# Stoney units

In physics the Stoney units form a system of units named after the Irish physicist George Johnstone Stoney, who first proposed them in 1881. They are the first historical example of natural units, i.e. units of measurement designed so that certain fundamental physical constants serve as base units.

## Units

Quantity Expression Metric Value
Length (L) ${\displaystyle l_{\text{S}}={\sqrt {\frac {Ge^{2}}{c^{4}(4\pi \epsilon _{0})}}}}$ 1.3807×10−36 m
Mass (M) ${\displaystyle m_{\text{S}}={\sqrt {\frac {e^{2}}{G(4\pi \epsilon _{0})}}}}$ 1.8592×10−9 kg
Time (T) ${\displaystyle t_{\text{S}}={\sqrt {\frac {Ge^{2}}{c^{6}(4\pi \epsilon _{0})}}}}$ 4.6054×10−45 s
Electric charge (Q) ${\displaystyle q_{\text{S}}=e\ }$ 1.6022×10−19 C
Temperature (Θ) ${\displaystyle T_{\text{S}}={\sqrt {\frac {c^{4}e^{2}}{G(4\pi \epsilon _{0}){k_{\text{B}}}^{2}}}}}$ 1.2103×1031 K

The set of the constants that Stoney used as base units is the following:[1][2]

This means that, in terms of Stoney units, the numerical values of all these constants equal one:

${\displaystyle c=G=k_{\text{e}}=e=1.}$

Stoney's set of base units is similar to the one used in Planck units, proposed independently by Planck thirty years later, but Planck normalized the reduced Planck constant in place of the elementary charge.[3] In Stoney units, the numerical value of the reduced Planck constant is not 1, but is

${\displaystyle \hbar ={\frac {1}{\alpha }}\approx 137.0359991,}$

where α is the fine-structure constant. Planck units are more commonly used than Stoney units in modern physics, especially quantum gravity (including string theory). Rarely, Planck units are referred to as Planck–Stoney units.[3]

## History

George Stoney was one of the first scientists to understand that electric charge was quantized; from this quantization he deduced the units that are now named after him.[4][5] James G. O’Hara[6] pointed out in 1974 that Stoney’s derived estimate of the unit of charge, 10−20 ampere-second, was 116 of the modern value of the charge of the electron. The reason is that Stoney used the approximated value of 1018 for the number of molecules presented in one cubic millimetre of gas at standard temperature and pressure. Using the modern values for the Avogadro constant 6.02214×1023 mol−1 and for the volume of a gram-molecule (at s.t.p.) of 22.4146×106 mm3, the modern value is 2.687×1016, instead of Stoney's 1018.

## Stoney units and Planck units

The Stoney length and the Stoney energy, collectively called the Stoney scale, are not far from the Planck length and the Planck energy, the Planck scale. The Stoney scale and the Planck scale are the length and energy scales at which quantum processes and gravity occur together. At these scales, a unified theory of physics is thus required. The only notable attempt to construct such a theory from the Stoney scale was that of H. Weyl, who associated a gravitational unit of charge with the Stoney length[7][8][9] and who appears to have inspired Dirac's fascination with the large number hypothesis.[10] Since then, the Stoney scale has been largely neglected in the development of modern physics, although it is still occasionally discussed.[11][12][13] The Planck scale is valid for all known interactions, and does not give prominence to the electromagnetic interaction, as the Stoney scale does.

## Footnotes

1. The anthropic cosmological principle, by Barrow and Tipler, p291
2. Astrophysics, clocks and fundamental constants, by Karshenboim and Peik, p79
3. Space: in science, art and society, by Penz, Radick, Howell, p191
4. Stoney G. On The Physical Units of Nature, Phil.Mag. 11, 381391, 1881
5. G. Johnstone Stoney, On The Physical Units of Nature, The Scientific Proceedings of the Royal Dublin Society, 3, 5160, 1883. Retrieved 2010-11-28.
6. J.G. O’Hara (1993). George Johnstone Stoney and the Conceptual Discovery of the Electron, Occasional Papers in Science and Technology, Royal Dublin Society 8, 528.
7. K. Tomilin, "Natural System of Units", Proc. of the XX11 International Workshop on High Energy Physics and Field theory, (2 000) 289.
8. H. Weyl, "Gravitation und Elekrizitaet", Koniglich Preussische Akademie der Wissenschaften (1918) 46578
9. H. Weyl, "Eine Neue Erweiterung der Relativitaetstheorie", Annalen der Physik 59 (1919) 1013.
10. G. Gorelik, "Hermann Weyl and Large Numbers in Relativistic Cosmology", Einstein Studies in Russia, (ed. Y. Balashov and V. Vizgin), Birkhaeuser. (2002).
11. M. Castans and J. Belinchon(1998). "Enlargement of Planck's System of Absolute Units", preprint: physics/9811018
12. Jean-Philippe Uzan (2011), "Varying Constants, Gravitation and Cosmology", Living Rev. Relativ., 14 (1): 15–16, doi:10.12942/lrr-2011-2, PMC 5256069, PMID 28179829
13. Ross McPherson. Stoney Scale and Large Number Coincidences. Apeiron, Vol. 14, No. 3, July 2007