CGM Black Hole Aperture: Geometric Modifications to Horizon Thermodynamics

Citation: Korompilias, B. (2025). Common Governance Model: Mathematical Physics Framework. Zenodo. https://doi.org/10.5281/zenodo.17521384

Abstract

We present a systematic investigation of black hole thermodynamics within the Common Governance Model (CGM) framework, which introduces a geometric aperture parameter m_a = 1/(2√(2π)) ≈ 0.19947 derived from first principles. This parameter modifies the standard Bekenstein-Hawking relations through three scaling laws: S_CGM = S_BH(1 + m_a), T_CGM = T_H/(1 + m_a), and τ_CGM = τ_std(1 + m_a)^4. Our analysis spans 10 orders of magnitude in black hole mass, from Planck-scale to supermassive black holes, incorporating species-resolved emission using recent greybody factors. The resulting 19.947% entropy increase and 16.63% temperature decrease align with independent calculations from area quantization and f(R) gravity studies. We find that primordial black holes evaporating today have masses reduced to 1.357 × 10^11 kg, while detection prospects for Hawking radiation remain far below current experimental thresholds. The framework maintains exact thermodynamic consistency and energy conservation while providing testable predictions for future observations.

1. Introduction

The thermodynamics of black holes represents a crucial intersection between general relativity, quantum mechanics, and statistical physics. Since the foundational work establishing black hole entropy proportional to horizon area and temperature inversely proportional to mass, numerous approaches have sought corrections arising from quantum gravitational effects. These corrections typically manifest as modifications to the Bekenstein-Hawking formulae, with various theoretical frameworks predicting adjustments ranging from logarithmic corrections to multiplicative factors.

The Common Governance Model (CGM) provides a geometric framework for understanding physical phenomena through recursive self-reference and structural alignment. Within this framework, the concept of an aperture emerges naturally from the requirement that observational closure must be incomplete to permit observation itself. This aperture, characterized by the parameter m_a = 1/(2√(2π)), represents a fundamental balance: sufficient closure for structural stability (97.93%) with sufficient openness for observation (2.07%).

This study applies CGM principles to black hole thermodynamics, deriving modified relations for entropy, temperature, and evaporation time. We implement these modifications computationally across a comprehensive range of black hole masses and compare our results with both standard predictions and recent quantum gravity corrections reported in the literature.

2. Theoretical Framework

2.1 The Aperture Parameter

The CGM framework identifies a fundamental geometric parameter:

m_a = 1/(2√(2π)) ≈ 0.199471140201

This parameter emerges from the requirement that the complete solid angle 4π steradians necessary for three-dimensional observation must accommodate an aperture preventing total closure. The value derives from geometric principles rather than empirical fitting, specifically from the constraint that recursive operations must achieve closure while maintaining observability.

2.2 Modified Thermodynamic Relations

The aperture parameter modifies black hole thermodynamics through three primary relations:

Entropy Enhancement:
S_CGM = S_BH × (1 + m_a)

where S_BH = k_B c^3 A/(4ħG) is the standard Bekenstein-Hawking entropy. This 19.947% increase reflects additional information storage capacity arising from the aperture structure.

Temperature Reduction:
T_CGM = T_H / (1 + m_a)

where T_H = ħc^3/(8πGMk_B) is the standard Hawking temperature. The 16.63% decrease maintains the thermodynamic identity TdS = dM.

Lifetime Extension:
τ_CGM = τ_std × (1 + m_a)^4

where τ_std = 5120πG^2M^3/(ħc^4) is the standard evaporation time. The factor of (1 + m_a)^4 arises from the Stefan-Boltzmann scaling with temperature to the fourth power.

2.3 Conservation Laws

The framework preserves fundamental conservation principles:

1. Energy Conservation: The total radiated energy equals Mc^2 in both standard and CGM frameworks.
2. First Law: The relation dM = TdS remains valid with CGM-modified quantities.
3. Smarr Relation: The identity M = 2TS/(c^2) holds with appropriate CGM substitutions.

3. Methodology

3.1 Computational Implementation

We developed a Python implementation calculating both standard and CGM-modified thermodynamic quantities for black holes spanning masses from the Planck mass (2.176 × 10^-8 kg) to supermassive black holes (6.5 × 10^9 solar masses). The code maintains numerical precision to at least 12 significant figures and implements consistency checks for conservation laws.

3.2 Species-Resolved Emission

For emission calculations, we incorporate greybody factors from Oshita and Okabayashi (2024, arXiv:2403.17487v2) for photons and neutrinos. For Schwarzschild black holes (spin parameter a* = 0), the greybody factors are:

• Photons: ε_photon ≈ 0.905
• Neutrinos: ε_neutrino ≈ 0.742 per flavor (multiplied by 3 for three flavors)

3.3 Page Curve Parameters

Following recent developments in black hole information theory, we adopt:

• Page time: t_Page = 0.5406 × τ_evap (Penington 2020)
• Emitted entropy at Page time: S_em(Page) = 0.750 × S_total (Almheiri et al. 2019)

These values reflect insights from replica wormhole calculations and the island formula for entanglement entropy.

4. Results

4.1 Universal Scaling Behavior

Across all black hole masses examined, the CGM modifications produce consistent scaling:

Quantity	Standard	CGM	Ratio
Entropy S	S_BH	1.199 × S_BH	1.199
Temperature T	T_H	0.834 × T_H	0.834
Lifetime τ	τ_std	2.070 × τ_std	2.070
Power L	L_std	0.483 × L_std	0.483
Emission rate dN/dt	(dN/dt)_std	0.579 × (dN/dt)_std	0.579

These ratios remain constant across 10 orders of magnitude in mass, confirming the universal nature of the geometric correction.

4.2 Primordial Black Hole Constraints

For primordial black holes (PBHs) evaporating over the current age of the universe (13.797 Gyr), the critical mass shifts from:

• Standard: M_crit,std = 1.730 × 10^11 kg
• CGM: M_crit,CGM = 1.357 × 10^11 kg

This 21.5% reduction modifies the lower bound of the PBH dark matter window, now spanning 1.357 × 10^11 kg to 10^15 kg.

4.3 Detection Prospects

For a 10^12 kg PBH at 10 kpc distance, the predicted fluxes are:

• Photon flux: 3.41 × 10^-27 ph/s/cm^2 (CGM) versus Fermi threshold ~10^-9 ph/s/cm^2
• Neutrino flux: 8.39 × 10^-27 nu/s/cm^2 (CGM) versus IceCube threshold ~10^-8 nu/s/cm^2

Both fluxes fall approximately 18 orders of magnitude below current detection capabilities, with CGM reducing fluxes by an additional factor of 0.579.

4.4 Supermassive Black Hole Radiation

For Sagittarius A* (4 × 10^6 solar masses), the Hawking radiation frequency of 8.68 × 10^-4 Hz falls far below the interstellar plasma cutoff frequency of approximately 8.98 kHz for electron density n_e = 1 cm^-3. The ratio f_Hawking/f_plasma ≈ 10^-7 confirms that electromagnetic Hawking radiation cannot propagate through the interstellar medium.

4.5 Information Content

Using Bekenstein-Mukhanov area quantization (ΔA = 8πl_P^2), the information content per area quantum is approximately 1.4 bits for CGM-modified black holes, compared to 1.2 bits in the standard case. This increase reflects enhanced information storage capacity consistent with the aperture concept.

5. Discussion

5.1 Comparison with Literature

Our results align remarkably with independent quantum gravity studies. A 2025 area quantization study based on Landauer's principle reports an identical 19-20% entropy increase for Schwarzschild black holes. The temperature-entropy anticorrelation we observe appears consistently in f(R) gravity and other modified theories. The 20-25% reduction in critical PBH mass matches calculations from multiple groups introducing similar entropy corrections.

5.2 Physical Interpretation

The CGM aperture parameter can be understood as encoding the fundamental incompleteness required for observation. The 2.07% aperture prevents complete closure while maintaining 97.93% structural stability. This balance manifests in black hole thermodynamics as enhanced entropy (more information storage) coupled with reduced temperature (slower information release).

5.3 Observational Implications

Current observations cannot distinguish CGM predictions from standard theory due to the extreme weakness of Hawking radiation. Future improvements in sensitivity would need to exceed current capabilities by many orders of magnitude. The most promising avenue for testing may be precision measurements of analog black hole systems where the aperture effect could be more accessible.

5.4 Extension to Other Horizons

The framework extends naturally to other horizon types. For Rindler horizons associated with uniform acceleration, CGM predicts a 16.63% reduction in Unruh temperature. For de Sitter horizons, the cosmological horizon entropy increases by the same factor (1 + m_a). These extensions provide additional avenues for testing the framework.

6. Conclusions

The CGM aperture framework provides a geometrically motivated modification to black hole thermodynamics that produces specific, testable predictions. The three scaling laws derived from the single parameter m_a = 1/(2√(2π)) maintain complete thermodynamic consistency while modifying observable quantities in ways that align with independent quantum gravity calculations.

Key findings include:

1. Universal 19.947% entropy enhancement and 16.63% temperature reduction across all black hole masses
2. Modification of the primordial black hole mass window with the lower bound reduced to 1.357 × 10^11 kg
3. Further suppression of already undetectable Hawking radiation fluxes
4. Preservation of all conservation laws and thermodynamic identities

The quantitative agreement with independent studies using different approaches suggests that the geometric aperture may represent a fundamental aspect of quantum gravitational corrections to black hole physics. While current observations cannot test these predictions directly, the framework provides clear targets for future theoretical and experimental investigations.

Acknowledgments

We acknowledge valuable discussions on the implementation of greybody factors and the use of recent results from the replica wormhole program.

Appendix A: Fundamental Assumptions

1. Geometric Origin: The aperture parameter m_a derives from geometric requirements for observation rather than empirical fitting.

2. Universal Application: The same scaling laws apply to all black hole types (Schwarzschild, Kerr, Reissner-Nordström) with modifications affecting only thermodynamic quantities, not the background geometry.

3. Thermodynamic Consistency: All standard thermodynamic relations remain valid with appropriate CGM substitutions.

4. Species Independence: The emission suppression factor applies equally to all particle species, preserving their relative abundances.

5. Classical Spacetime: The background metric remains classical; only horizon thermodynamics receives quantum corrections.

6. Weak Field Propagation: Emitted radiation propagates through standard vacuum or plasma according to conventional physics once away from the horizon.

Appendix B: Numerical Methods

The computational implementation employs IEEE 754 double precision arithmetic with explicit verification of conservation laws at each calculation. Greybody factors are interpolated linearly from tabulated values in the referenced literature. All physical constants follow CODATA 2018 recommended values except where noted. The code is available for verification and reproduction of all reported results.