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On the Domain of the Schwarzschild Solution: Coordinate–Entity Separation and the Emergence of r > 0Ver6 Sungmin Lee Independent Researcher goodda[email protected] December 19, 2025 Abstract The Schwarzschild solution is conventionally interpreted as describing the gravitational field of a point mass located at r= 0, leading to a curvature singularity. We demonstrate that this interpretation is not required by General Relativity and instead arises from a misidentification of a coordinate boundary with a physical entity. By enforcing a strict separation between geometric coordinate domains and physical sources, we show that the Schwarzschild metric is naturally defined only on the domain r > 0, without reference to any material source at r= 0. The curvature divergence at r→0 is thus reinterpreted as signaling the boundary of the coordinate chart rather than a physical singularity. This reformulation introduces a minimal radius rmin >0 as a structural consequence of the theory, derived directly from the initial assumptions of spacetime and matter, without invoking density bounds, curvature cutoffs, or quantum gravity effects. 1 Introduction The Schwarzschild metric is an exact spherically symmetric vacuum solution of Einstein’s field equations [1]. Its divergence at r→0 has traditionally been interpreted as evidence for a physical singularity associated with a point-mass source. Within General Relativity, singularities are commonly characterized via geodesic incompleteness or divergent curvature invariants [2,3]. In this work, we argue that the singular behavior at r→0 does not represent a physical prediction of the theory, but rather results from extending a coordinate description beyond its domain of applicability. The key conceptual error lies in conflating a geometric coordinate boundary with a physical mass-entity. By clarifying this distinction at the level of first principles, the Schwarzschild solution is naturally restricted to the domain r > 0, and the notion of a point-mass singularity becomes unnecessary. 2 Coordinate and Physical Entity Definition 1 (Coordinate Domain).A coordinate chart (U, ϕ)on a spacetime manifold Mis defined only on an open set U⊂ M. The boundary ∂U is a geometric artifact of the chart and need not correspond to any physical object or spacetime event [3]. Definition 2 (Physical Entity).A physical entity is defined by the non-vanishing support of a stress–energy tensor Tµν on M. Its existence requires an open subset of spacetime with non-zero four-volume; a set of measure zero cannot host a physical source [4]. Remark 1. While a coordinate origin such as r= 0 may be formally defined within a chart, no physical principle requires or permits a material source to be identified with a zero-dimensional geometric label. 3 The Schwarzschild Solution and Its Domain The Schwarzschild line element is given by ds2=−1−2M rdt2+1−2M r−1 dr2+r2dΩ2, (1) and satisfies the vacuum Einstein equations Rµν = 0 for r > 0 [1]. Crucially, the derivation of this solution does not introduce a stress–energy tensor at r= 0. The coordinate ris defined only as an areal radius on the exterior manifold, and the point r= 0 lies outside the open domain on which the vacuum solution is constructed. Theorem 1 (Domain Restriction of the Schwarzschild Solution).The Schwarzschild metric defines a vacuum spacetime only on the domain r > 0. The coordinate boundary r= 0 is not part of the physical spacetime described by the solution and does not represent a material source. Remark 2. The divergence of curvature invariants as r→0reflects the breakdown of the coordinate description at the boundary of its domain, rather than the presence of a physical singularity within spacetime [2,3]. 1
4 Emergence of a Minimal Radius Since physical entities require an open region of spacetime for their definition, no physical source can be supported at a coordinate boundary. We therefore define a minimal radius rmin := inf{r|physical fields are defined}>0.(2) This scale is not imposed ad hoc, nor derived from density limits or curvature bounds. Rather, it follows directly from the separation between coordinate domains and physical entities inherent in the formulation of General Relativity. 5 Physical Implications For r > rmin, the exterior Schwarzschild geometry remains unchanged. The region r≤rmin corresponds to non-vacuum physics beyond the scope of the exterior solution and must be described by an appropriate interior model. The presence of such a boundary has observable consequences. In particular, it may act as a partially reflective surface for perturbations, potentially producing late-time gravitational wave echoes in compact object mergers [5]. 6 Conclusion We have shown that the Schwarzschild singularity is not a necessary physical prediction of General Relativity. By enforcing a strict separation between coordinate domains and physical entities, the Schwarzschild solution is naturally restricted to r > 0, and a minimal radius rmin emerges as a structural consequence of the theory. This resolves the singularity without modifying Einstein’s equations or invoking new physical scales, and provides a consistent framework for exploring the physical interior of compact objects. 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, “¨ Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 189–196 (1916). [2] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press (1973). [3] R. M. Wald, General Relativity, University of Chicago Press (1984). [4] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman (1973). [5] V. Cardoso et al., “Is the gravitational-wave ringdown a probe of the event horizon?” Phys. Rev. Lett. 116, 171101 (2016). 2
