Dunstan Low

Dunstan Low

dunstanlow@gmail.com

October 31, 2025

COSMOLOGY

Singularity-Free Spacetime: Collapse Cutoff Without Measurement

Abstract. General relativity predicts spacetime singularities where curvature diverges. We prove that singularities are unphysical: collapse requires measurement, and no measurement is possible in isolated cores. Thus, collapse density $$\rho_C = 0$$, scalar curvature $$R = 0$$, and all invariants vanish. The proof is self-contained, relies only on GR and QM, and requires no Planck scale. Information is preserved. The result is falsifiable via BMV/QGEM (2026). This resolves black hole and Big Bang singularities without quantizing gravity.

1. Introduction

The Schwarzschild metric is

\[ ds^2 = -\left(1 - \frac{2GM}{r}\right)dt^2 + \left(1 - \frac{2GM}{r}\right)^{-1}dr^2 + r^2 d\Omega^2. \]

At $$r \to 0$$, the Kretschmann scalar diverges:

\[ R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} = \frac{48G^2M^2}{r^6} \to \infty. \]

The Hawking--Penrose theorems prove geodesic incompleteness under reasonable energy conditions.

We resolve singularities using the Gravity as Collapse (GAC) model, where:

2. Collapse and Measurement

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

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

Collapse occurs only via measurement (QM). In realized spacetime, $$\rho_C > 0$$. In isolated regions (e.g., black hole cores), no measurement is possible:

\[ \rho_C = 0. \]

Thus,

\[ R = -8\pi G \rho_C = 0. \]

3. Theorem: No Curvature Singularity

Theorem. Spacetime admits no curvature singularity.

Proof.
Let $$\mathcal{M}$$ be the spacetime manifold with metric $$g_{\mu\nu}$$. Then:

  2. The scalar curvature is $$R = -8\pi G \rho_C$$ (GR).
  3. $$\rho_C = 0$$ where no collapse occurs.
  4. No collapse occurs without measurement (QM).
  5. No measurement is possible in an isolated core.
  6. Thus, $$R = 0$$ at $$r = 0$$.
  7. All curvature invariants vanish when $$R = 0$$.

No divergence occurs. Spacetime is geodesically complete. \(\quad \blacksquare\)

4. Singularity-Free Interior Metric

The metric is

\[ ds^2 = \begin{cases} -\left(1 - \frac{2GM}{r}\right)dt^2 + \left(1 - \frac{2GM}{r}\right)^{-1}dr^2 + r^2 d\Omega^2 & \text{measurable region} \\ \text{undefined} & \text{no measurement} \end{cases} \]

The core is a timeless void — no geometry, no physics.

5. Information Paradox

Matter collapses at the measurable shell. No information enters the core. Hawking radiation carries collapse history. Information is preserved.

5.1 The Phantom Singularity

We have been chasing a ghost.

If information could reach $$r=0$$, it would trigger collapse. Collapse would generate gravity ($$R = -8\pi G \rho_C$$). But the core is causally isolated - no signal, no measurement, no collapse. Thus, no information ever entered.

Engelhardt et al.(2019) confirm this: Hawking radiation follows the Page curve, recovering black hole information via quantum extremal surfaces Engelhardt et al. (2019). In GAC, this recovery is natural - the information never entered the core. It was encoded at the collapse front (Σ) and leaked outward.

The "singularity" is not a physical point. It is the absence of information. A phantom.

6. Cosmological Singularity

Pre-Bang: $$\rho_C = 0$$ $$\to$$ $$R = 0$$. The Big Bang is the first collapse front. No initial singularity.

7. Experimental Tests

Experiment Prediction Capability
LIGO ringdown No infinite mode aLIGO+
BMV/QGEM No pre-collapse gravity 2026

Assumptions and Falsifiability

Key assumption:

Falsifiable via BMV/QGEM (2026).

8. Conclusion

Singularities are not physical. They are regions without measurement. Spacetime ends where collapse cannot occur. This is the minimal resolution using only GR and QM.

References

    1. D.Low, ``Gravity as Collapse,'' Zenodo (2025), doi.org/10.5281/zenodo.17359070.
    2. A.Almheiri, N.Engelhardt, D.Marolf, and H.Maxfield, ``The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,'' JHEP 12 (2019) 063, doi:10.1007/JHEP12(2019)063.