Hyperbolic Rapidity Parametrization of the Coulomb Field: Gaussian Envelope, Two-Regime Electrostatics, and the Fine-Structure Constant as a Rest-Frame Ratio
Abstract
We examine whether the hyperbolic rapidity parametrization developed in earlier work for the interior metric of a Schwarzschild black hole can also parametrize the electrostatic field of a charged particle described by a Gaussian envelope. The construction is motivated by the observation that the Lorentz-contracted width of a Gaussian wavepacket, σ(φ) = σ₀ sech φ, acts as a linear amplitude whose Born-like conjugate square matches the functional form of the interior metric of a Schwarzschild black hole, r/rₛ = sech² β. Treating the Compton width σ₀ as a boundary between an exterior Coulomb regime and an interior regime, we clarify how the linear exterior coordinate rₑₓₜ(φ) = σ₀ cosh φ combines with the inverse-square Coulomb law to yield Eₑₓₜ ∝ sech² φ, while the interior profile is algebraically equivalent to that of a uniformly charged sphere. The electrostatic self-energy of this configuration is (3/5)αmₑc², where α is the fine-structure constant. We also observe that the ratio of the classical electron radius rₑ = ασ₀ to the dynamical Compton horizon σ(φ) equals α cosh φ in a frame of rapidity φ, a purely kinematic identity that reduces to α at rest. Finally, using the exterior gravitational metric rₑₓₜ = rₛ cosh² β to derive the exact Schwarzschild lapse factor √(1 − rₛ/rₑₓₜ) = tanh β in the Tolman temperature, we justify the physical domain |β| > 0 on which ψ(β) = sech β satisfies a differential identity algebraically identical to the Ginzburg–Landau field equation. The mathematical identities are exact; the physical interpretations are offered as conjectures.
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BibTeX
@misc{barros2026hyperbolic,
title = {{Hyperbolic Rapidity Parametrization of the Coulomb Field: Gaussian Envelope, Two-Regime Electrostatics, and the Fine-Structure Constant as a Rest-Frame Ratio}},
author = {Rodrigo Silva Barros},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23651339},
url = {https://zenodo.org/records/23651339},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hyperbolic Rapidity Parametrization of the Coulomb Field: Gaussian Envelope, Two-Regime Electrostatics, and the Fine-Structure Constant as a Rest-Frame Ratio
AU - Rodrigo Silva Barros
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23651339
UR - https://zenodo.org/records/23651339
AB - We examine whether the hyperbolic rapidity parametrization developed in earlier work for the interior metric of a Schwarzschild black hole can also parametrize the electrostatic field of a charged particle described by a Gaussian envelope. The construction is motivated by the observation that the Lorentz-contracted width of a Gaussian wavepacket, σ(φ) = σ₀ sech φ, acts as a linear amplitude whose Born-like conjugate square matches the functional form of the interior metric of a Schwarzschild black hole, r/rₛ = sech² β. Treating the Compton width σ₀ as a boundary between an exterior Coulomb regime and an interior regime, we clarify how the linear exterior coordinate rₑₓₜ(φ) = σ₀ cosh φ combines with the inverse-square Coulomb law to yield Eₑₓₜ ∝ sech² φ, while the interior profile is algebraically equivalent to that of a uniformly charged sphere. The electrostatic self-energy of this configuration is (3/5)αmₑc², where α is the fine-structure constant. We also observe that the ratio of the classical electron radius rₑ = ασ₀ to the dynamical Compton horizon σ(φ) equals α cosh φ in a frame of rapidity φ, a purely kinematic identity that reduces to α at rest. Finally, using the exterior gravitational metric rₑₓₜ = rₛ cosh² β to derive the exact Schwarzschild lapse factor √(1 − rₛ/rₑₓₜ) = tanh β in the Tolman temperature, we justify the physical domain |β| > 0 on which ψ(β) = sech β satisfies a differential identity algebraically identical to the Ginzburg–Landau field equation. The mathematical identities are exact; the physical interpretations are offered as conjectures.
ER -