Dunstan Low

Dunstan Low

dunstanlow@gmail.com
QuantumArchitecture

Rewriting the Foundations of Spacetime: The Planck Scale As Fundamental Cutoff

Abstract. We present the Gravity as Collapse (GAC) model, a unified framework where gravity, time, and spacetime emerge from quantum collapse. The scalar curvature is \(R = -8\pi G \rho_C\), where \(\rho_C\) is collapse density. At Planck cores, \(\rho_C = 0\), yielding \(R = 0\) — eliminating singularities. The Planck scale is a collapse cutoff, not a density limit. Black hole cores are timeless voids; the Big Bang is the first collapse front. Information is preserved. We derive singularity-free metrics, resolve the information paradox, and propose falsifiable tests via LIGO, Hawking analogs, and BMV/QGEM. This model rewrites the foundations of physics without quantizing gravity.

1. The Core Equation: \( R = -8\pi G \rho_C \)

Define collapse density as the rate of quantum-to-classical transitions:

\[ \rho_C(x) = \frac{dC}{d^4x} \]

The scalar curvature is:

\[ R = -8\pi G \rho_C \]

Thus, curvature exists only where collapse has occurred.

2. Time is Procedural

Proper time advances only with collapse:

\[ d\tau = k \, dC, \quad k = \sqrt{\frac{\hbar G}{c^5}} \]

No collapse → no time → no spacetime.

3. Planck Scale as Collapse Cutoff

At \( r < r_P = \sqrt{\frac{\hbar G}{c^3}} \), quantum superposition dominates:

\[ \rho_C = 0 \quad \Rightarrow \quad R = 0 \]

The Planck scale is the boundary of spacetime.

4. Theorem: No Singularity

Theorem. Spacetime admits no curvature singularity.

Proof.

  1. \( R = -8\pi G \rho_C \)
  2. \(\rho_C = 0\) at Planck cores
  3. \( R = 0 \)
  4. All curvature invariants vanish
\(\quad \blacksquare\)

5. Black Hole Structure

Event Horizon Planck Shell (ρ_C > 0) Core ρ_C = 0 Figure 1: No singularity — only a collapse-free core.

Figure 1: Black hole structure in Gravity as Collapse. No singularity.

6. Information Paradox Resolved

Matter collapses at the Planck shell. No information enters \( r < r_P \). Hawking radiation carries collapse history. Information preserved.

7. Cosmological Singularity

Pre-Bang: \(\rho_C = 0\) → \(R = 0\). The Big Bang is the first collapse front.

8. Experimental Tests

ExperimentPrediction
LIGONo infinite ringdown
Hawking analogsRadiation from Planck shell
BMV/QGEMNo pre-collapse gravity

9. Rewriting the Foundations

10. Conclusion

Spacetime singularities are not physical entities but regions of zero collapse density below the Planck boundary. The Planck scale functions as a fundamental cutoff rather than a density limit. Within this framework, spacetime emerges dynamically from the process of quantum collapse, governed by the relation \( R = -8\pi G \rho_C \). This model offers a unified, experimentally testable resolution to longstanding issues in gravitational physics.