Golden Ratio Subdivision in Black Hole Formation and Accretion: A Topological Hypothesis

Ritesh January 8, 2025

Abstract

This paper proposes that gravitational collapse, black hole formation, and accretion dynamics follow a recursive subdivision pattern governed by the golden ratio (φ ≈ 1.618). Inspired by the topological properties of Möbius strip subdivision, I hypothesize that stellar collapse partitions mass approximately 38:62 (golden ratio proportions), with similar patterns manifesting in accretion disk structure and ultimately in quantum field dissolution at the event horizon. Specific testable predictions for Event Horizon Telescope observations at 0.87mm wavelength are provided.

1. Introduction

The observation that initiated this hypothesis was a simple physical experiment: cutting a Möbius strip longitudinally at 1/3 of its width produces one small Möbius strip (retaining the topological twist) and one larger twisted loop. This 1/3 subdivision ratio, which approximates the golden ratio division (φ⁻¹ ≈ 0.618 for the larger piece, 1-φ⁻¹ ≈ 0.382 for the smaller), suggested a potential universal principle for systems undergoing topological transformation.

The golden ratio appears frequently in self-organizing systems where stability requires avoiding resonant interference - from phyllotaxis in plant growth to spiral galaxy arm spacing. This pattern recognition led to the hypothesis that gravitational collapse and accretion, as transformations involving extreme geometry and information encoding, might follow similar principles.

2. Theoretical Framework

2.1 Stellar Collapse and Mass Partition

During core-collapse supernova, the stellar core undergoes catastrophic compression while outer layers are ejected. Current models show highly variable mass retention (15-40% depending on progenitor mass, metallicity, and rotation). However, if golden ratio subdivision governs this process, we would expect:

• Core retention: ~38% of initial stellar mass • Ejection (supernova ejecta, neutrinos, jets): ~62% • This partition minimizes resonant instabilities while allowing sufficient mass loss to shed angular momentum

The Möbius analogy: The topological twist (complexity of stellar structure) is 'retained' in the compact remnant (preserving quantum information on the event horizon), while the bulk material (classical information) is dispersed.

2.2 Accretion Disk Structure

Accretion onto black holes exhibits spiral density wave structures. If these follow golden ratio spacing:

• Material spirals inward with each orbital shell at radius r/φ relative to the previous • Density waves form at golden ratio intervals to avoid destructive resonances • Angular momentum transport occurs most efficiently at these geometrically optimal spacings • The observed brightness asymmetry in M87* (~60:40 distribution) approximates golden ratio

This 'polite accretion' mechanism explains why matter doesn't fall in catastrophically but rather spirals gradually, shedding angular momentum through structured density waves.

2.3 Multi-Wavelength Ring Structure

Event Horizon Telescope observations at different wavelengths reveal nested emission rings:

• 3.5mm observation (2018): Ring diameter ~8.4 Schwarzschild radii (Rs) • 1.3mm observation (2017): Ring diameter ~5.2 Rs • Ratio: 8.4/5.2 ≈ 1.615 (within 0.2% of φ = 1.618)

This remarkable agreement suggests the emission regions are organized at golden ratio radial intervals, with different wavelengths probing different layers of the accretion flow structure.

2.4 Quantum Field Dissolution

Within the event horizon, recursive subdivision continues at quantum scales. In quantum field theory, particles are excitations in underlying fields. As matter approaches the singularity, recursive golden ratio partitioning continues until excitation energy falls below threshold - the 'wave function collapses' back to vacuum field state. This prevents true singularities and provides a mechanism for information encoding compatible with the holographic principle: the 2D horizon surface encodes 3D volume information according to the subdivision history.

3. Testable Predictions

3.1 Event Horizon Telescope 0.87mm Observations

Primary prediction: When EHT releases M87* images at 0.87mm wavelength (expected 2025-2026), the emission ring diameter should be approximately 3.2 ± 0.3 Schwarzschild radii.

Calculation: 5.2 Rs / φ = 5.2 / 1.618 = 3.21 Rs

This would confirm the golden ratio progression: 8.4 → 5.2 → 3.2 (each step divided by φ).

3.2 Brightness Asymmetry Quantification

Quantitative photometric analysis of the M87* ring should reveal brightness distribution of 61.8% ± 2% in the brighter region versus 38.2% ± 2% in the dimmer region, integrated azimuthally around the ring.

3.3 Supernova Remnant Statistics

Statistical analysis of core-collapse supernovae black hole remnants should show a peak in mass retention around 38% of progenitor core mass, with standard deviation reflecting variations in metallicity and rotation.

3.4 Spiral Density Wave Structure

High-resolution radio interferometry of accretion disks should reveal spiral density waves with Fourier mode analysis showing peaks at m=2, m=3, m=5, m=8 (Fibonacci sequence), corresponding to golden angle phase relationships.

4. Discussion

4.1 Connection to Existing Theory

This hypothesis is consistent with:

• General relativity predictions for ring structure • Holographic principle (information encoding on horizon surface) • Quantum field theory (particle/field duality) • Observed multi-wavelength emission structures • Spiral density wave theory in accretion disks

The golden ratio constraint adds a specific geometric principle explaining WHY certain configurations are preferred over others - namely, because they minimize resonant instabilities while maximizing information preservation.

4.2 Physical Mechanism

The golden ratio is mathematically unique as the 'most irrational' number - it has the slowest convergence in continued fraction expansion. In physical systems:

• Orbital resonances at simple ratios (1:2, 2:3) create instabilities • Golden ratio spacing maximally avoids these resonances • Self-organizing systems naturally settle into golden ratio configurations • This principle appears in phyllotaxis, spiral galaxies, and potentially black hole physics

4.3 Limitations and Uncertainties

This hypothesis is speculative and faces several challenges:  • Current supernova data shows wide variation in mass retention (15-40%), with no published analysis for clustering near 38% • The 8.4/5.2 ratio is suggestive but requires the 0.87mm observation for confirmation • Brightness asymmetry appears qualitatively consistent but lacks quantitative photometric analysis • The physical mechanism driving golden ratio partitioning in extreme gravity requires theoretical development • Alternative explanations for observed structures exist and must be ruled out

5. Conclusion

The observation that Möbius strip topology preserves itself through 1/3 subdivision inspired a hypothesis connecting this geometric principle to black hole physics via the golden ratio. The suggestive agreement between observed multi-wavelength ring structures (8.4/5.2 ≈ φ) and brightness asymmetry patterns warrants serious investigation.  The primary test is straightforward: upcoming 0.87mm EHT observations of M87* should reveal an emission ring at ~3.2 Schwarzschild radii if this hypothesis is correct. A significant deviation would falsify the model.  Regardless of outcome, this exercise demonstrates how physical intuition from simple experiments (cutting paper strips) can generate testable hypotheses about extreme astrophysical phenomena. The golden ratio may represent a fundamental organizational principle in systems undergoing topological transformation under extreme gravity.

6. Key References

Event Horizon Telescope Collaboration (2019). First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. Astrophysical Journal Letters, 875, L1.

Lu, R.-S., et al. (2023). A ring-like accretion structure in M87 connecting its black hole and jet. Nature, 616, 686-690.

Raymond, A.W., et al. (2024). First Very Long Baseline Interferometry Detections at 870 μm. The Astronomical Journal.

Event Horizon Telescope Collaboration (2024). Broadband Multi-wavelength Properties of M87 during the 2018 EHT Campaign. Astronomy & Astrophysics (in press).

Bekenstein, J.D. (1973). Black Holes and Entropy. Physical Review D, 7, 2333-2346.

't Hooft, G. (1993). Dimensional Reduction in Quantum Gravity. arXiv:gr-qc/9310026.

Document prepared January 8, 2025 For discussion and review