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{{Short description|Early system of natural units}}
In [[physics]], the '''Stoney units''' form a [[System of measurement|system of units]] named after the Irish physicist [[George Johnstone Stoney]], who first proposed them in 1881. They are the earliest example of [[natural units]], i.e., a [[Coherence (units of measurement)|coherent set]] of units of measurement designed so that chosen [[physical constant]]s fully define and are included in the set.

== Units ==

{| class="wikitable floatright"
! Quantity
! Expression
! Value in SI units
|- align="left"
| [[Length]] (L)
| <math>l_\text{S} = \sqrt{\frac{G e^2}{4\pi\epsilon_0 c^4}}</math>
| {{val|1.3807|e=-36|u=m}}
|-
| [[Mass]] (M)
| <math>m_\text{S} = \sqrt{\frac{e^2}{4\pi\epsilon_0 G}}</math>
| {{val|1.8592|e=-9|u=kg}}
|-
| [[Time]] (T)
| <math>t_\text{S} = \sqrt{\frac{G e^2}{4\pi\epsilon_0 c^6}}</math>
| {{val|4.6054|e=-45|u=s}}
|-
| [[Electric charge]] (Q)
| <math>q_\text{S} = ~ e </math>
| {{physconst|e|round=4|ref=no}}
|-
|}

The [[physical constant|constants]] that Stoney used to define his set of units is the following:<ref>{{citation |last=Ray |first=T. P. |title=Stoney's fundamental units |journal=Irish Astronomical Journal |volume=15 |pages=152 |year=1981|bibcode=1981IrAJ...15..152R }}</ref><ref>{{citation |last=Stoney |first=G. Johnstone |date=May 1881 |title=LII. On the physical units of nature |url=https://www.tandfonline.com/doi/full/10.1080/14786448108627031 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |language=en |volume=11 |issue=69 |pages=381–390 |doi=10.1080/14786448108627031 |issn=1941-5982}}</ref>
* {{math|''c''}}, the [[speed of light]] in vacuum,
* {{math|''G''}}, the [[gravitational constant]],
* {{math|''k''<sub>e</sub>}}, the [[Coulomb's law#Coulomb constant|Coulomb constant]],
* {{math|''e''}}, the charge on the [[electron]].

Later authors typically replace the Coulomb constant with {{math|{{sfrac|1|4''πε''{{sub|0}}}}}}.<ref>{{citation |last1=Barrow |first1=John D. |author-link1=John D. Barrow |last2=Tipler |first2=Frank |author-link2=Frank J. Tipler |year=1988 |title=The anthropic cosmological principle |url=https://books.google.com/books?id=uSykSbXklWEC |publisher=Oxford University Press |page=291 |isbn=978-0-192-82147-8}}</ref><ref>{{citation |chapter=Constants, Units and Standards |first1=Jeff |last1=Flowers |first2=Brian |last2=Petley |page=79 |title=Astrophysics, clocks and fundamental constants |url=https://books.google.com/books?id=sGk05bF9KJEC&pg=PA79 |editor-first1=Savely G. |editor-last1=Karshenboim |editor-first2=Ekkehard |editor-last2=Peik |publisher=Springer |year=2004 |isbn=978-3-540-21967-5}}</ref>

This means that the numerical values of all these constants, when expressed in coherent Stoney units, is equal to one:

<math display="block">\begin{align}
c &= 1\ l_\text{S} \cdot t_\text{S}^{-1} \\
G &= 1\ l_\text{S}^3 \cdot t_\text{S}^{-2} \cdot m_\text{S}^{-1} \\
k_\text{e} &= 1\ l_\text{S}^3 \cdot t_\text{S}^{-2} \cdot m_\text{S} \cdot q_\text{S}^{-2}\\
e &= 1\ q_\text{S}
\end{align}</math>

In Stoney units, the numerical value of the [[reduced Planck constant]] is

<math display="block">\hbar = \frac{1}{\alpha} ~l_\text{S}^2 \cdot t_\text{S}^{-1} \cdot m_\text{S} \approx 137.036 ~l_\text{S}^2 \cdot t_\text{S}^{-1} \cdot m_\text{S}^{~},</math>

where {{mvar|α}} is the [[fine-structure constant]].

== History ==

[[George Johnstone Stoney|George Stoney]] was one of the first scientists to understand that electric charge was quantized; from this quantization and three other constants that he perceived as being universal (a speed from electromagnetism, and the coefficients in the electrostatic and gravitational force equations) he derived the units that are now named after him.<ref>{{citation |last=Stoney |first=G. |author-link=George Johnstone Stoney |title=On The Physical Units of Nature |journal=Phil. Mag. |volume=11 |pages=381–391 | year=1881 |issue=69 |doi=10.1080/14786448108627031 }}</ref><ref>{{citation |first=G. Johnstone |last=Stoney |author-link=George Johnstone Stoney |url=https://books.google.com/books?id=R79WAAAAIAAJ&pg=PA51 |title=On The Physical Units of Nature |journal=The Scientific Proceedings of the Royal Dublin Society |volume=3 |pages=51–60 |year=1883 |access-date=2010-11-28 }}</ref>
Stoney's derived estimate of the ''unit of charge'', 10<sup>−20</sup> ampere-second, was {{frac|1|16}} of the modern value of the charge of the electron<ref name = "O'Hara1">{{citation |first1=J. G. |last1=O'Hara |year=1993 |title=George Johnstone Stoney and the Conceptual Discovery of the Electron |journal=Occasional Papers in Science and Technology |volume=8 |pages=5–28 }}</ref> due to Stoney using the approximated value of 10<sup>18</sup> 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]] {{physconst|NA|round=5|ref=no}} and for the volume of a [[Mole (unit)|gram-molecule]] under these conditions of {{val|22.4146|e=6|u=mm3}}, the modern value is {{val|2.687|e=16}}, instead of Stoney's 10<sup>18</sup>.

== Stoney units and Planck units ==

Stoney's set of base units is similar to the one used in [[Planck units]], proposed independently by Planck thirty years later, in which Planck normalized the [[Planck constant]]{{efn|In modern usage, Planck units are understood to normalize the [[reduced Planck constant]] in place of the Planck constant.}} in place of the elementary charge.<ref name=Penz>{{citation |url=https://books.google.com/books?id=nzbuV_WWS5EC&pg=PA191 |title=Space: in science, art and society |page=191 |editor-first1=François |editor-last1=Penz |editor-first2=Gregory |editor-last2=Radick |editor-first3=Robert |editor-last3=Howell |chapter=Outer Space |first=John D. |last=Barrow |author-link=John D. Barrow |publisher=Cambridge University Press |year=2004 |isbn=978-0-521-82376-0}}</ref>

Planck units are more commonly used than Stoney units in modern physics, especially for [[quantum gravity]] (including [[string theory]]). Rarely, Planck units are referred to as Planck–Stoney units.<ref name=Penz/>

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 [[Hermann Weyl]], who associated a gravitational unit of charge with the Stoney length<ref name = Tomilin1>{{citation 
|first1=K. |last1=Tomilin |title=Natural System of Units |journal=Proc. of the XX11 International Workshop on High Energy Physics and Field Theory |year=2000 |page=289 }}</ref><ref name = Weyl1>{{citation |first=H. |last=Weyl |author-link=Hermann Weyl |title=Gravitation und Elekrizitaet |journal=Koniglich Preussische Akademie der Wissenschaften |year=1918 |pages=465–78}}</ref><ref name = Weyl2>{{citation
|author-link=Hermann Weyl |first=H. |last=Weyl |title=Eine Neue Erweiterung der Relativitaetstheorie |journal=Annalen der Physik |volume=59 |year=1919 |issue=10 |pages=101–103|doi=10.1002/andp.19193641002 |bibcode=1919AnP...364..101W }}</ref>
and who appears to have inspired Dirac's fascination with the [[Dirac large numbers hypothesis|large numbers hypothesis]].<ref name = Gorelik1>
{{citation|first=G. |last=Gorelik |author-link=Gennady Gorelik |chapter=Hermann Weyl and Large Numbers in Relativistic Cosmology |title=Einstein Studies in Russia |editor-first1=Y. |editor-last1=Balashov |editor-first2=V. |editor-last2=Vizgin |publisher=Birkhaeuser |year=2002}}</ref>
Since then, the Stoney scale has been largely neglected in the development of modern physics, although it is still occasionally discussed.<ref>{{citation |first=Jean-Philippe |last=Uzan |author-link=Jean-Philippe Uzan |year=2011 |title=Varying Constants, Gravitation and Cosmology |pages=15–16 |journal=Living Rev. Relativ. |volume=14 |issue=1 |doi=10.12942/lrr-2011-2 |doi-access=free |pmid=28179829 |pmc=5256069 |arxiv=1009.5514 |bibcode=2011LRR....14....2U }}</ref>

The ratio of Stoney units to Planck units of length, time and mass is <math>\sqrt{\alpha}</math>, where <math>\alpha</math> is the [[fine-structure constant]]:<ref>{{Citation |last1=Duff |first1=M. J. |author-link1=Michael Duff (physicist) |last2=Okun |first2=L. B. |author-link2=Lev Okun |last3=Veneziano |first3=G. |author-link3=Gabriele Veneziano |date=2002-03-09 |title=Trialogue on the number of fundamental constants |journal=Journal of High Energy Physics |volume=2002 |issue=3 |page=3 |doi=10.1088/1126-6708/2002/03/023 |arxiv=physics/0110060 |bibcode=2002JHEP...03..023D |s2cid=15806354 |issn=1029-8479 }}</ref> <math>l_\text{S} = \sqrt{\alpha}\,l_\text{P};</math> <math>m_\text{S} = \sqrt{\alpha}\,m_\text{P};</math> <math>t_\text{S} = \sqrt{\alpha}\,t_\text{P};</math> <math>q_\text{S} = \sqrt{\alpha}\,q_\text{P}.</math>

== See also ==
{{Portal|Physics}}

* [[Dimensional analysis]]
* [[Natural units]]
* [[Physical constant]]s

== Notes ==
{{notelist}}

== References ==
{{reflist}}

== External links ==

{{Systems of measurement}}
{{Authority control}}

{{DEFAULTSORT:Stoney Scale Units}}
[[Category:Natural units|*]]