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.

• The paper demonstrates that quantum metric fluctuations truncate the gravitational path integral at a finite boundary, thereby avoiding classical singularities.
• It reveals that the breakdown of Sobolev regularity at the Planck scale, marked by a vanishing wavefunctional, signals a robust quantum-geometric cutoff.
• The study quantifies boundary radii in Schwarzschild and Kerr black holes, reinforcing cosmic censorship and informing black hole evolution models.

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

Summary and Framework

This paper establishes that the classical black hole singularity predicted by general relativity is never physically realized. Instead, the domain of the gravitational functional integral, as prescribed by the Wheeler-DeWitt (WDW) equation and the Feynman sum over geometries, terminates at a finite, positive radius corresponding to the Planck curvature threshold ($\mathcal{K} \sim \ell_P^{-4}$). This quantum boundary, denoted $\mathcal{B}_Q$, is not an artifact of extended physics or novel degrees of freedom but emerges directly from mathematical failure within the Einstein-Hilbert action due to quantum metric fluctuations.

The core argument links canonical ADM uncertainty to the probabilistic width of the functional integral, and formalizes how quantum fluctuations force the metric to become non-differentiable at Planck scale, violating the $W^{2,2}$ Sobolev regularity required for the action’s definition. Beyond $\mathcal{B}_Q$, the phase amplitude is undefined and the wavefunctional has zero support ($\Psi=0$), signaling a genuine quantum-geometric boundary of spacetime.

Mathematical Foundation and Threshold Mechanism

The Wheeler-DeWitt equation provides the non-perturbative basis for quantum geometry, where the wavefunctional $\Psi[h_{ij}]$ vanishes for inadmissible metrics. Within the Feynman sum, as the Kretschmann scalar approaches the Planck threshold, the metric fluctuations dictated by the ADM commutation relations inflate to order unity per Planck cell, destroying differentiability and causing the Sobolev failure of the action.

The sharp mathematical cutoff is supported by the bounded $L^2$ curvature theorem (Klainerman et al.), which constrains the well-posedness of the Cauchy problem to the $W^{2,2}$ regularity class. When the curvature exceeds $\ell_P^{-4}$, this threshold is violated and the sum over geometries loses domain of definition.

This phenomenon is universal: even homogeneous collapse inevitably reaches the threshold before Planck density ($\rho_B \simeq 0.034\,\rho_P$), and matter fields dissolve as the curvature radius falls below the scale of excitation wavelengths, but the fundamental breakdown occurs strictly at $\mathcal{B}_Q$0. The matter functional measure itself ceases to be defined on the background geometry.

Physical Instantiations in Black Hole Interiors

In Schwarzschild black holes, $\mathcal{B}_Q$1 truncates the manifold at radius $\mathcal{B}_Q$2. For solar-mass black holes, $\mathcal{B}_Q$3 m, completely excising the classical singularity. This boundary is self-stabilizing: moving inward invokes Sobolev failure, forbidding further collapse; moving outward restores support in the functional integral.

Rotating black holes (Kerr geometry) undergo the internal mass-inflation instability at the inner Cauchy horizon, driving the mass parameter to exponential growth and causing sphericalization. The quantum boundary forms when mass inflation caps the internal amplification at $\mathcal{B}_Q$4, with the truncation radius for maximally spinning black holes $\mathcal{B}_Q$5. The interior Cauchy horizon, theoretical ring singularity, and non-unique Kerr extensions are structurally excised.

Anisotropic BKL oscillations and Kasner transitions in the interior are locally capped as metric fluctuations reach the Planck threshold, erasing continuous symmetries and conservation laws patch by patch without violating exterior ADM invariants.

Boundary Action and Implications for Black Hole Evolution

The Gibbons-Hawking-York (GHY) boundary term acquires a significant role: the terminal boundary at $\mathcal{B}_Q$6 yields a finite macroscopic action, $\mathcal{B}_Q$7, per boundary segment, and summing over the black hole lifetime $\mathcal{B}_Q$8 gives $\mathcal{B}_Q$9. The boundary acts as a bookkeeping device for the excised bulk, tracking the total enclosed mass.

The terminal quantum boundary enforces cosmic censorship: mass inflation and the quantum-geometric cutoff prevent non-unique extensions and naked singularity exposure, closing any classical loopholes in the Kerr interior.

Implications and Future Directions

This approach requires no trans-Planckian extensions or discrete spacetime structures. It leverages only the standard functional integral and the continuous metric manifold, conferring parsimony and internal consistency. The quantum boundary is a generic feature, not dependent on specific initial conditions, the equation of state, or the presence/absence of matter fields. It provides a complete framework for singularity avoidance strictly within the classical and standard quantum field-theoretic domains.

Future lines of work should explore cosmological ramifications, possible observable consequences in gravitational wave phenomena, and implications for information retention in black hole evaporation. Advanced tests of continuous versus discrete spacetime at the Planck scale could critically discriminate between this framework and other approaches like loop quantum cosmology or Planck star proposals.

Conclusion

The paper demonstrates that quantum fluctuations at the Planck scale inevitably cause the gravitational functional integral to lose domain of definition, establishing a robust quantum boundary $W^{2,2}$0 that terminates black hole interiors at finite radius. All classical singularity, mass-inflation, and non-unique interior extensions are physically excluded by the vanishing of the wavefunctional. The approach maintains ultraviolet finiteness without recourse to speculative extensions, suggesting general relativity combined with quantum mechanics is a self-truncating theory. This quantum boundary concept has broad implications for the theoretical structure of quantum gravity and black hole physics.