Black Hole Singularities and the Limits of the Spacetime Continuum
Abstract
Classical general relativity predicts curvature singularities within black holes, mathematical infinities widely regarded as artifacts signaling breakdown of the geometric description. While the Schwarzschild and Kerr solutions match all external observations, no consensus exists on the physical interior. Existing approaches to singularity resolution, including limiting curvature hypotheses, emergent spacetime models, phase transition analogies, and elastic medium formulations, address aspects of this problem but remain disconnected. This paper proposes a mechanical interpretation. Spacetime is treated as a finite-strength substrate supporting metric relations up to a critical stress threshold $\sigma_c$. The Kretschmann curvature invariant $K = R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$ is reinterpreted as substrate stress $\sigma = \sqrt{K}$, and when $\sigma$ reaches $\sigma_c$, the continuum approximation fails and the medium transitions to a non-metric phase rather than infinite curvature. Event horizons thus mark mechanical failure boundaries where geometric description terminates. All external predictions of general relativity remain unchanged, while the interior is reframed as beyond the domain of continuum geometry. This framework synthesizes geometric, thermodynamic, and mechanical perspectives under a single substrate paradigm, anchoring singularity avoidance to the expected Planck-scale breakdown of spacetime as a continuum.
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