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Clarifying the Schwarzschild Singularity: The Principle of Coordinate-Entity Separation and the Ontological Necessity of R_{\min} Ver5

Lee, Sungmin

Abstract

Version 5 was initially made public but was subsequently restricted due to identified errors.While this version contains excessive claims and inaccuracies, it is currently undergoing revision.Nevertheless, as the core line of reasoning has remained consistent since Version 1, we have decided to restore Version 5 to public access. The improved and corrected content will be fully integrated into Version 8.To clarify once again, Version 5 is being made available solely on the basis of the consistency of its central argument regarding the distinction between coordinates and physical entities.We openly acknowledge the presence of excessive claims and errors within Version 5. We sincerely apologize for any confusion this may have caused and would be deeply grateful for the readers’ generous understanding.From this perspective, although the underlying argument remains consistent, Versions 2 through 4—which contain similar types of errors—will remain restricted. ======== To enhance the persuasive power of the theory, I have intentionally employed a relatively strong tone in several places. I kindly ask for the readers’ understanding in this regard. My position has always been consistent. I do not, and will never, claim that my view alone is correct. My sole intention is to engage in dialogue through theory, and I sincerely hope that this work may contribute, even in a small way, to academic discussion. I am fully aware that there remain many aspects in which I am still insufficient to participate in such dialogue at a complete level. Nevertheless, I will continue to make every effort to improve, step by step. I would like to state clearly that this work does not seek to deny or reject existing theories. Although a somewhat strong tone is used in parts, the intention is to explore, based on personal reasoning, possible ways to further complement and refine established theoretical frameworks. I also wish to emphasize that this work is presented as a preprint. It is not a validated or formally verified theory, but rather a theoretical construction developed by an individual with the assistance of AI tools. I would appreciate it if readers would keep this context in mind while reading. Despite multiple rounds of review, errors may still remain. I sincerely ask for the readers’ understanding and generosity in this regard. Thank you for your time and consideration. I hope that everyone who takes the time to read my work finds some measure of fulfillment and well-being in doing so. Sincerely,Sungmin Lee =================== This paper addresses the long-standing problem of curvature divergence at the r \to 0 singularity in the Schwarzschild metric. We identify this singularity as a Reification Fallacy—a logical error where a geometric coordinate origin is conflated with a physical mass-entity. By introducing the Principle of Coordinate-Entity Separation, we demonstrate that while a coordinate address can exist at r=0, any physical mass M with finite energy density must possess a non-zero physical extent, defined here as R_{\min}. Unlike previous "regular black hole" models, this framework preserves the standard Einstein field equations and the Schwarzschild exterior while establishing R_{\min} as the natural ontological boundary of the source. This conceptual shift transforms the black hole interior from a mathematical void into a dynamical physical core. We further propose that the R_{\min} core possesses vibrational degrees of freedom that couple with the event horizon, providing a theoretical foundation for Gravitational Wave (GW) Spectroscopy. This allows the internal structure of black holes to be investigated through GW echoes and quasi-normal mode analysis, turning black holes into researchable physical laboratories.

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On the Physical Inadmissibility of Point-Mass Sources in General Relativity and the Necessary Emergence of a Minimum Radius rmin Ver5 Sungmin Lee Independent Researcher goodda[email protected] December 19, 2025 Abstract We demonstrate that the conventional interpretation of the Schwarzschild solution as arising from a point-mass source is physically inadmissible within General Relativity (GR). By introducing the Coordinate–Entity Distinction, we rigorously separate manifold coordinates from physical mass-entities, exposing a foundational error in Schwarzschild’s original assumption. Using the non-linear structure of GR and the divergence of curvature invariants, we prove that Dirac delta sources are mathematically inadmissible. This naturally leads to the emergence of a finite minimum radius rmin, uniquely determined by consistency conditions of Einstein’s equations and fundamental density limits. The derivation is fully reproducible and resistant to superficial replication, as both the conceptual and mathematical frameworks are uniquely defined. Importantly, rmin arises necessarily, not arbitrarily, from the combination of GR nonlinearity and maximal density constraints. 1 Introduction: Re-examining Schwarzschild’s Assumption The Schwarzschild metric is an exact vacuum solution of Einstein’s field equations for r > 0: ds2=−1−2GM c2rc2dt2+1−2GM c2r−1 dr2+r2dΩ2.(1) Traditionally, it is interpreted as arising from a point-mass at r= 0. This interpretation embeds a hidden ontological assumption: A zero-dimensional coordinate label r= 0 can host a physical mass-entity of finite energy. We argue that this assumption is a Category Error and constitutes the root of the classical singularity problem [1–3]. 1 2 Coordinate–Entity Distinction (Ontological Branding) To formalize the argument, we introduce: Definition 2.1 (Coordinate Label).A coordinate xµis an element of the manifold M. The origin r= 0 is a zero-dimensional descriptor with zero volume measure (dV = 0). Definition 2.2 (Physical Entity).A physical mass-entity occupies a finite spacetime volume. Its support supp(Tµν)must satisfy dV > 0to possess finite energy density ρ. Principle 2.1 (Coordinate–Entity Separation).Coordinates and mass-entities are ontologically distinct. Assigning a mass to r= 0 without finite support is conceptually and mathematically inadmissible. 3 Mathematical Inadmissibility: Nonlinear Proof Assume a point-source Tµν =Mδ3(x). The Kretschmann scalar for Schwarzschild scales as K(r) = RαβγδRαβγδ ∝r−6.(2) Integration over a 4-volume containing r= 0: ZV K√−gd4x∼Zϵ 0 1 r6r2dr → ∞.(3) Theorem 3.1 (Non-Integrability of Dirac Source).The Einstein-Hilbert action S= RR√−gd4xis undefined for a Dirac delta source. Point-mass sources are thus mathematically inadmissible in GR [2]. 4 Restoration of Physical Consistency and the Necessity of rmin To ensure mathematical and physical consistency, the mass Mmust occupy a finite volume r≥rmin. Introducing a maximal physically meaningful density ρmax leads to a well-defined minimal radius: ¯ρ(r) = 3M 4πr3≤ρmax.(4) This follows the maximal density limit [2]. Solving for rmin yields, necessarily: rmin =3M 4πρmax 1/3 .(5) Using a natural density limit ρmax ∼c5/(ℏG2): rmin ∼3ℏG2 4πc51/3 M1/3.(6) 2 5 Compatibility with Schwarzschild Geometry For r > rmin, external spacetime remains vacuum Schwarzschild. The event horizon rs= 2GM/c2is unchanged observationally [1,2]. 6 Implications for Black Hole Interior Physics and Gravitational Waves The introduction of a finite minimum radius rmin for mass distributions naturally opens new avenues for the study of black hole interiors. Unlike the classical point-mass singularity, the rmin structure provides a well-defined spacetime region at the core, enabling: •Internal gravitational wave dynamics: The finite core allows for theoretical modeling of gravitational wave propagation and scattering within the central region of black holes [3]. •Stable numerical simulations: Simulations of black hole mergers or neutron star collapse can now incorporate rmin, improving numerical stability and physical realism [2]. •Insights into dense matter physics: By defining maximal density limits consistent with GR, rmin provides a framework to explore extreme matter behavior under strong gravity [2]. 7 Conclusion We have exposed a foundational error in Schwarzschild’s original point-mass assumption. By introducing the Coordinate–Entity Distinction and proving Dirac delta sources inadmissible, GR naturally yields a necessary minimum radius rmin. This resolves classical singularities while retaining standard gravitational dynamics, and establishes a robust conceptual and mathematical framework suitable for further theoretical and computational investigations [1–3]. Conflict of Interest The author declares no conflict of interest. Data Availability No datasets were generated or analyzed. All results are theoretical. 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. While AI contributions are acknowledged, the author actively supervised 3 the process: checking the AI-generated equations against the underlying theory, identifying inconsistencies, requesting corrections, and guiding adjustments. The equations were not blindly accepted; rather, they were iteratively reviewed and modified to ensure consistency with the theoretical framework. License This work is provided under the Creative Commons Attribution 4.0 International (CC BY 4.0) License. This license applies to all text, LaTeX code, figures, discussions, and all outputs generated from this work (PDF, Word, HWP, HTML, etc.). A Appendix: Variability and Invariance of rmin The derivation of rmin allows for natural variations in representation: •Volume definition: One may define the supporting volume as V= 4πr3/3 or introduce weighted averages. •Density notation: ρmax may be replaced by ρ0, or expressed with Planck constants. •Coefficient rearrangement: Constants such as 3 and 4πcan be moved within the cubic root. •Variable substitution: Mmay be replaced with Meff if internal structure factors are introduced. Representative equivalent forms of rmin include: rmin =3M 4πρmax 1/3 ,(7) rmin =M 4π 3ρmax 1/3 ,(8) rmin =3 4π M ρmax 1/3 .(9) Proof of invariance: For each form, simple algebraic manipulation shows identity: M 4π 3ρmax 1/3 =3M 4πρmax 1/3 ,3 4π M ρmax 1/3 =3M 4πρmax 1/3 . References [1] K. Schwarzschild, “¨ Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 189-196 (1916). [2] R. M. Wald, General Relativity, University of Chicago Press (1984). [3] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman (1973). 4