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

Physical Completeness of Classical Black Hole Spacetimes: Domain-Restricted Interpretation of Schwarzschild and Kerr Geometries Sungmin Lee Independent Author [email protected] December 13, 2025 Abstract We present a conservative and fully analytical examination of classical black hole spacetimes, demonstrating that curvature divergence at the center of Schwarzschild and Kerr solutions is not required to possess physical interpretation. Without modifying Einstein’s field equations or introducing new physics, we show that restricting attention to geometrically and operationally well-defined domains yields physically complete descriptions. This interpretation is shown to be consistent with Penrose diagram analysis, Israel’s uniqueness theorem, and the classical formulation of the black hole information problem. 1 Introduction Singularities occupy a central role in the interpretation of classical general relativity. While the Einstein field equations provide precise local constraints on spacetime geometry, their global interpretation has often relied on maximal analytic extension, leading to curvature divergences traditionally regarded as physical singularities. This work reexamines this practice. We argue that the appearance of central curvature divergence reflects an extension beyond the physical domain of applicability of classical geometry rather than a dynamical prediction of Einstein’s equations. No modification of general relativity is proposed. All observationally verified predictions remain unchanged. 1 2 Schwarzschild Geometry and Physical Domain The Schwarzschild solution, ds2=−1−2GM rc2c2dt2+1−2GM rc2−1 dr2+r2dΩ2,(1) solves the vacuum Einstein equations for r > 0. The radial coordinate ris defined invariantly by the area of symmetry spheres: A= 4πr2.(2) At r= 0, this definition degenerates and no physical two-surface exists. Proposition 1 The Schwarzschild solution is physically complete on the domain r≥rmin >0. Proof Einstein’s equations impose no boundary condition at r= 0. All invariant observables are finite for r > 0, and the manifold structure required for physical interpretation ceases to exist at r= 0. □ 3 Formal Clarification of Physical Domain Restriction Definition 1 (Physical Domain) A physical domain Dphys of a spacetime solution is the maximal subset of the manifold on which: 1. Geometric invariants are well-defined, 2. Timelike observers can be locally defined, 3. Einstein’s equations admit operational interpretation. Proposition 2 The maximal analytic extension of a solution need not coincide with its physical domain. Proof Analytic continuation is a mathematical procedure unconstrained by operational interpretability. Einstein’s equations restrict local geometry but do not mandate physical realization beyond regions where manifold structure degenerates. □ 2 4 Extension to Rotating Spacetimes: Kerr Geometry The Kerr metric in Boyer–Lindquist coordinates is ds2=−1−2Mr Σc2dt2−4Mar sin2θ Σc dt dϕ +Σ ∆dr2+ Σdθ2 +r2+a2+2Ma2rsin2θ Σsin2θ dϕ2,(3) with Σ=r2+a2cos2θ, ∆=r2−2Mr +a2.(4) Curvature divergence occurs only where Σ = 0. Proposition 3 The Kerr spacetime admits a physically complete interpretation on the domain Σ ≥Σmin >0. Proof The set Σ = 0 corresponds to r= 0, θ=π/2, which has measure zero and does not support observer congruences, finite proper time, or operational geometry. Vacuum field equations remain valid throughout Σ >0. □ 5 Penrose Diagrams and Physical Interpretation Penrose diagrams encode global causal structure but do not enforce physical realizability of all analytically extended regions. Truncation of the diagram at a boundary of classical applicability preserves all causal relations relevant to external observers. Thus, causal completeness does not imply physical completeness. 6 Relation to Penrose–Hawking Singularity Theorems The Penrose–Hawking singularity theorems establish geodesic incompleteness under broad conditions. Proposition 4 Geodesic incompleteness does not imply the existence of a physical singularity. Proof The theorems assert the impossibility of extending geodesics, not the realization of divergent physical states. A boundary of classical description satisfies the theorem without requiring physical divergence. □ 3 7 Relation to Israel’s Uniqueness Theorem Israel’s theorem guarantees uniqueness of stationary black hole solutions under assumptions of asymptotic flatness and horizon regularity. The present interpretation: •Preserves the exterior Kerr geometry, •Leaves horizon structure unchanged, •Makes no claim about interior matter. Thus, Israel’s theorem remains fully satisfied. 8 Minimal Connection to the Information Problem Classical formulations of the black hole information problem assume physical destruction of information at a singular endpoint. Under domain-restricted interpretation, classical evolution terminates at a geometric boundary, not a destructive physical state. Einstein’s equations do not mandate information loss; the paradox arises from extrapolation beyond classical applicability. Interpretational Comparison +----------------------+-------------------------+--------------------------+ | Aspect | Standard Interpretation | Domain-Restricted View | +----------------------+-------------------------+--------------------------+ | Schwarzschild r->0 | Physical singularity | Applicability boundary | | Kerr ring | Physical object | Geometric limit | | Penrose diagram | Fully physical | Causal representation | | Geodesic end | Destruction | Classical termination | | Information fate | Lost/undefined | Not classically fixed | +----------------------+-------------------------+--------------------------+ Conceptual Summary Mathematical extension : Permitted Physical interpretation: Restricted Einstein equations : Local constraints Singularity : Indicator of breakdown, not an object 4 9 Conclusion We have shown that classical black hole spacetimes are physically complete when interpreted within their geometrically meaningful domains. Curvature divergence marks the boundary of classical description rather than the existence of a physical singularity. This interpretation preserves all empirical successes of general relativity, respects established theorems, and removes internal inconsistencies arising from unjustified extrapolation. Author Contributions The author conducted all aspects of this study independently. This study is based on the author’s theory, mechanisms, and models, with AI assistance in equation formulation and LaTeX editing. The author actively supervised the process, ensuring consistency with the underlying theoretical framework. This work is provided under the Creative Commons Attribution 4.0 International (CC BY 4.0) License. References [1] A. Einstein, Annalen der Physik 49, 769 (1916). [2] K. Schwarzschild, Sitzungsberichte der K¨oniglich Preußischen Akademie (1916). [3] R. P. Kerr, Phys. Rev. Lett. 11, 237 (1963). [4] R. Penrose, Phys. Rev. Lett. 14, 57 (1965). [5] W. Israel, Phys. Rev. 164, 1776 (1967). [6] R. M. Wald, General Relativity, University of Chicago Press (1984). 5