Physical Completeness of Classical Black Hole Spacetimes: Domain-Restricted Interpretation of Schwarzschild and Kerr Geometries Ver2

Physical Completeness of Classical Black Hole Spacetimes: A Domain-Restricted Interpretation of Schwarzschild and Kerr Geometries Ver2 Sungmin Lee Independent Author [email protected] December 13, 2025 Abstract This paper presents a conservative and analytical examination of classical black hole spacetimes, proposing that the curvature divergences inherent in Schwarzschild and Kerr solutions do not necessitate physical interpretation. By strictly defining the physical domain of applicability for Einstein’s field equations, we demonstrate that these geometries can be viewed as physically complete without invoking singularities as actual points of infinite density. We argue that the breakdown of the manifold structure at r= 0 represents a natural boundary of the classical theory rather than a dynamical prediction of physical singularity. This domain-restricted view is shown to be consistent with the Penrose-Hawking singularity theorems, Israel’s uniqueness theorem, and provides a novel perspective on the black hole information paradox by reclassifying the ”endpoint” of gravitational collapse. 1 Introduction The existence of singularities in general relativity (GR) has long been viewed as a sign of the theory’s incompleteness. Traditionally, the Schwarzschild and Kerr solutions are analytically extended to their maximal manifolds, where curvature invariants such as the Kretschmann scalar K=Rµνρσ Rµνρσ diverge at r= 0. In this work, we propose a shift in perspective. We argue that the physical domain of a spacetime solution is not necessarily identical to its maximal analytic extension. By restricting the interpretation of the metric to the domain where the manifold and its operational observables remain well-defined, we show that classical black hole spacetimes are physically complete. This approach does not modify the Einstein field equations (EFE) but refines the criteria for mapping mathematical solutions to physical reality. 1 2 The Schwarzschild Geometry and the Physical Domain The Schwarzschild metric in standard coordinates is given by: ds2=−1−2GM rc2c2dt2+1−2GM rc2−1 dr2+r2(dθ2+ sin2θdϕ2) (1) which is a vacuum solution for all r > 0. The radial coordinate ris defined through the area of a sphere of symmetry, A= 4πr2. As r→0, the surface area vanishes, and the metric components become ill-defined. Proposition 1: Completeness through Restriction The Schwarzschild spacetime is physically complete on the domain D={p∈M|r(p)> ϵ}, where ϵis an arbitrarily small positive constant corresponding to the limit of classical continuum applicability. Justification Einstein’s equations govern the local curvature of a manifold. At r= 0, the manifold structure itself degenerates. Since the EFE require a smooth Lorentzian manifold to be defined, the point r= 0 lies outside the theory’s domain. Therefore, the divergence at the origin is not a physical prediction but a boundary of the theory’s validity. 3 Formal Definition of the Physical Domain To generalize this view, we define the criteria for a physically interpretable region of a solution. Definition 1 (Physical Domain Dphys): A subset of a mathematical manifold Mis considered the physical domain if: 1. All curvature invariants (e.g., R, Rµν Rµν , K) remain finite. 2. The metric tensor gµν maintains a Lorentzian signature (−+ ++). 3. Operational measurements (proper time and distance) are definable for all timelike observers. Proposition 2: The maximal analytic extension of a solution is a mathematical construction that may exceed Dphys. Physical completeness is satisfied if all geodesics within Dphys either extend infinitely or terminate at the boundary ∂Dphys. 4 Extension to Rotating Spacetimes: Kerr Geometry The Kerr metric describes a rotating mass and exhibits a ring-shaped singularity where Σ = r2+a2cos2θ= 0. 2 Proposition 3: Kerr Physicality The Kerr spacetime is physically complete on the domain Σ >0. Justification The region Σ = 0 corresponds to a set of measure zero in the manifold. In Boyer-Lindquist coordinates, this occurs at r= 0, θ =π/2. Since no physical observer can occupy or traverse a state of infinite curvature, the ”ring singularity” acts as a geometric termination point for the classical description, not a physical object within the universe. 5 Reconciling with Singularity Theorems The Penrose-Hawking singularity theorems prove that under certain energy conditions, spacetimes are geodesically incomplete. Proposition 4: Incompleteness vs. Singularity Geodesic incompleteness does not necessitate the existence of a physical singularity. Justification A geodesic may terminate because it reaches the boundary of the theory’s applicability (e.g., where quantum effects dominate or the manifold description fails). The theorem guarantees that geodesics cannot be extended, but it does not mandate that the termination point must possess infinite physical density. Truncating the manifold at ∂Dphys satisfies the theorem while maintaining physical consistency. 6 Implications for the Information Paradox The black hole information paradox relies on the premise that information is ”crushed” at the singularity. Under a domain-restricted interpretation: 1. The ”singularity” is merely the boundary ∂Dphys. 2. Evolution of information within Dphys remains unitary according to classical GR. 3. The paradox is reframed as a boundary condition problem rather than an internal inconsistency of the field equations. 3 Table 1: Comparison of Spacetime Interpretations Feature Standard Interpretation Domain-Restricted View r→0 Limit Physical Singularity Applicability Boundary Kerr Ring Singular Physical Object Geometric Limit Point Penrose Diagram Fully Physical Extension Causal Map of Dphys Information Fate Potential Destruction Classically Undefined at Boundary Metric gµν Globally Valid Valid on Dphys only 7 Interpretational Comparison 8 Conclusion We have argued that the perceived ”pathologies” of black hole spacetimes arise from the unjustified extrapolation of classical geometry into domains where the manifold structure itself ceases to be operationally meaningful. By restricting our interpretation to the well-defined physical domain Dphys, we find that Schwarzschild and Kerr geometries provide a complete and consistent description of black holes. This perspective respects the mathematical rigors of general relativity while acknowledging its limits as a classical field theory. 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. License This work is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) License. References [1] A. Einstein, ”Die Grundlage der allgemeinen Relativit¨atstheorie,” Annalen der Physik, 49, 769 (1916). [2] K. Schwarzschild, ”¨ Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie,” Sitzungsber. Preuss. Akad. Wiss., 189-196 (1916). [3] R. P. Kerr, ”Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics,” Phys. Rev. Lett., 11, 237 (1963). 4 [4] R. Penrose, ”Gravitational Collapse and Space-Time Singularities,” Phys. Rev. Lett., 14, 57 (1965). [5] S. W. Hawking and R. Penrose, ”The Singularities of Gravitational Collapse and Cosmology,” Proc. Roy. Soc. Lond. A, 314, 529-548 (1970). [6] R. M. Wald, General Relativity, University of Chicago Press (1984). 5