Regularizing Relativistic Singularities through a C ∞ -Smooth Extension of the d'Alembert Invariance: Resolving the Hubble Tension and the Non-Zero Photon Mass
Description
Manuscript Structure
Regularizing Relativistic Singularities through a C -Smooth Extension of the d'Alembert
Invariance: Resolving the Hubble Tension and the Non-Zero Photon Mass
1. Introduction
2. Baseline Mathematics of the Kinematic Boost
2.1 The Lorentz Function Family: Denition and Kinematic Limit
2.2 The Lorentz Function Family: Intrinsic Geometric Properties
3. Regularized Relativistic Kinematics
3.1 Asymptotic Invariance of the d'Alembert Operator
3.2 The Deviation Factor D(β): Higher-Order Regularization
3.3 Preservation of the Causal Light-Cone Structure
3.4 Inheritance of Classical Relativity
3.4.1 Relativistic Velocity Composition: The Illusion of Group Closure
3.4.2 Hierarchy of Relativities: Truncation of Kinematic Information
4. ALR Dynamics at Light Barrier
4.1 Kinematic Tunneling: Transition at the Light-Barrier
4.2 The Superluminal Regime: Four Stable States
4.3 Convergence of Trajectories and Finite Temporal Flow
4.4 The Architecture of Finite Energy States
4.4.1 Bounded Mass, Energy and Momentum at the β = 1 Limit
4.4.2 Preservation of the Finite Length Contraction
4.5 ALR-Upgrade of Lemaître-Einstein Law
5. Application 1: Resolving the Hubble Tension
5.1 The Kinematic Gap: Signicance of Truncated Energy in the Limit Regime
5.2 Quantifying the Hubble Matching Scale (HMS): The Kinematic Breaking Point
5.3 Corresponding Distances: The Hubble-Lemaître Relation at the HMS
5.4 The Hubble Tension in a Nutshell: No Physical Discrepancies
5.5 The Invariance Boundary: Regime Bifurcation at the HMS
6. Application 2: The Non-Zero Photon Mass
6.1 Beyond the 0 · ∞ Ambiguity
6.2 Numerical Estimation of Dynamic Photon Mass Benchmark
6.3 The Footnote Scale of Photon Mass
6.4 Photon in a Nutshell:Comparisson with Electron and Neutrino
6.5 Consequences for the Fine-Structure Constant α
6.6 The Footnote-Scale Photon Mass in the LQG Big Bounce
7. Conclusion and Outlook
Methods (English)
The manuscript introduces the Asymptotic Lorentz Relativity (ALR) framework. The core of this work is to present a new, $C^\infty$-smooth regularized relativistic kinematics that remains valid even at the light-speed limit ($\beta \to 1$). While this framework approximates Einsteinian relativity up to order $\mathcal{O}(\beta^4)$ in the intermediate velocity regime, it substitutes the standard singular behavior with a regularized description as $\beta$ approaches 1, where classical theory ceases to be valid.
Key scientific highlights of this manuscript include:
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First-Principles Resolution of the Hubble Tension: A geometric reconciliation of the $5\sigma$ Hubble tension between Planck and SH0ES data by identifying the Hubble Matching Scale (HMS) at $\beta_{\text{HMS}} \approx 0.527$ ($z_{\text{HMS}} \approx 0.726$).
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Non-Zero Dynamic Photon Mass ($\S 6$): Resolves the $0 \cdot \infty$ singularity of the photon by establishing a frequency-dependent dynamic photon mass benchmark ($m_0 \approx 3.11 \times 10^{-36}\text{ kg}$) and a fundamental cosmological vacuum floor ($m_{\text{foot}} \approx 4.89 \times 10^{-47}\text{ kg}$).
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d’Alembert Form Invariance: Demonstrates full asymptotic invariance of the differential wave operator across all velocity regimes, eliminating unphysical coordinate and boost divergences without modifying the standard Einsteinian domain at intermediate velocities ($\beta \ll 1$).
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Finite Physical Observables at the Luminal Boundary: Establishes bounded dynamic energy and momentum states at $v = c$ ($E_{\text{max}} = \sqrt{2} \cdot m \cdot c^2$) and proves the preservation of a finite temporal flow $d\tau_{\text{ALR}} = dt / \sqrt{2} > 0$.
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Applications to Foundational Field Theories & Quantum Cosmology: Provides the mathematical basis for an analytically continuous transition at boundary limits, offering a non-singular background compatible with Big Bounce models (Loop Quantum Gravity) and finite vacuum impedance structures (aligning with research on electromagnetic field bounds by Prof. Gerd Leuchs, FAU Erlangen).
Notes (English)
Update Note (July 2026):
This version restores the complete, extended manuscript text. Section 6 (Non-zero photon mass) has been fully reintegrated into the core framework, providing the exact geometric derivation of the finite, frequency-dependent dynamic photon mass benchmark ($m_0 \approx 3.11 \times 10^{-36}$ kg) and the fundamental cosmological vacuum floor ($m_{\text{foot}} \approx 4.89 \times 10^{-47}$ kg). The text is structurally complete, aligned with current journal submission guidelines, and presented in a clean preprint layout.
Files
ALR_Hubble_Tension_and_Photon_Mass_Delinic_2026.pdf
Additional details
Related works
- Is supplement to
- Preprint: 10.5281/zenodo.21217485 (DOI)