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A Causal Re-examination of the Schwarzschild Domain: Coordinate-Entity Separation and the Emergence of rmin Ver7 Sungmin Lee Independent Researcher goodda[email protected] December 19, 2025 Abstract The Schwarzschild singularity at r= 0 has historically been modeled as a point-mass using Dirac-delta distributions, yet this approach faces causal and mathematical challenges within the non-linear framework of General Relativity. This paper proposes a formal distinction between coordinate domains and physical entities, re-evaluating the singularity problem through a ”Causal Theory of Time,” where temporal flow is viewed as a derivative of material change (entropy). By identifying physical mechanisms, such as the nuclear repulsive core, that halt gravitational collapse at extreme densities, we demonstrate that the causal arrow of time directed toward the origin terminates at a minimal radius rmin >0. This result suggests that the singularity is not a physical reality but a boundary of the coordinate description, providing a potential resolution to the information paradox by preserving matter within a finite volume. 1 Introduction The Schwarzschild solution is a cornerstone of General Relativity [1]. Traditionally, the divergence at r→0 has been interpreted as a physical singularity, supported by the Penrose singularity theorems which demonstrate geodesic incompleteness under certain energy conditions [2]. However, these geometric proofs often prioritize spacetime curvature as an independent driver of collapse, potentially bypassing the underlying material mechanisms required to sustain it. In this work, we argue that the r= 0 singularity may be re-evaluated by considering an implicit ”geometric determinism” that occasionally neglects the causal priority of material interaction. By strictly separating the mathematical coordinate chart from the physical entity, we propose that rmin emerges as a structural necessity. When material changes reach a causal equilibrium, the ”flow” of time toward the origin—defined by those very changes—must naturally find a boundary. 2 Coordinate Domain and Physical Entity To resolve the singularity problem, it is essential to distinguish between the analytical tools of description and the physical objects being described. Definition 1 (Coordinate Domain).A coordinate chart (U, ϕ)is an analytical mapping defined on an open set. Its boundary, r= 0, is a geometric limit and not necessarily a physical event or an element of the spacetime manifold M[3]. Definition 2 (Physical Entity).A physical entity requires the non-vanishing support of the stress-energy tensor Tµν . Since a physical source must occupy a nonzero volume in classical field theory to avoid a measurezero set, identifying a finite mass with a zero-dimensional point (r= 0) introduces an inherent causal discontinuity [4]. The Schwarzschild vacuum solution (Rµν = 0) is mathematically defined on the domain r > 0. The divergence as r→0 signals the breakdown of the coordinate chart’s applicability rather than a physical prediction of infinite density within the manifold. 3 The Causal Nature of Time 3.1 Temporal Flow and Entropy We posit that the physical arrow of time is a consequence of the rate of material change, as reflected in entropy production (dS > 0). dt ∝dS(Material Change) (1) Time is not an independent background stage but an emergent phenomenon. In a state of static equilibrium where no further material transition or contraction occurs, the causal basis for extending the temporal coordinate vanishes. 1
3.2 Limitations of Dirac-Delta Modeling Historically, modeling the singularity with a Dirac-delta source attempted to maintain the r= 0 origin within the theory. However, if a physical mechanism halts the collapse, the causal path toward r= 0 is severed. The singularity is prevented not by a mathematical cutoff, but by the cessation of the causal process that defines the direction of time toward the origin. 4 The Emergence of rmin 4.1 The Nuclear Repulsive Core The assumption of universal gravitational attraction is a simplification at extreme densities. Nuclear physics demonstrates a powerful **Repulsive Core** at distances below 0.5 fm [5]. This provides a concrete causal mechanism to halt contraction, counteracting the inward gravitational pull. 4.2 Equilibrium and rmin As the inward gravitational force is balanced by shortrange repulsive interactions, the system reaches a state of equilibrium at a minimal radius rmin >0. At this boundary, the material change (dS) associated with the collapse process becomes zero. Consequently, the causal arrow of time directed toward r= 0 terminates. This resolves the infinite curvature problem by ensuring that matter is always contained within a finite radius, a hypothesis that may be tested through observations of gravitational-wave echoes [6]. 5 Conclusion This study has re-examined the Schwarzschild singularity by integrating the principle of coordinate-entity separation with a causal theory of time. We conclude that r= 0 is a coordinate boundary, and rmin is the physical terminus of causal evolution. By recognizing that time is an emergent result of material change, the singularity is naturally replaced by a physical material core. This approach respects the mathematical achievements of existing theorems while providing a refined causal interpretation that aligns with material reality and avoids the information paradox. Acknowledgments The author wishes to express sincere gratitude to Karl Schwarzschild, Roger Penrose, Robert Wald, Stephen Hawking, George Ellis, and the authors of the Bonn meson-exchange model for their extraordinary contributions and dedicated efforts toward the advancement of science. This research is deeply grateful for their remarkable achievements. The author respectfully requests that this study be viewed as a productive academic dialogue, discussing various possibilities within the expansive landscape of theoretical physics. 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, Sitzungsber. Preuss. Akad. Wiss. Berlin, 189 (1916). [2] R. Penrose, Phys. Rev. Lett. 14, 57 (1965). [3] R. M. Wald, General Relativity, Univ. of Chicago Press (1984). [4] C. W. Misner et al., Gravitation, W. H. Freeman (1973). [5] R. Machleidt et al., Phys. Rep. 149, 1 (1987). [6] V. Cardoso et al., Phys. Rev. Lett. 116, 171101 (2016). 2
