Metre

=== Metrology and paradigm shift in physics ===
[[File:Komplet invarskih žica.png|thumb|Invar wire baseline apparatus]]

The comparison of the new prototypes of the metre with each other involved the development of special measuring equipment and the definition of a reproducible temperature scale. The BIPM's [[Temperature measurement|thermometry]] work led to the discovery of special alloys of iron–nickel, in particular [[invar]], whose practically negligible coefficient of expansion made it possible to develop simpler baseline measurement methods, and for which its director, the Swiss physicist [[Charles Édouard Guillaume|Charles-Edouard Guillaume]], was granted the [[Nobel Prize in Physics]] in 1920. Guillaume's Nobel Prize marked the end of an era in which [[metrology]] was leaving the field of [[geodesy]] to become a [[Technology|technological]] application of [[physics]].<ref>{{Cite web |title=BIPM – la définition du mètre |url=https://www.bipm.org/fr/measurement-units/history-si/evolution_metre.html |access-date=2019-05-15 |website=www.bipm.org}}</ref><ref>{{Cite journal |date=1934-12-01 |title=Dr. C. E. Guillaume |journal=Nature |language=en |volume=134 |issue=3397 |pages=874 |doi=10.1038/134874b0 |bibcode=1934Natur.134R.874. |s2cid=4140694 |issn=1476-4687|doi-access=free }}</ref><ref>{{Cite journal |last=Guillaume |first=C.-H.-Ed |date=1906-01-01 |title=La mesure rapide des bases géodésiques |journal=Journal de Physique Théorique et Appliquée |volume=5 |pages=242–263 |url=https://zenodo.org/record/2007289 |doi=10.1051/jphystap:019060050024200}}</ref>

In 1921, the Nobel Prize in Physics was awarded to another Swiss scientist, [[Albert Einstein]], who following [[Michelson–Morley experiment]] had questioned the [[luminiferous aether]] in 1905, just as [[Isaac Newton|Newton]] had questioned [[Mechanical explanations of gravitation|Descartes' Vortex theory]] in 1687 after [[Jean Richer]]'s pendulum experiment in [[Cayenne]], [[French Guiana]].<ref>{{Cite web |last=Huet |first=Sylvestre |title=Einstein, le révolutionnaire de la lumière |url=https://www.liberation.fr/week-end/2005/02/12/einstein-le-revolutionnaire-de-la-lumiere_509445/ |access-date=2023-10-07 |website=Libération |language=fr}}</ref><ref>{{Cite book |last=Ferreiro |first=Larrie D. |url=https://play.google.com/store/books/details?id=p-Y3DgAAQBAJ |title=Measure of the Earth: The Enlightenment Expedition That Reshaped Our World |date=2011-05-31 |publisher=Basic Books |isbn=978-0-465-02345-5 |pages=19–23 |language=en}}</ref><ref name=":3">{{Cite web |title=Lettres philosophiques/Lettre 15 - Wikisource |url=https://fr.wikisource.org/wiki/Lettres_philosophiques/Lettre_15 |access-date=2023-10-07 |website=fr.wikisource.org |language=fr}}</ref><ref name="Earth-1911" />

Furthermore, [[special relativity]] changed conceptions of [[time]] and [[mass]], while [[general relativity]] changed that of [[space]]. According to Newton, space was [[Euclidean geometry|Euclidean]], infinite and without boundaries and bodies gravitated around each other without changing the structure of space. [[Introduction to general relativity|Einstein's theory of gravity]] states, on the contrary, that the mass of a body has an effect on all other bodies while modifying the structure of space. A massive body induces a curvature of the space around it in which the path of light is inflected, as was demonstrated by the displacement of the position of a star observed near the Sun during an eclipse in 1919.<ref>Stephen Hawking, Paris, Dunod, 2003, 2014, 929 <abbr>p.</abbr>, p. 816–817</ref>

==== Wavelength definition ====
In 1873, [[James Clerk Maxwell]] suggested that light emitted by an element be used as the standard both for the unit of length and for the second. These two quantities could then be used to define the unit of mass.<ref>{{cite book |url=https://archive.org/details/electricandmagne01maxwrich |title=A Treatise On Electricity and Magnetism |first=James Clerk |last=Maxwell |author-link=James Clerk Maxwell |publisher=MacMillan and Co. |location=London |volume=1 |year=1873 |page=3}}</ref> About the unit of length he wrote:

{{blockquote|text=In the present state of science the most universal standard of length which we could assume would be the wave length in vacuum of a particular kind of light, emitted by some widely diffused substance such as sodium, which has well-defined lines in its spectrum. Such a standard would be independent of any changes in the dimensions of the earth, and should be adopted by those who expect their writings to be more permanent than that body.|author=James Clerk Maxwell|title=''[[A Treatise on Electricity and Magnetism]]''|source=3rd edition, Vol. 1, p. 3}}

[[Charles Sanders Peirce]]’s work promoted the advent of American science at the forefront of global metrology. Alongside his intercomparisons of artifacts of the metre and contributions to gravimetry through improvement of reversible pendulum, Peirce was the first to tie experimentally the metre to the wave length of a spectral line. According to him the standard length might be compared with that of a wave of light identified by a line in the [[Sunlight|solar spectrum]]. Albert Michelson soon took up the idea and improved it.<ref name=":4">{{Cite journal |last=Crease |first=Robert P. |date=2009-12-01 |title=Charles Sanders Peirce and the first absolute measurement standard |url=https://doi.org/10.1063/1.3273015 |journal=Physics Today |volume=62 |issue=12 |pages=39–44 |doi=10.1063/1.3273015 |bibcode=2009PhT....62l..39C |s2cid=121338356 |issn=0031-9228}}</ref><ref>{{Cite journal |last=Lenzen |first=Victor F. |date=1965 |title=The Contributions of Charles S. Peirce to Metrology |url=https://www.jstor.org/stable/985776 |journal=Proceedings of the American Philosophical Society |volume=109 |issue=1 |pages=29–46 |jstor=985776 |issn=0003-049X}}</ref>

In 1893, the standard metre was first measured with an [[interferometer]] by [[Albert Abraham Michelson|Albert A. Michelson]], the inventor of the device and an advocate of using some particular [[wavelength]] of [[light]] as a standard of length. By 1925, [[interferometry]] was in regular use at the BIPM. However, the International Prototype Metre remained the standard until 1960, when the eleventh CGPM defined the metre in the new [[International System of Units]] (SI) as equal to {{val|1650763.73}} [[wavelength]]s of the [[orange (colour)|orange]]-[[red]] [[emission line]] in the [[electromagnetic spectrum]] of the [[krypton-86]] [[atom]] in [[vacuum]].<ref name="Marion">{{cite book |last=Marion |first=Jerry B. |title=Physics For Science and Engineering |year=1982 |publisher=CBS College Publishing |isbn=978-4-8337-0098-6 |page=3}}</ref>

==== Speed of light definition ====
To further reduce uncertainty, the 17th CGPM in 1983 replaced the definition of the metre with its current definition, thus fixing the length of the metre in terms of the [[second]] and the [[speed of light]]:<ref name="Res1" /><ref>{{Cite web|last=BIPM|date=20 May 2019|title=Mise en pratique for the definition of the meter in the SI|url=https://www.bipm.org/documents/20126/41489670/SI-App2-metre.pdf/0e011055-9736-d293-5e56-b8b1b267fd68?version=1.8&t=1637238031486&download=false|website=BIPM}}</ref>
:: The metre is the length of the path travelled by light in vacuum during a time interval of {{gaps|1|/|299|792|458}} of a second.

This definition fixed the speed of light in [[vacuum]] at exactly {{val|299792458}} metres per second<ref name="Res1">{{cite web |url=https://www.bipm.org/en/committees/cg/cgpm/17-1983/resolution-1 |title=17th General Conference on Weights and Measures (1983), Resolution 1. |access-date=2022-12-07}}</ref> (≈{{val|300000|u=km/s}} or ≈1.079 billion km/hour<ref>The exact value is {{val|299792458|u=m/s}} = {{val|1079252848.8|u=km/h}}.</ref>). An intended by-product of the 17th CGPM's definition was that it enabled scientists to compare lasers accurately using frequency, resulting in wavelengths with one-fifth the uncertainty involved in the direct comparison of wavelengths, because interferometer errors were eliminated. To further facilitate reproducibility from lab to lab, the 17th CGPM also made the iodine-stabilised [[helium–neon laser]] "a recommended radiation" for realising the metre.<ref name="recommendations-2" /> For the purpose of delineating the metre, the BIPM currently considers the HeNe laser wavelength, {{nowrap|''λ''{{sub|HeNe}}}}, to be {{val|632.99121258|u=nm}} with an estimated relative standard uncertainty (''U'') of {{val|2.1|e=-11}}.<ref name="recommendations-2" /><ref name="uncertainty">The term "relative standard uncertainty" is explained by NIST on their web site: {{cite web |title=Standard Uncertainty and Relative Standard Uncertainty |work=The NIST Reference on constants, units, and uncertainties: Fundamental physical constants |url=http://physics.nist.gov/cgi-bin/cuu/Info/Constants/definitions.html |publisher=NIST |access-date=2011-12-19}}</ref><ref>[[#NRC2010|National Research Council 2010]].</ref>

This uncertainty is currently one limiting factor in laboratory realisations of the metre, and it is several orders of magnitude poorer than that of the second, based upon the caesium fountain [[atomic clock]] ({{nowrap|1=''U'' = {{val|5|e=-16}}}}).<ref>[[#NIST2011|National Institute of Standards and Technology 2011]].</ref> Consequently, a  realisation of the metre is usually delineated (not defined) today in labs as {{val|1579800.762042|(33)}} wavelengths of helium–neon laser light in vacuum, the error stated being only that of frequency determination.<ref name="recommendations-2">{{cite web |title=Iodine (&lambda; ≈ 633 nm) |publisher=BIPM |url=http://www.bipm.org/utils/common/pdf/mep/M-e-P_I2_633.pdf |work=Mise en Pratique |year=2003 |access-date=2011-12-16}}</ref> This bracket notation expressing the error is explained in the article on [[Standard uncertainty#Measurements|measurement uncertainty]].

Practical realisation of the metre is subject to uncertainties in characterising the medium, to various uncertainties of interferometry, and to uncertainties in measuring the frequency of the source.<ref name="Beers2" /> A commonly used medium is air, and the [[National Institute of Standards and Technology]] (NIST) has set up an online calculator to convert wavelengths in vacuum to wavelengths in air.<ref name="NIST_calculator">The formulas used in the calculator and the documentation behind them are found at {{cite web |url=http://emtoolbox.nist.gov/Wavelength/Documentation.asp |title=Engineering metrology toolbox: Refractive index of air calculator |date=23 September 2010 |publisher=NIST |access-date=2011-12-16}} The choice is offered to use either the [http://emtoolbox.nist.gov/Wavelength/Edlen.asp modified Edlén equation] or the [http://emtoolbox.nist.gov/Wavelength/Ciddor.asp Ciddor equation]. The documentation provides [http://emtoolbox.nist.gov/Wavelength/Documentation.asp#EdlenorCiddor a discussion of how to choose] between the two possibilities.</ref> As described by NIST, in air, the uncertainties in characterising the medium are dominated by errors in measuring temperature and pressure. Errors in the theoretical formulas used are secondary.<ref name="errors">{{cite web |url=http://emtoolbox.nist.gov/Wavelength/Documentation.asp#UncertaintyandRangeofValidity |title=§VI: Uncertainty and range of validity |work=Engineering metrology toolbox: Refractive index of air calculator |date=23 September 2010 |publisher=NIST |access-date=2011-12-16}}</ref>

By implementing a refractive index correction such as this, an approximate realisation of the metre can be implemented in air, for example, using the formulation of the metre as {{val|1579800.762042|(33)}} wavelengths of helium–neon laser light in vacuum, and converting the wavelengths in vacuum to wavelengths in air. Air is only one possible medium to use in a realisation of the metre, and any [[partial vacuum]] can be used, or some inert atmosphere like helium gas, provided the appropriate corrections for refractive index are implemented.<ref name="Dunning">{{cite book |title=Atomic, molecular, and optical physics: electromagnetic radiation, Volume 29, Part 3 |chapter=Physical limits on accuracy and resolution: setting the scale |chapter-url=https://books.google.com/books?id=FV4Y39AGYuYC&pg=PA316 |page=316 |first1=F. B. |last1=Dunning |first2=Randall G. |last2=Hulet |isbn=978-0-12-475977-0 |publisher=Academic Press |year=1997 |quote=The error [introduced by using air] can be reduced tenfold if the chamber is filled with an atmosphere of helium rather than air.}}</ref>

The metre is ''defined'' as the path length travelled by light in a given time, and practical laboratory length measurements in metres are determined by counting the number of wavelengths of laser light of one of the standard types that fit into the length,{{#tag:ref|The BIPM maintains a list of recommended radiations on their web site.<ref name="recommendations-1">{{cite web |title=Recommended values of standard frequencies |url=http://www.bipm.org/en/publications/mep.html |publisher=BIPM |date=9 September 2010 |access-date=2012-01-22}}</ref><ref>[[#NPL2010|National Physical Laboratory 2010]].</ref>}} and converting the selected unit of wavelength to metres. Three major factors limit the accuracy attainable with laser [[Interferometry|interferometers]] for a length measurement:<ref name="Beers2">
A more detailed listing of errors can be found in {{cite web |work=NIST length scale interferometer measurement assurance; NIST document NISTIR 4998 |title=§4 Re-evaluation of measurement errors |first1=John S |last1=Beers |first2=William B |last2=Penzes |url=https://www.nist.gov/calibrations/upload/4998.pdf  |access-date=2011-12-17 |date=December 1992 |pages=9 ''ff'' }}
</ref><ref name="Webster2">[[#Zagar1999|Zagar, 1999, pp. 6–65''ff'']].</ref>
* uncertainty in vacuum wavelength of the source,
* uncertainty in the refractive index of the medium,
* [[least count]] resolution of the interferometer.

Of these, the last is peculiar to the interferometer itself. The conversion of a length in wavelengths to a length in metres is based upon the relation
: <math> \lambda = \frac{c}{n f} ,</math>
which converts the unit of wavelength ''&lambda;'' to metres using ''c'', the speed of light in vacuum in m/s. Here ''n'' is the [[refractive index]] of the medium in which the measurement is made, and ''f'' is the measured frequency of the source. Although conversion from wavelengths to metres introduces an additional error in the overall length due to measurement error in determining the refractive index and the frequency, the measurement of frequency is one of the most accurate measurements available.<ref name="Webster2" />

The CIPM issued a clarification in 2002:
{{Blockquote|text=Its definition, therefore, applies only within a spatial extent sufficiently small that the effects of the non-uniformity of the gravitational field can be ignored (note that, at the surface of the Earth, this effect in the vertical direction is about 1 part in {{val|e=16}} per metre). In this case, the effects to be taken into account are those of special relativity only.}}