a finite hypersurface. A coordinate chart around an extendible horizon can produce a finite edge that disappears in a larger chart. The envelopment remembers how $M$ sits in the auxiliary manifold; it does not automatically declare the boundary physical.

Scott and Szekeres define the construction and its classifications in Scott and Szekeres 1994, Definitions 20–22 and §§4–5. The key move is to compare boundary sets across envelopments by the physical sequences or neighborhoods that approach them.

The covering relation

Let $B\subset\partial\phi(M)$ and $B'\subset\partial\phi'(M)$ be boundary sets in two envelopments. Informally, $B$ covers $B'$ when every sequence of physical points whose image approaches $B'$ also approaches $B$ after translating back through $M$. A neighborhood formulation avoids dependence on first-countability details: every neighborhood of $B$ must contain the physical preimage of a sufficiently small neighborhood of $B'$.

Covering is reflexive and transitive but need not be symmetric. When $B$ covers $B'$ and $B'$ covers $B$, the two boundary sets are equivalent. Their equivalence class $[B]$ is an abstract boundary set. An abstract boundary set is an abstract boundary point precisely when it has a singleton representative $\{p\}$; that class is denoted $[p]$. The abstract boundary, or a-boundary, $\mathcal B(M)$ is the collection of all such abstract boundary points—not the collection of every abstract boundary set.

Asymmetry carries information. One boundary point can cover an entire set in another embedding because the first representation has collapsed several approach directions. Equivalence prevents this coordinate-dependent splitting or merging from being mistaken for distinct physical endpoints, while the one-way relation still records relative resolution.

The covering construction and $\mathcal B(M)$ depend only on the smooth manifold and its envelopments; neither requires a metric, affine connection, or curve family. Those extra structures enter only when representatives are classified. A label passes to an a-boundary point only if it is invariant under equivalence: raw regularity is not, whereas essential singularity and being an essential point at infinity are. This separation is valuable because a removable coordinate boundary and a curvature-singular boundary may look identical as subsets until extension regularity and curve behavior are tested.

Regular, infinite, and singular classes

A boundary point is regular at differentiability level $C^k$ if the metric can be extended with that regularity to a neighborhood of the point in the envelopment. A nonregular point may be approachable by curves in a chosen family $\mathcal C$. The family is required to satisfy a bounded-parameter property so that finite and infinite parameter approach is invariant under permitted reparametrizations.

With those choices fixed, a nonregular boundary point is classified schematically as:

• a point at infinity if it is approachable and every approaching curve in $\mathcal C$ has unbounded parameter;
• a singular boundary point if at least one approaching curve has bounded parameter; or
• unapproachable when no curve in $\mathcal C$ reaches it.

Further qualifiers distinguish removable from essential points according to whether equivalent or covering representations are regular. The exact definitions manage cases in which a singular-looking point is covered by a regular boundary set or mixed behavior is present.

The choices are substantive. If $\mathcal C$ contains affinely parametrized causal geodesics, the classification emphasizes the endpoints used by singularity theorems. If it includes bounded-acceleration causal curves, it can detect endpoints missed by geodesics. Changing the regularity threshold from $C^2$ to $C^0$ can make a representative regular and thereby change whether its a-boundary point qualifies as an abstract singularity at that threshold; it does not change the underlying a-boundary point. Thus “this a-boundary point is an abstract singularity” is incomplete without the curve family and regularity threshold.

Comparison with other ideal boundaries

A conformal boundary uses a metric rescaling and can carry differentiable conformal geometry, normals, generators, and asymptotic field data. It is especially effective for regular infinity. A causal boundary uses indecomposable chronological pasts and futures, preserving causal order with little metric input. The abstract boundary instead compares general manifold embeddings and supplies a flexible regularity/curve classification.

These constructions answer overlapping but different questions:

Construction	Primary input	Natural strength	Principal qualification
Conformal	Metric and conformal factor	Smooth causal/asymptotic geometry	Completion and gauge may not exist uniquely
Causal	Chronological relation	Endpoints from causal behavior	Pairing and topology can be delicate
Abstract	All admissible envelopments	Embedding-independent ideal boundary points	Underlying boundary is differential-topological; later classification uses geometry, curve family, and regularity threshold

Agreement in Minkowski or Schwarzschild examples is a consistency check, not a theorem of universal equivalence. An endpoint can be visible in one construction and represented less finely or not at all in another. The intended theorem determines which information must be retained.

Interpretive limits

An a-boundary point $[p]$ is an equivalence class with a singleton representative, not an added spacetime event with a tangent space; other representatives of the same class need not themselves be singletons. Tensor fields do not automatically possess values there. A classification as singular does not specify curvature strength, physical formation, stability under perturbation, or quantum resolution. Those require separate estimates and dynamics.

Nor does the construction guarantee a single completed Hausdorff manifold containing $M$ and every abstract endpoint. Its power is precisely that it can compare boundary behavior without forcing incompatible extensions into one manifold. This also means it is not a drop-in domain for evolving fields past the boundary.

The punctured Minkowski example calibrates the language. In the identity envelopment of the punctured plane into the full plane, the missing origin is regular and approachable by bounded-parameter geodesics. A coordinate representation that makes it look singular is covered by this regular representative; its apparent singularity is removable. For a curvature-divergent endpoint, the corresponding a-boundary point can be an abstract $C^2$ singularity while failing to be an abstract $C^0$ singularity if it has a $C^0$ regular representative.

Exercise

Why must a classification of an a-boundary point state both a curve family and a differentiability threshold? Give one change caused by varying each choice.

Solution

The curve family determines approach and boundedness. An a-boundary point missed by all causal geodesics may be reached by an accelerated causal curve; expanding the family can change it from unapproachable to approachable or singular. The bounded-parameter property ensures this is not an arbitrary reparametrization effect.

The differentiability threshold determines regularity of representatives. A representative across which the metric extends continuously but not twice differentiably is $C^0$ regular but $C^2$ nonregular; this can change whether the a-boundary point is an abstract singularity at the stated threshold. Neither choice changes the physical interior or the underlying a-boundary point.

The next chapter begins with Anatomy of a Singularity Theorem, whose conclusions use incomplete physical curves rather than any particular attached boundary.

References

• Scott, Susan M., and Peter Szekeres, “The Abstract Boundary—A New Approach to Singularities of Manifolds”, Journal of Geometry and Physics 13, §§2–5, pp. 223–253, 1994.
• Steinbauer, Roland, “The Singularity Theorems of General Relativity and Their Low Regularity Extensions”, Jahresbericht der Deutschen Mathematiker-Vereinigung 125, §2.3, pp. 8–9, 2023.