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	{{short description\|Mathematical model combining space and time}}		{{short description\|Mathematical model combining space and time}}
	{{Use dmy dates\|date=April 2019}}		{{Use dmy dates\|date=April 2019}}
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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.		[[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-date=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 }}</ref> Even as late as 1909, Poincaré continued to describe the dynamical interpretation of the Lorentz transform.<ref name="Pais" />{{rp\|163–174}}		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-date=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 }}</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>}}		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>}}
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	: <math>\tan(\theta) = v/c.</math>		: <math>\tan(\theta) = v/c.</math>

	The ''x''′ axis is also tilted with respect to the ''x'' axis. To determine the angle of this tilt, we recall that the slope of the world line of a light pulse is always ±1. Fig. 2-3c presents a spacetime diagram from the viewpoint of observer O′. Event P represents the emission of a light pulse at {{math\|1=''x''′ = 0,}} {{math\|1=''ct''′ = −''a''.}} The pulse is reflected from a mirror situated a distance ''a'' from the light source (event Q), and returns to the light source at {{math\|1=''x''′ = 0, ''ct''′ = ''a''}} (event R).		The ''x''′ axis is also tilted with respect to the ''x'' axis. To determine the angle of this tilt, we recall that the slope of the world line of a light pulse is always ±1. Fig. 2-3c presents a spacetime diagram from the viewpoint of observer O′. Event P represents the emission of a light pulse at {{math\|1=''x''′ = 0,}} {{math\|1=''ct''′ = −''a''.}} The pulse is reflected from a mirror situated at distance ''a'' from the light source (event Q), and returns to the light source at {{math\|1=''x''′ = 0, ''ct''′ = ''a''}} (event R).

	The same events P, Q, R are plotted in Fig. 2-3b in the frame of observer O. The light paths have {{nowrap\|1=slopes = 1}} and −1, so that △PQR forms a right triangle with PQ and QR both at 45 degrees to the ''x'' and ''ct'' axes. Since {{nowrap\|1=OP = OQ = OR,}} the angle between {{mvar\|x′}} and {{mvar\|x}} must also be ''θ''.<ref name="Collier" />{{rp\|113–118}}		The same events P, Q, R are plotted in Fig. 2-3b in the frame of observer O. The light paths have {{nowrap\|1=slopes = 1}} and −1, so that △PQR forms a right triangle with PQ and QR both at 45 degrees to the ''x'' and ''ct'' axes. Since {{nowrap\|1=OP = OQ = OR,}} the angle between {{mvar\|x′}} and {{mvar\|x}} must also be ''θ''.<ref name="Collier" />{{rp\|113–118}}
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	=== Asymptotic symmetries ===		=== Asymptotic symmetries ===

	{{Main\|Bondi–Metzner–Sachs group}}		{{Main\|Bondi–Metzner–Sachs group}}

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			{{pp\|small=yes}}
	{{short description\|Mathematical model combining space and time}}		{{short description\|Mathematical model combining space and time}}
	{{Use dmy dates\|date=April 2019}}		{{Use dmy dates\|date=April 2019}}
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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.		[[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-date=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 }}</ref> Even as late as 1909, Poincaré continued to describe the dynamical interpretation of the Lorentz transform.<ref name="Pais" />{{rp\|163–174}}		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-date=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 }}</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>}}		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>}}
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	: <math>\tan(\theta) = v/c.</math>		: <math>\tan(\theta) = v/c.</math>

	The ''x''′ axis is also tilted with respect to the ''x'' axis. To determine the angle of this tilt, we recall that the slope of the world line of a light pulse is always ±1. Fig. 2-3c presents a spacetime diagram from the viewpoint of observer O′. Event P represents the emission of a light pulse at {{math\|1=''x''′ = 0,}} {{math\|1=''ct''′ = −''a''.}} The pulse is reflected from a mirror situated a distance ''a'' from the light source (event Q), and returns to the light source at {{math\|1=''x''′ = 0, ''ct''′ = ''a''}} (event R).		The ''x''′ axis is also tilted with respect to the ''x'' axis. To determine the angle of this tilt, we recall that the slope of the world line of a light pulse is always ±1. Fig. 2-3c presents a spacetime diagram from the viewpoint of observer O′. Event P represents the emission of a light pulse at {{math\|1=''x''′ = 0,}} {{math\|1=''ct''′ = −''a''.}} The pulse is reflected from a mirror situated at distance ''a'' from the light source (event Q), and returns to the light source at {{math\|1=''x''′ = 0, ''ct''′ = ''a''}} (event R).

	The same events P, Q, R are plotted in Fig. 2-3b in the frame of observer O. The light paths have {{nowrap\|1=slopes = 1}} and −1, so that △PQR forms a right triangle with PQ and QR both at 45 degrees to the ''x'' and ''ct'' axes. Since {{nowrap\|1=OP = OQ = OR,}} the angle between {{mvar\|x′}} and {{mvar\|x}} must also be ''θ''.<ref name="Collier" />{{rp\|113–118}}		The same events P, Q, R are plotted in Fig. 2-3b in the frame of observer O. The light paths have {{nowrap\|1=slopes = 1}} and −1, so that △PQR forms a right triangle with PQ and QR both at 45 degrees to the ''x'' and ''ct'' axes. Since {{nowrap\|1=OP = OQ = OR,}} the angle between {{mvar\|x′}} and {{mvar\|x}} must also be ''θ''.<ref name="Collier" />{{rp\|113–118}}
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	=== Asymptotic symmetries ===		=== Asymptotic symmetries ===

	{{Main\|Bondi–Metzner–Sachs group}}		{{Main\|Bondi–Metzner–Sachs group}}

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