Physical Completeness of Classical Black Hole Spacetimes: Domain-Restricted Interpretation of Schwarzschild and Kerr Geometries Ver5

Physical Completeness of Classical Black Hole Spacetimes: A Domain-Restricted Interpretation of Schwarzschild and Kerr Geometries Ver5 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. We employ the Israel junction conditions to demonstrate that the termination of the manifold is mathematically reconcilable with local conservation laws. This perspective reconciles the Penrose-Hawking singularity theorems with physical reality by distinguishing between mathematical geodesic incompleteness and physical termination, while addressing the information paradox as a boundary condition problem. 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 [5]. 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→ϵ(where ϵ>0 as K→Λ) is not a point in the physical universe but the terminal boundary ∂Dphys of the theory. Figure 1: Conformal diagram of the domain-restricted Schwarzschild spacetime. Future Infinity (I+) / \ / \ <-- Physical Domain (D_phys) / \ (i+) *-------* (i0) Spacelike Infinity \ / \ / <--- Boundary: d(D_phys) at r = epsilon > 0 \ / (Classical manifold terminates here) Past Infinity (I-) 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 Mathematical Consistency at the Boundary To ensure the integrity of the theory at ∂Dphys, we invoke the Israel junction conditions [6]. Consider a hypersurface Σ defined by r=ϵ. The conservation of the stress-energy tensor ∇µTµν = 0 inside Dphys is supplemented by the boundary condition on Σ. 2 The jump in the extrinsic curvature [Kab] across the boundary relates to the surface stress-energy tensor Sab via the Lanczos equation: Sab =−1 8πG ([Kab]−hab[K]) (3) In our interpretation, since the region r < ϵ is non-physical, [Kab] represents the structural termination of the field. This transition satisfies the mathematical requirements of local conservation within an effective field theory (EFT) framework [7]. 4 Kerr Geometry and Causal Consistency The Kerr metric introduces additional complexities: the ring singularity and Closed Timelike Curves (CTCs). 4.1 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}(4) By excluding the region where gϕϕ <0 (the CTC region) and the ring Σ = 0, we preserve the causal integrity. The ”singularity” is not an object within the spacetime but the geometric edge of the rotating system. 5 Reinterpreting Singularity Theorems The Penrose-Hawking theorems prove that gµν is geodesically incomplete [4]. In our domain-restricted view, this is a ”feature” defining the theory’s range. 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. 6 Implications for the Information Paradox If the singularity is a boundary rather than a point of destruction, the ”loss of information” can be reframed. Information approaching ∂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 to a boundary condition problem, aligning with the Holographic Principle and the membrane paradigm [8]. 3 7 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 8 Conclusion The ”singularities” of black holes are artifacts of extending General Relativity beyond its operational domain. By adopting a domain-restricted interpretation supported by junction condition formalisms, we find that Schwarzschild and Kerr geometries are physically complete descriptions of the classical gravitational field. This approach treats the limits of the manifold as natural boundaries, 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. 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). 4 [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). [6] E. Poisson, A Relativist’s Toolkit, Cambridge University Press (2004). [7] S. Weinberg, The Quantum Theory of Fields, Vol. 1, Cambridge University Press (1995). [8] K. S. Thorne, et al., Black Holes: The Membrane Paradigm, Yale University Press (1986). 5