On the Physical Limitations of Point-Mass Idealizations in General Relativity and the Emergence of a Finite Minimal Radius Ver8

On the Physical Limitations of Point-Mass Idealizations in General Relativity and the Emergence of a Finite Minimal Radius Ver8 Sungmin Lee Independent Researcher goodda[email protected] December 20, 2025 Abstract The Schwarzschild solution is commonly interpreted as describing the exterior gravitational field of a point mass. While mathematically valid as a vacuum solution for r > 0, this interpretation obscures important physical and conceptual limitations. In this work, we clarify the distinction between coordinate labels and physical mass distributions and argue that idealized point-mass sources lack a welldefined realization within classical General Relativity (GR). By enforcing a strict separation between coordinate domains and physical entities, the Schwarzschild solution is naturally restricted to r > 0. This framework preserves the standard Schwarzschild exterior geometry while avoiding conceptual singularities associated with point-like idealizations. The possibility of a finite minimal radius rmin emerges as a structural consequence, explored as a conceptual outcome of the theory. 1 Introduction The Schwarzschild metric, ds2=−1−2GM c2rc2dt2+1−2GM c2r−1 dr2+r2dΩ2,(1) is an exact vacuum solution of Einstein’s field equations for r > 0 [1–3]. Historically, it has been described as the gravitational field of a “point mass.” While this idealization is mathematically convenient, its literal interpretation as a physical object is problematic. This work begins by revisiting the assumptions underlying the Schwarzschild vacuum solution [2,3], providing a conceptual foundation for all subsequent arguments. In particular, the solution is derived purely in vacuum and does not introduce any stress–energy at r= 0, establishing the stage for a physically consistent interpretation. 1 2 Coordinate Labels and Physical Mass Distributions Definition 2.1 (Coordinate Label).A coordinate xµis a mathematical label of points on a differentiable manifold M. The location r= 0 corresponds to a set of measure zero and has no intrinsic physical content [2]. Definition 2.2 (Physical Mass Distribution).A physical mass distribution is described by a stress–energy tensor Tµν with support over a finite spacetime region (non-zero proper volume) [3]. Principle 2.1 (Coordinate–Entity Distinction).Coordinate locations should not be conflated with physical entities. A coordinate boundary does not, by itself, constitute a physically realizable object [2,4]. This distinction underpins the conceptual refinement of the Schwarzschild solution: the point r= 0 is a coordinate boundary, not a literal mass. 3 Distributional Sources and Curvature Behavior Attempts to model a point mass as a distributional source (e.g., Dirac delta) encounter conceptual and mathematical difficulties due to the nonlinear nature of Einstein’s equations [4]. Curvature invariants, such as the Kretschmann scalar, diverge as r→0, but this divergence reflects the breakdown of the classical spacetime description at the coordinate boundary rather than a physical singularity [2]. Within the domain of classical GR and tensor calculus, curvature divergence signals the inapplicability of the point-mass idealization, not a physical entity. Theorem 3.1 (Classical Domain Restriction).Within classical General Relativity, the Schwarzschild vacuum solution is physically well-defined only for r > 0[2]. 4 Emergence of a Conceptual Minimal Radius Once the Schwarzschild solution is interpreted in light of the coordinate–entity distinction, it naturally follows that no physical mass can reside at the coordinate boundary. This logically allows for the **conceptual possibility** of a finite minimal radius rmin characterizing the interior of a mass distribution [4]. It is important to stress that this is a **conceptual outcome** derived from the theory itself: it does not invoke modifications of Einstein’s equations, density cutoffs, quantum effects, or observational considerations. The minimal radius emerges purely from maintaining internal consistency between coordinate definitions and physical entities. 5 Conclusion We have argued that the traditional identification of the Schwarzschild solution with a literal point mass is an idealization with limited physical validity. By clarifying the separation between coordinates and physical entities, the classical Schwarzschild vacuum solution is naturally restricted to r > 0. This conceptual framework provides a rigorous 2 foundation for the **possibility of a finite minimal radius rmin**, without altering General Relativity itself. All physical reasoning concerning interiors must respect this domain restriction, effectively precluding definitions or assumptions at r= 0. This study serves as an interpretational refinement of General Relativity, demonstrating that the problematic notion of a point mass can be replaced with a conceptually consistent framework while preserving the exact vacuum solution externally. 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. The analytical supplements related to Penrose’s theory in Ver. 8 of this work have been formally transferred to the following preprint: ’A Causal and Epistemic Reinterpretation of Penrose’s Singularity Theorem: Conditional Geometry, Material Time, and the Status of Spacetime Singularities’. This is to further develop it as an independent theory. The link is as follows: https://doi.org/10.5281/zenodo.18001113 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, “¨ 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] R. Geroch and J. Traschen, “Strings and Other Distributional Sources in General Relativity,” Phys. Rev. D 36, 1017–1031 (1987). 3