Physical Completeness of Classical Black Hole Spacetimes: Domain-Restricted Interpretation of Schwarzschild and Kerr Geometries Ver3

Physical Completeness of Classical Black Hole Spacetimes: A Domain-Restricted Interpretation of Schwarzschild and Kerr Geometries Ver3 Sungmin Lee Independent Author [email protected] December 13, 2025 Abstract This paper proposes a domain-restricted interpretation of classical black hole solutions, arguing that Schwarzschild and Kerr geometries are physically complete within their operational domains. We contend that the curvature divergences at r= 0 (or Σ = 0) should be viewed as natural boundaries of the manifold structure rather than predicted physical entities. By defining a physical domain Dphys where the metric remains Lorentzian and curvature invariants are finite, we show that General Relativity (GR) remains a self-consistent effective theory. This perspective reconciles the Penrose-Hawking singularity theorems with physical reality by distinguishing between mathematical geodesic incompleteness and physical termination. Furthermore, we address the information paradox by reclassifying the black hole interior as a boundary condition problem, preserving the structural integrity of classical field equations. 1 Introduction The appearance of singularities in General Relativity (GR) is traditionally interpreted as a breakdown of the theory. The Schwarzschild and Kerr solutions, when analytically extended, lead to regions where the Kretschmann scalar K=RµνρσRµνρσ diverges, signifying a failure of the differentiable manifold. However, we propose that the physical domain of a solution need not coincide with its maximal analytic extension. In other field theories, we routinely disregard regions where the underlying assumptions (such as continuity) fail. By applying a similar domainrestriction to GR, we can interpret black hole spacetimes as physically complete ”effective” descriptions of gravity. This work refines the mapping between mathematical solutions and physical observables without modifying the Einstein field equations (EFE). 1 2 The Physical Domain and Effective Completeness Standard GR assumes a smooth, four-dimensional Lorentzian manifold (M, gµν). The EFE are defined only where this structure is intact. 2.1 Definition of the Physical Domain We define the physical domain Dphys as the subset of the mathematical manifold Mwhere the classical field description is operationally valid: Dphys ={p∈M|K(p)<Λ,and sig(gµν) = (−+ ++)}(1) where Λ represents an upper bound of curvature (potentially at the Planck scale l−4 P). 2.2 Schwarzschild Spacetime The Schwarzschild metric, ds2=−1−2GM rc2c2dt2+1−2GM rc2−1 dr2+r2dΩ2(2) is mathematically defined for r∈(0,∞). We argue that the limit r→0 is not a point in the physical universe but the terminal boundary ∂Dphys of the theory. Proposition 1: Schwarzschild spacetime is physically complete because every timelike geodesic within Dphys either extends to infinity or terminates at the boundary where the manifold structure itself ceases to exist. Thus, no ”physical” particle ever occupies a state of infinite density. 3 Kerr Geometry and Causal Consistency The Kerr metric introduces additional complexities: the ring singularity and Closed Timelike Curves (CTCs). 3.1 Causal Domain Restriction In the rotating case, the physical domain must be further restricted to ensure causality. DKerr phys ={p∈M|Σ>0 and gϕϕ >0}(3) By excluding the region where gϕϕ <0 (the CTC region) and the ring Σ = 0, we preserve the causal integrity of the solution. The ”singularity” is not an object within the spacetime but the geometric edge of the rotating system. 4 Reinterpreting Singularity Theorems The Penrose-Hawking theorems prove that gµν = geodesically incomplete. In the standard view, this is a ”flaw.” In our domain-restricted view, this is a ”feature” defining the theory’s range. 2 Proposition 2: Geodesic incompleteness is a statement about the limit of the manifold description. If a geodesic reaches the boundary ∂Dphys in finite proper time, it implies the particle has exited the domain of classical gravity, not that it has reached a point of infinite physical reality. This truncation satisfies the mathematical requirements of the theorems while avoiding the ontological paradox of ”infinite density.” 5 Implications for the Information Paradox If the singularity at r= 0 is a boundary rather than a point of destruction, the ”loss of information” can be reframed. Information approaching the boundary ∂Dphys becomes classically inaccessible as the manifold description fails. However, since the EFE remain valid everywhere inside Dphys, the evolution remains unitary within the valid domain. The paradox is shifted from ”how is information destroyed?” to ”how is information encoded at the boundary of a classical field?”—a question that aligns naturally with the Holographic Principle. 6 Comparison of Interpretations Table 1: Traditional vs. Domain-Restricted View Feature Traditional Interpretation Domain-Restricted (Proposed) r= 0 Status Singular Point in Space Theory Boundary (∂Dphys) Infinite Curvature Physical Reality Mathematical Extrapolation Limit Geodesic End Pathological Failure Natural Termination of Classicality Completeness Incomplete (Needs UV completion) Effective Completeness 7 Conclusion The ”singularities” of black holes are artifacts of extending General Relativity beyond its operational domain. By adopting a domain-restricted interpretation, we find that the Schwarzschild and Kerr geometries are physically complete descriptions of the classical gravitational field. This approach treats the limits of the manifold as natural boundaries of the theory, preserving the predictive power of GR while removing the need for a physical interpretation of mathematical infinities. Author Contributions The author conducted all aspects of this study independently. This study is based on the author’s theory, with AI assistance in LaTeX structuring and formalizing equation justifications. The author supervised the entire process to ensure theoretical alignment. 3 License This work is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) License. References References [1] A. Einstein, ”Die Grundlage der allgemeinen Relativit¨atstheorie,” Annalen der Physik, 49, 769 (1916). [2] K. Schwarzschild, ”¨ Uber das Gravitationsfeld eines Massenpunktes,” Sitzungsber. Preuss. Akad. Wiss., 189 (1916). [3] R. P. Kerr, ”Gravitational Field of a Spinning Mass,” Phys. Rev. Lett., 11, 237 (1963). [4] R. Penrose, ”Gravitational Collapse and Space-Time Singularities,” Phys. Rev. Lett., 14, 57 (1965). [5] R. M. Wald, General Relativity, University of Chicago Press (1984). 4