Ideal points, directed completion and the case of the maximally extended Schwarzschild spacetime

Technical Summary: Ideal points, directed completion and the case of the maximally extended Schwarzschild spacetime

Problem Statement
The paper addresses the mathematical formulation of boundaries and completions for spacetimes within Lorentzian geometry and General Relativity. While various definitions exist—such as Penrose's conformal boundary, the geodesic boundary, and the abstract boundary—the causal completion introduced by Geroch, Kronheimer, and Penrose (GKP) in 1972 remains a central concept. The GKP construction adds "ideal points" at infinity to a spacetime by considering terminal indecomposable past (TIP) and future (TIF) sets. However, this construction relies on specific causal structures and is not always canonical or well-defined for non-smooth geometries.

Motivated by the need for a robust framework applicable to non-smooth Lorentzian geometry, the first author previously introduced the notion of directed completion for partially ordered sets (posets) in 2025. This completion adds suprema to directed sets that lack them, defined via a universal property in the category of partial orders. The central problem investigated in this work is the relationship between the future causal completion (GKP) and the directed completion (poset-theoretic). Specifically, the authors seek to determine if these two distinct constructions coincide under natural assumptions covering smooth globally hyperbolic spacetimes, and to explicitly characterize this completion for the maximally extended Schwarzschild spacetime.

Methodology
The authors employ a synthesis of order theory and Lorentzian geometry. The methodology proceeds in two main stages:

1. Theoretical Equivalence:

   • The authors work within the framework of Lorentzian pre-length spaces (Kunzinger and Sämann), which generalize smooth Lorentzian manifolds to non-smooth settings.
   • They define the directed completion of a poset $(X, \leq)$ as the smallest directed-complete poset containing $X$ via a map with the monotone convergence property (Mcp).
   • They define the future causal completion $C_+(X)$ as the set of all indecomposable past sets (IPs) of $X$, ordered by inclusion.
   • A key technical step involves introducing the supremum-compatibility condition (SC). This condition requires that for any directed set $D \subseteq X$ admitting a supremum $x$, the chronological past of the set equals the chronological past of the supremum ($I^-(D) = I^-(x)$).
   • The authors prove that if a Lorentzian pre-length space is approximable, past-distinguishing, and satisfies (SC), then its directed completion is isomorphic to its future causal completion. They further show that forward completeness (a Dedekind-type order condition) implies (SC), thereby establishing the equivalence for globally hyperbolic spacetimes.

2. Explicit Characterization of Schwarzschild Spacetime:

   • The authors apply their theoretical results to the maximally extended Schwarzschild spacetime ($M_{Krus}$), which is globally hyperbolic and thus falls within the scope of their equivalence theorem.
   • Instead of relying solely on the abstract GKP construction, they explicitly characterize the directed completion by analyzing the structure of directed sets in $M_{Krus}$.
   • The analysis relies on a case-by-case study of directed sets based on the behavior of their radial null geodesics, specifically tracking the limits of the Kruskal-Szekeres coordinates $U$ and $V$.
   • They classify directed sets into four categories: those converging to a point in the spacetime, those diverging to future timelike infinity ($i^+$), those diverging to future null infinity ($I^+$), and those terminating at the black hole singularity ($S^+$).

Key Contributions and Results

• Equivalence of Completions: The primary theoretical result (Theorem 3.15) establishes that for approximable, past-distinguishing Lorentzian pre-length spaces satisfying the supremum-compatibility condition (SC), the directed completion coincides with the future causal completion. Since global hyperbolicity implies forward completeness, which in turn implies (SC), this result confirms that the directed completion is the appropriate generalization of the future causal completion to non-smooth settings (Proposition 3.17).
• Characterization of Directed Sets in Schwarzschild: The paper provides a complete classification of directed sets in the Kruskal-Szekeres spacetime (Proposition 4.2). It demonstrates that every directed set corresponds to the causal past of a specific radial null geodesic (or a limit thereof), characterized by the limits of its $U$ and $V$ coordinates and a limiting angular coordinate $\Omega^*$.
• Explicit Structure of the Schwarzschild Completion: The authors construct the directed completion of $M_{Krus}$ explicitly (Theorem 4.1). The completion consists of the original spacetime augmented by:
  • The black hole singularity ($S^+$), parameterized by $(U^*, \Omega^*)$ where $U^* > 0$.
  • Future timelike infinity ($i^+$ and $i'^+$).
  • Future null infinity ($I^+$ and $I'^+$), parameterized by $(U^*, \Omega^*)$ or $(V^*, \Omega^*)$ with $U^*, V^* < 0$.
  • The partial order on this extended set is defined by the inclusion of the corresponding causal pasts (or their closures), which translates into explicit inequalities involving the coordinates $U, V$ and the angular parameters.
• Alternative Viewpoint: The paper offers an alternative derivation of the causal completion of the maximally extended Schwarzschild spacetime. By utilizing the directed completion framework and radial null geodesics, the authors recover the standard GKP boundary structure without relying on the original TIP/TIF construction directly, but rather through the analysis of directed sets.

Significance
The paper claims significance in two main areas:

1. Foundational Rigor in Non-Smooth Geometry: By proving the equivalence between the directed completion and the future causal completion under natural assumptions (specifically global hyperbolicity), the work validates the directed completion as a robust tool for studying boundaries in non-smooth Lorentzian pre-length spaces. It suggests that the directed completion, which relies only on the partial order structure, is the correct generalization of the causal completion when smoothness is lost.
2. Concrete Application to Schwarzschild Spacetime: The explicit characterization of the Schwarzschild completion serves as a concrete test case. It demonstrates how the abstract theory of directed sets and suprema can be applied to a physically relevant, maximally extended spacetime. The result provides a clear, geometric description of the ideal points (singularities and infinities) in terms of the limits of radial null geodesics, offering a fresh perspective on the causal structure of black hole spacetimes.

The authors do not propose new experimental applications or future extensions beyond the scope of the current mathematical framework, maintaining a focus on the structural properties of Lorentzian pre-length spaces and their completions.