Editing Spacetime (section)

== History ==

{{anchor|History}}
{{Main|History of special relativity|History of Lorentz transformations}}

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 | caption2          = Figure 1-2. Michelson and Morley expected that motion through the aether would cause a differential phase shift between light traversing the two arms of their apparatus. The most logical explanation of their negative result, aether dragging, was in conflict with the observation of stellar aberration.
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By the mid-1800s, various experiments such as the observation of the [[Arago spot]] and [[Foucault's measurements of the speed of light|differential measurements of the speed of light in air versus water]] were considered to have proven the wave nature of light as opposed to a [[Corpuscular theory of light|corpuscular theory]].<ref>{{cite book|last1=Hughes|first1=Stefan|title=Catchers of the Light: Catching Space: Origins, Lunar, Solar, Solar System and Deep Space|date=2013|publisher=ArtDeCiel Publishing|location=Paphos, Cyprus|isbn=978-1-4675-7992-6|pages=202–233|url=https://books.google.com/books?id=iZk5OOf7fVYC|access-date=7 April 2017|archive-date=17 January 2023|archive-url=https://web.archive.org/web/20230117023500/https://books.google.com/books?id=iZk5OOf7fVYC|url-status=live}}</ref> Propagation of waves was then assumed to require the existence of a ''waving'' medium; in the case of light waves, this was considered to be a hypothetical [[luminiferous aether]].<ref group=note>''luminiferous'' from the Latin ''lumen'', light, + ''ferens'', carrying; ''aether'' from the Greek αἰθήρ (''aithēr''), pure air, clear sky</ref> The various attempts to establish the properties of this hypothetical medium yielded contradictory results. For example, the [[Fizeau experiment]] of 1851, conducted by French physicist [[Hippolyte Fizeau]], demonstrated that the speed of light in flowing water was less than the sum of the speed of light in air plus the speed of the water by an amount dependent on the water's index of refraction.<ref>{{Cite web |last=Williams |first=Matt |date=2022-01-28 |title=What is Einstein's Theory of Relativity? |url=https://www.universetoday.com/45484/einsteins-theory-of-relativity-1/ |access-date=2022-08-13 |website=Universe Today |language=en-US |archive-date=3 August 2022 |archive-url=https://web.archive.org/web/20220803084231/https://www.universetoday.com/45484/einsteins-theory-of-relativity-1/ |url-status=live }}</ref>

Among other issues, the dependence of the partial [[aether-dragging]] implied by this experiment on the index of refraction (which is dependent on wavelength) led to the unpalatable conclusion that aether ''simultaneously'' flows at different speeds for different colors of light.<ref name="Stachel">{{cite book |last=Stachel |first=John |title=The Universe of General Relativity |date=2005 |publisher=Birkhäuser |isbn=0-8176-4380-X |editor1-last=Kox |editor1-first=A. J. |location=Boston, Massachusetts |pages=1–13 |language=en-us |chapter=Fresnel's (Dragging) Coefficient as a Challenge to 19th Century Optics of Moving Bodies. |editor2-last=Eisenstaedt |editor2-first=Jean |chapter-url=http://www.bu.edu/cphs/files/2015/04/2005_Fresnel.pdf |archive-url=https://web.archive.org/web/20170415200532/http://www.bu.edu/cphs/files/2015/04/2005_Fresnel.pdf |archive-date=15 April 2017 |url-status=dead}}</ref> The [[Michelson–Morley experiment]] of 1887 (Fig.&nbsp;1-2) showed no differential influence of Earth's motions through the hypothetical aether on the speed of light, and the most likely explanation, complete aether dragging, was in conflict with the observation of [[stellar aberration]].<ref name="French" />

[[George Francis FitzGerald]] in 1889,<ref>{{Cite web |title=George Francis FitzGerald |url=https://www.lindahall.org/about/news/scientist-of-the-day/george-francis-fitzgerald |access-date=2022-08-13 |website=The Linda Hall Library |language=en-US |archive-date=17 January 2023 |archive-url=https://web.archive.org/web/20230117023441/https://www.lindahall.org/about/news/scientist-of-the-day/george-francis-fitzgerald |url-status=live }}</ref> and [[Hendrik Lorentz]] in 1892, independently proposed that material bodies traveling through the fixed aether were physically affected by their passage, contracting in the direction of motion by an amount that was exactly what was necessary to explain the negative results of the Michelson–Morley experiment. No length changes occur in directions transverse to the direction of motion.

By 1904, Lorentz had expanded his theory such that he had arrived at equations formally identical with those that Einstein was to derive later, i.e. the [[Lorentz transformation]].<ref>{{Cite web |title=The Nobel Prize in Physics 1902 |url=https://www.nobelprize.org/prizes/physics/1902/lorentz/biographical/ |access-date=2022-08-13 |website=NobelPrize.org |language=en-US |archive-date=23 June 2017 |archive-url=https://web.archive.org/web/20170623231447/http://www.nobelprize.org/nobel_prizes/physics/laureates/1902/lorentz-bio.html |url-status=live }}</ref> As a theory of [[Dynamics (mechanics)|dynamics]] (the study of forces and torques and their effect on motion), his theory assumed actual physical deformations of the physical constituents of matter.<ref name="Pais">{{cite book |last1=Pais |first1=Abraham |url=https://archive.org/details/subtleislordscie00pais |title='Subtle is the Lord–': The Science and the Life of Albert Einstein |date=1982 |publisher=Oxford University Press |isbn=0-19-853907-X |edition=11th |location=Oxford |language=en}}</ref>{{rp|163–174}} Lorentz's equations predicted a quantity that he called ''local time'', with which he could explain the [[aberration of light]], the Fizeau experiment and other phenomena.

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[[Henri Poincaré]] was the first to combine space and time into spacetime.<ref>{{Citation |author=Darrigol, O. |title=The Genesis of the theory of relativity |year=2005 |journal=Séminaire Poincaré |volume=1 |pages=1–22 |url=http://www.bourbaphy.fr/darrigol2.pdf |doi=10.1007/3-7643-7436-5_1 |isbn=978-3-7643-7435-8 |bibcode=2006eins.book....1D |access-date=17 July 2017 |archive-date=28 February 2008 |archive-url=https://web.archive.org/web/20080228124558/http://www.bourbaphy.fr/darrigol2.pdf |url-status=live }}</ref><ref name="Miller">{{cite book|last1=Miller|first1=Arthur I.|title=Albert Einstein's Special Theory of Relativity|date=1998|publisher=Springer-Verlag|location=New York|isbn=0-387-94870-8}}</ref>{{rp|73–80,93–95}} He argued in 1898 that the simultaneity of two events is a matter of convention.<ref name=Galison2003>{{cite book |last1=Galison |first1=Peter |title=Einstein's Clocks, Poincaré's Maps: Empires of Time |date=2003 |publisher=W. W. Norton & Company, Inc. |location=New York |isbn=0-393-02001-0 |pages=[https://archive.org/details/einsteinsclocksp00gali/page/13 13–47] |url=https://archive.org/details/einsteinsclocksp00gali/page/13 }}</ref>{{refn|group=note|By stating that simultaneity is a matter of convention, Poincaré meant that to talk about time at all, one must have synchronized clocks, and the synchronization of clocks must be established by a specified, operational procedure (convention). This stance represented a fundamental philosophical break from Newton, who conceived of an absolute, true time that was independent of the workings of the inaccurate clocks of his day. This stance also represented a direct attack against the influential philosopher [[Henri Bergson]], who argued that time, simultaneity, and duration were matters of intuitive understanding.<ref name=Galison2003 />}} In 1900, he recognized that Lorentz's "local time" is actually what is indicated by moving clocks by applying an explicitly ''operational definition'' of clock synchronization assuming constant light speed.{{refn|group=note|The operational procedure adopted by Poincaré was essentially identical to what is known as [[Einstein synchronization]], even though a variant of it was already a widely used procedure by telegraphers in the middle 19th century. Basically, to synchronize two clocks, one flashes a light signal from one to the other, and adjusts for the time that the flash takes to arrive.<ref name=Galison2003 />}} In 1900 and 1904, he suggested the inherent undetectability of the aether by emphasizing the validity of what he called the [[principle of relativity]]. In 1905/1906<ref>{{cite journal |last1=Poincare |first1=Henri |title=On the Dynamics of the Electron (Sur la dynamique de l'électron) |journal=Rendiconti del Circolo Matematico di Palermo |date=1906 |volume=21 |pages=129–176 |url=https://en.wikisource.org/wiki/Translation:On_the_Dynamics_of_the_Electron_(July)#.C2.A7_9._.E2.80.94_Hypotheses_on_gravitation |access-date=15 July 2017 |doi=10.1007/bf03013466 |bibcode=1906RCMP...21..129P |hdl=2027/uiug.30112063899089 |s2cid=120211823 |hdl-access=free |archive-date=11 July 2017 |archive-url=https://web.archive.org/web/20170711124425/https://en.wikisource.org/wiki/Translation:On_the_Dynamics_of_the_Electron_(July)#.C2.A7_9._.E2.80.94_Hypotheses_on_gravitation |url-status=live }}</ref> he mathematically perfected Lorentz's theory of electrons in order to bring it into accordance with the postulate of relativity.

While discussing various hypotheses on Lorentz invariant gravitation, he introduced the innovative concept of a 4-dimensional spacetime by defining various [[four vector]]s, namely [[four-position]], [[four-velocity]], and [[four-force]].<ref>{{Citation |author=Zahar |first=Elie |title=Einstein's Revolution: A Study in Heuristic |year=1989 |chapter=Poincaré's Independent Discovery of the relativity principle |location=Chicago, Illinois |publisher=Open Court Publishing Company |isbn=0-8126-9067-2 |orig-year=1983}}</ref><ref name=Walter /> He did not pursue the 4-dimensional formalism in subsequent papers, however, stating that this line of research seemed to "entail great pain for limited profit", ultimately concluding "that three-dimensional language seems the best suited to the description of our world".<ref name="Walter">{{cite book |author1=Walter |first=Scott A. |title=The Genesis of General Relativity, Volume 3 |date=2007 |publisher=Springer |editor1-last=Renn |editor1-first=Jürgen |location=Berlin, Germany |pages=193–252 |chapter=Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910 |access-date=15 July 2017 |editor2-last=Schemmel |editor2-first=Matthias |chapter-url=http://scottwalter.free.fr/papers/2007-genesis-walter.html |archive-url=https://archive.today/20240528051526/https://www.webcitation.org/6rxvbrr7g?url=http://scottwalter.free.fr/papers/2007-genesis-walter.html |archive-date=28 May 2024 |url-status=dead}}</ref> Even as late as 1909, Poincaré continued to describe the dynamical interpretation of the Lorentz transform.<ref name="Pais" />{{rp|163–174}}

In 1905, [[Albert Einstein]] analyzed special relativity in terms of [[kinematics]] (the study of moving bodies without reference to forces) rather than dynamics. His results were mathematically equivalent to those of Lorentz and Poincaré. He obtained them by recognizing that the entire theory can be built upon two postulates: the principle of relativity and the principle of the constancy of light speed. His work was filled with vivid imagery involving the exchange of light signals between clocks in motion, careful measurements of the lengths of moving rods, and other such examples.<ref name="Einstein1905">{{cite journal|last1=Einstein|first1=Albert|title=On the Electrodynamics of Moving Bodies ( Zur Elektrodynamik bewegter Körper)|journal=Annalen der Physik|date=1905|volume=322|issue=10|pages=891–921|url=https://en.wikisource.org/wiki/On_the_Electrodynamics_of_Moving_Bodies_(1920_edition)|access-date=7 April 2018|bibcode=1905AnP...322..891E|doi=10.1002/andp.19053221004|doi-access=free|archive-date=6 November 2018|archive-url=https://web.archive.org/web/20181106132340/https://en.wikisource.org/wiki/On_the_Electrodynamics_of_Moving_Bodies_(1920_edition)|url-status=live}}</ref>{{refn|group=note|A hallmark of Einstein's career, in fact, was his use of visualized [[thought experiment]]s (Gedanken–Experimente) as a fundamental tool for understanding physical issues. For special relativity, he employed moving trains and flashes of lightning for his most penetrating insights. For curved spacetime, he considered a painter falling off a roof, accelerating elevators, blind beetles crawling on curved surfaces and the like. In his great [[Bohr–Einstein debates|Solvay Debates]] with [[Niels Bohr|Bohr]] on the nature of reality (1927 and 1930), he devised multiple imaginary contraptions intended to show, at least in concept, means whereby the [[Heisenberg uncertainty principle]] might be evaded. Finally, in a profound contribution to the literature on quantum mechanics, Einstein considered two particles briefly interacting and then flying apart so that their states are correlated, anticipating the phenomenon known as [[quantum entanglement]].<ref name="Isaacson2007">{{cite book |last1=Isaacson |first1=Walter |title=Einstein: His Life and Universe |url=https://archive.org/details/einsteinhislifeu0000isaa |url-access=registration |date=2007 |publisher=Simon & Schuster |isbn=978-0-7432-6473-0 |pages=26–27;122–127;145–146;345–349;448–460}}</ref>}}

Einstein in 1905 superseded previous attempts of an [[electromagnetic mass]]–energy relation by introducing the general [[equivalence of mass and energy]], which was instrumental for his subsequent formulation of the [[equivalence principle]] in 1907, which declares the equivalence of inertial and gravitational mass. By using the mass–energy equivalence, Einstein showed that the gravitational mass of a body is proportional to its energy content, which was one of the early results in developing [[general relativity]]. While it would appear that he did not at first think geometrically about spacetime,<ref name="Schutz">{{cite book |last1=Schutz |first1=Bernard |title=Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity |date=2004 |publisher=[[Cambridge University Press]] |location=Cambridge |isbn=0-521-45506-5 |edition=Reprint |url=https://books.google.com/books?id=P_T0xxhDcsIC |access-date=24 May 2017 |language=en |archive-date=17 January 2023 |archive-url=https://web.archive.org/web/20230117023501/https://books.google.com/books?id=P_T0xxhDcsIC |url-status=live }}</ref>{{rp|219}} in the further development of general relativity, Einstein fully incorporated the spacetime formalism.

When Einstein published in 1905, another of his competitors, his former mathematics professor [[Hermann Minkowski]], had also arrived at most of the basic elements of special relativity. [[Max Born]] recounted a meeting he had made with Minkowski, seeking to be Minkowski's student/collaborator:<ref name="Weinstein">{{cite arXiv|last1=Weinstein |first1=Galina |title=Max Born, Albert Einstein and Hermann Minkowski's Space–Time Formalism of Special Relativity |eprint=1210.6929 |class=physics.hist-ph |year=2012 }}</ref>

{{cquote|I went to Cologne, met Minkowski and heard his celebrated lecture 'Space and Time' delivered on 2 September 1908. [...] He told me later that it came to him as a great shock when Einstein published his paper in which the equivalence of the different local times of observers moving relative to each other was pronounced; for he had reached the same conclusions independently but did not publish them because he wished first to work out the mathematical structure in all its splendor. He never made a priority claim and always gave Einstein his full share in the great discovery.}}

Minkowski had been concerned with the state of electrodynamics after Michelson's disruptive experiments at least since the summer of 1905, when Minkowski and [[David Hilbert]] led an advanced seminar attended by notable physicists of the time to study the papers of Lorentz, Poincaré et al. Minkowski saw Einstein's work as an extension of Lorentz's, and was most directly influenced by Poincaré.<ref name="Galison">{{cite journal|last1=Galison |first1=Peter Louis|title=Minkowski's space–time: From visual thinking to the absolute world |journal=Historical Studies in the Physical Sciences |date=1979 |volume=10 |pages=85–121 |doi=10.2307/27757388 |jstor=27757388 }}</ref>

[[File:Minkowski Diagram from 1908 'Raum und Zeit' lecture.jpg|thumb|upright=1.5|Figure 1–4. Hand-colored transparency presented by Minkowski in his 1908 ''Raum und Zeit'' lecture]]
On 5 November 1907 (a little more than a year before his death), Minkowski introduced his geometric interpretation of spacetime in a lecture to the Göttingen Mathematical society with the title, ''The Relativity Principle'' (''Das Relativitätsprinzip'').{{refn|group=note|In the original version of this lecture, Minkowski continued to use such obsolescent terms as the ether, but the posthumous publication in 1915 of this lecture in the ''Annals of Physics'' (''Annalen der Physik'') was edited by Sommerfeld to remove this term. Sommerfeld also edited the published form of this lecture to revise Minkowski's judgement of Einstein from being a mere clarifier of the principle of relativity, to being its chief expositor.<ref name=Weinstein />}} On 21 September 1908, Minkowski presented his talk, ''Space and Time'' (''Raum und Zeit''),<ref name="Minkowski_Raum_und_Zeit">{{cite journal |last1=Minkowski |first1=Hermann |date=1909 |title=Raum und Zeit |trans-title=Space and Time |url=https://en.wikisource.org/wiki/Translation:Space_and_Time |url-status=live |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |publisher=B. G. Teubner |pages=1–14 |archive-url=https://web.archive.org/web/20170728154753/https://en.wikisource.org/wiki/Translation:Space_and_Time |archive-date=28 July 2017 |access-date=17 July 2017}}</ref> to the German Society of Scientists and Physicians. The opening words of ''Space and Time'' include Minkowski's statement that "Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence." ''Space and Time'' included the first public presentation of spacetime diagrams (Fig.&nbsp;1-4), and included a remarkable demonstration that the concept of the ''invariant interval'' ([[#Spacetime in special relativity|discussed below]]), along with the empirical observation that the speed of light is finite, allows derivation of the entirety of special relativity.{{refn|group=note|''(In the following, the group'' '''''G''<sub>∞</sub>''' ''is the Galilean group and the group'' '''''G''<sub>c</sub>''' ''the Lorentz group.)'' "With respect to this it is clear that the group '''''G''<sub>c</sub>''' in the limit for {{nowrap|1='''''c'' = ∞'''}}, i.e. as group '''''G''<sub>∞</sub>''', exactly becomes the full group belonging to Newtonian Mechanics. In this state of affairs, and since '''''G''<sub>c</sub>''' is mathematically more intelligible than '''''G''<sub>∞</sub>''', a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena actually possess an invariance, not for the group '''''G''<sub>∞</sub>''', but rather for a group '''''G''<sub>c</sub>''', where '''''c''''' is definitely finite, and only exceedingly large using the ordinary measuring units."<ref name="Minkowski_Raum_und_Zeit" />}}

The spacetime concept and the Lorentz group are closely connected to certain types of [[Lie sphere geometry|sphere]], [[hyperbolic geometry|hyperbolic]], or [[conformal geometry|conformal geometries]] and their transformation groups already developed in the 19th century, in which [[History of Lorentz transformations|invariant intervals analogous to the spacetime interval]] are used.{{refn|group=note|For instance, the Lorentz group is a subgroup of the [[spherical wave transformation|conformal group in four dimensions]].<ref name=cartan>{{Cite journal|author1=Cartan, É.|author2=Fano, G.|year=1955|orig-year=1915|journal=Encyclopédie des Sciences Mathématiques Pures et Appliquées|volume=3|issue=1|title=La théorie des groupes continus et la géométrie|pages=39–43|url=http://gallica.bnf.fr/ark:/12148/bpt6k29100t/f194.image|access-date=6 April 2018|archive-date=23 March 2018|archive-url=https://web.archive.org/web/20180323032943/http://gallica.bnf.fr/ark:/12148/bpt6k29100t/f194.image|url-status=live}} (Only pages 1–21 were published in 1915, the entire article including pp.&nbsp;39–43 concerning the groups of Laguerre and Lorentz was posthumously published in 1955 in Cartan's collected papers, and was reprinted in the Encyclopédie in 1991.)</ref>{{rp|41–42}}
The Lorentz group is isomorphic to the [[Spherical wave transformation#Laguerre group isomorphic to Lorentz group|Laguerre group]] transforming planes into planes,<ref name=cartan />{{rp|39–42}}
it is isomorphic to the [[Möbius group]] of the plane,<ref>{{Cite journal|author=Kastrup, H. A.|title=On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics|journal=Annalen der Physik|volume=520|issue=9–10|year=2008|pages=631–690|arxiv=0808.2730|doi=10.1002/andp.200810324|bibcode = 2008AnP...520..631K |s2cid=12020510}}</ref>{{rp|22}}
and is isomorphic to the group of isometries in [[hyperbolic space]] which is often expressed in terms of the [[hyperboloid model]].<ref>{{Cite book|author=Ratcliffe, J. G.|year=1994|title=Foundations of Hyperbolic Manifolds|chapter=Hyperbolic geometry|pages=[https://archive.org/details/foundationsofhyp0000ratc/page/56 56–104]|location=New York|isbn=0-387-94348-X|chapter-url=https://archive.org/details/foundationsofhyp0000ratc/page/56}}</ref>{{rp|3.2.3}} }}

Einstein, for his part, was initially dismissive of Minkowski's geometric interpretation of special relativity, regarding it as ''überflüssige Gelehrsamkeit'' (superfluous learnedness). However, in order to complete his search for general relativity that started in 1907, the geometric interpretation of relativity proved to be vital. In 1916, Einstein fully acknowledged his indebtedness to Minkowski, whose interpretation greatly facilitated the transition to general relativity.<ref name="Pais" />{{rp|151–152}} Since there are other types of spacetime, such as the curved spacetime of general relativity, the spacetime of special relativity is today known as ''Minkowski spacetime.''