Finding the eleven: the atom and the nucleus as test beds
Abstract
The relativity at eleven parameters bounds the proper acceleration of a body of mass m by a max (m) = mc 3 /ℏ, and its new effects are of the order of the square of the dimensionless acceleration χ = a/a max (m). This paper asks where, among the best-measured systems of physics, the eleven could be found, and takes the two natural test beds, the hydrogen atom and the atomic nucleus. In hydrogen the electron's acceleration at the Bohr radius is exactly α 3 a max (e), the Coulomb field reaches the Schwinger field at √ α Compton lengths from the proton, and the pointwise acceleration of an S electron has no finite mean square; read at the electron's own resolution, its Compton length, the acceleration in the field of a charge Ze is at most Zα a max (e), with the mean square (24/5)α 5 in the ground state. The two transcriptions of the bound into a term of the atomic Hamiltonian, a saturation of the Coulomb force at the pointwise or at the Compton-resolved acceleration, shift the nS levels by (2/3)α 5 mc 2 /n 3 and α 6 mc 2 /(2n 3 ), that is 213 MHz and 1.2 MHz on the 2S state, and both are excluded by the Lamb shift, measured and computed to a few kHz, by four and two orders of magnitude; hydrogen-like uranium excludes them by more. The atom therefore decides how the bound may be read: not as a term in the Hamiltonian of a bound state, whose stationary states have no mean acceleration, but categorically, as the statement that a body pushed to its bound is replaced by copies. In the atom that statement is the critical charge: the Compton-resolved acceleration of the electron reaches its bound at Zα = 1, exactly where the Dirac equation with a point nucleus loses its ground state, and at Z ≃ 173 with extended nuclei, where quantum electrodynamics predicts the spontaneous emission of positrons. In the nucleus the nucleon's acceleration is 4 × 10 -3 of its bound in the mean field and of order one half in the repulsive core, where nucleons cease to be nucleons; the nuclear spectrum is computed too coarsely to test the square of the first, and the second is again the categorical reading. The eleven are not found in the spectra of the two test beds, and the paper says why: they are found where χ reaches one and a body stops being one, at the critical charge, in the core, in the strong-field laser and at the horizon.
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BibTeX
@misc{reynaud2026finding,
title = {{Finding the eleven: the atom and the nucleus as test beds}},
author = {Patrick .B Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05749766v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finding the eleven: the atom and the nucleus as test beds
AU - Patrick .B Reynaud
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05749766v2
AB - The relativity at eleven parameters bounds the proper acceleration of a body of mass m by a max (m) = mc 3 /ℏ, and its new effects are of the order of the square of the dimensionless acceleration χ = a/a max (m). This paper asks where, among the best-measured systems of physics, the eleven could be found, and takes the two natural test beds, the hydrogen atom and the atomic nucleus. In hydrogen the electron's acceleration at the Bohr radius is exactly α 3 a max (e), the Coulomb field reaches the Schwinger field at √ α Compton lengths from the proton, and the pointwise acceleration of an S electron has no finite mean square; read at the electron's own resolution, its Compton length, the acceleration in the field of a charge Ze is at most Zα a max (e), with the mean square (24/5)α 5 in the ground state. The two transcriptions of the bound into a term of the atomic Hamiltonian, a saturation of the Coulomb force at the pointwise or at the Compton-resolved acceleration, shift the nS levels by (2/3)α 5 mc 2 /n 3 and α 6 mc 2 /(2n 3 ), that is 213 MHz and 1.2 MHz on the 2S state, and both are excluded by the Lamb shift, measured and computed to a few kHz, by four and two orders of magnitude; hydrogen-like uranium excludes them by more. The atom therefore decides how the bound may be read: not as a term in the Hamiltonian of a bound state, whose stationary states have no mean acceleration, but categorically, as the statement that a body pushed to its bound is replaced by copies. In the atom that statement is the critical charge: the Compton-resolved acceleration of the electron reaches its bound at Zα = 1, exactly where the Dirac equation with a point nucleus loses its ground state, and at Z ≃ 173 with extended nuclei, where quantum electrodynamics predicts the spontaneous emission of positrons. In the nucleus the nucleon's acceleration is 4 × 10 -3 of its bound in the mean field and of order one half in the repulsive core, where nucleons cease to be nucleons; the nuclear spectrum is computed too coarsely to test the square of the first, and the second is again the categorical reading. The eleven are not found in the spectra of the two test beds, and the paper says why: they are found where χ reaches one and a body stops being one, at the critical charge, in the core, in the strong-field laser and at the horizon.
ER -