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Clarifying the Schwarzschild Singularity: The Principle of Coordinate-Entity Separation and the Ontological Necessity of Rmin Sungmin Lee1 1Independent Researcher, goodda[email protected] December 17, 2025 Abstract The curvature divergence at r→0in the Schwarzschild solution remains one of the greatest challenges in modern physics. This paper identifies this singularity not as a physical reality, but as a result of a Reification Fallacy—the conflation of a geometric coordinate origin with a physical mass-entity. By introducing the Principle of Coordinate-Entity Separation, we argue that any finite mass Mmust possess a non-zero physical extent Rmin to maintain a finite energy density. This approach preserves the standard Schwarzschild exterior solution while naturally removing the singularity by defining Rmin as the ontological boundary of the source. Furthermore, the introduction of the Rmin core transforms the black hole interior from a mathematical void into a dynamical laboratory accessible via Gravitational Wave (GW) Spectroscopy. This framework establishes a theoretical foundation for probing the internal structure of black holes through GW echoes and vibrational modes. 1 Introduction: The Reification Fallacy In General Relativity, the point-mass hypothesis is a useful mathematical tool for simplifying calculations [1]. However, interpreting the mathematical origin r= 0 of a coordinate system as the actual location where physical mass is compressed constitutes a “Reification Fallacy”— mistaking a geometric abstraction for a physical entity [2]. The singularity at r= 0 is not a failure of General Relativity (GR), but an indication of the domain limits of the point-mass idealization. This paper demonstrates the necessity of Rmin based solely on the consistency of classical continuum mechanics and ontological analysis, without relying on specific quantum gravity models. 2 The Principle of CoordinateEntity Separation The core defense of this framework rests on the distinction between the “Label” (Coordinate) and the “Object” (Mass-Entity). Definition 1 (Principle of Coordinate-Entity Separation).The geometric origin of a coordinate system is a zero-dimensional reference point. In contrast, a physical entity is a three-dimensional occupancy state with finite energy density. While a coordinate address can be defined at r= 0, a physical mass Mmust occupy a domain r≥Rmin to remain physically consistent. Theorem 1 (Non-Existence of Point-Mass Singularity). For the stress-energy tensor Tµν to be physically valid, the mass density must be finite [3]. For any finite mass M, there must exist a minimum radius Rmin >0such that M=RρdV is satisfied. Consequently, the Schwarzschild vacuum solution is valid only for r > Rmin. At r=Rmin, we encounter the physical source itself, rendering the mathematical divergence of the vacuum solution at r→0physically irrelevant. 3 Modified Domain and Metric Consistency The standard Schwarzschild metric is a perfect description of the exterior region (r > Rmin). Our framework redefines the physical domain using an effective radius reff =r−Rmin to maintain consistency: ds2=−f(r)dt2+f(r)−1dr2 eff + (reff +Rmin)2dΩ2(1) 1
where f(r)=1−2M reff +Rmin . In this formulation, the limit reff →0(i.e., r→Rmin) represents the boundary where the exterior vacuum solution meets the physical core, rather than a breakdown of spacetime. 4 Comparison with Existing Paradigms Our approach differs from prior attempts in several key aspects: i) vs. Quantum Gravity: Unlike Loop Quantum Gravity or String Theory, we do not invoke Planckscale physics. Rmin is derived from the classical requirement of finite energy density. ii) vs. Modified GR: We do not modify Einstein’s field equations (Gµν = 8πTµν ). We simply clarify that the “vacuum” ends where the “source” begins. iii) Model Independence: This principle establishes the ontological necessity of a core first, independent of internal state equations. 5 Internal Dynamics and GW Spectroscopy A physical core Rmin, unlike a static singularity, possesses dynamical degrees of freedom. 5.1 Vibrational Modes of the Core Perturbations such as matter accretion or mergers excite the vibrational modes of the Rmin core. Since these oscillations occur at the boundary of the mass source, they represent fluctuations of the spacetime metric itself and are not “trapped” in the classical sense. 5.2 Coupled Oscillations and Echoes The vibration of the core couples with the spacetime geometry between Rmin and the event horizon Rs. •GW Echoes: The existence of a physical boundary Rmin provides a reflective surface, potentially generating gravitational wave echoes [5]. •Spectroscopy: Analyzing these echoes allows us to encode internal information into the Quasi-Normal Modes (QNMs), transforming GW detectors into telescopes for black hole interiors [4]. 6 Conclusion The Rmin framework resolves the singularity problem without sacrificing the elegance of General Relativity. By separating the coordinate origin from the physical entity, the black hole interior is transformed into a researchable dynamical laboratory. Rmin serves as the ontological key to opening a new chapter in physics: the exploration of black hole interiors via gravitational wave data. 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 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.). References [1] K. Schwarzschild, “Über das Gravitationsfeld eines Massenpunktes nach 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] B. P. Abbott et al. (LIGO/Virgo), “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, 061102 (2016). [5] V. Cardoso et al., “Is the Gravitational-Wave Ringdown a Probe of the Event Horizon?,” Phys. Rev. Lett. 116, 171101 (2016). 2
