The Quantum Boundary of Black Hole Interiors: Termination of the Sum over Geometries at Planck Curvature

Abstract

Classical general relativity predicts a singularity at the center of every black hole. We argue that this singularity is never reached. Operating purely within the standard framework of quantum mechanics and the Feynman sum over geometries, we demonstrate that the gravitational functional integral loses support at the Planck curvature threshold ($\mathcal{K} \sim \ell_P^{-4}$). This forms a quantum boundary, $\mathcal{B}_Q$, that truncates the spacetime manifold at a finite, positive radius ($r_\mathcal{B} \approx 10^{-22}$\,m for a solar-mass black hole). The suppression is driven by the mathematical Sobolev failure of the Einstein-Hilbert action; at Planck curvature, Heisenberg uncertainty in the ADM conjugate variables dictates that quantum metric fluctuations render the manifold non-differentiable, making the action mathematically undefined. Because the phase amplitude is undefined, the wavefunctional evaluates identically to zero ($Ψ= 0$), formally marking where physical spacetime cannot exist. For realistic rotating black holes, we demonstrate that $\mathcal{B}_Q$ acts as a quantum-geometric cutoff for the mass-inflation instability, capping the internal mass parameter at a finite amplification of $n_{max} \approx 0.67\,(r_g/\ell_P)^{1/5}$ and $r_B^{\rm max\, Kerr} = 1.67 r_g^{2/5} \ell_P^{3/5}$ for a maximally spinning black hole, and dynamically enforcing a universal, sphericalized core. Evaluating the Gibbons-Hawking-York boundary term over this terminal spacelike slice yields a finite, macroscopic interior action per boundary segment, $S_{GHY}^{\mathcal{B}} \approx \frac{3}{2}Mc^2\,Δt$. Operating without injecting novel trans-Planckian degrees of freedom, these results suggest the classical singularity is not a physical event, but the natural terminal boundary of the geometry's domain of definition.