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| [[Image:Minkowski lightcone lorentztransform.svg|thumb|right|Minkowski diagram with resting frame (x,t), moving frame (x',t'), light cone, and hyperbolas marking out time and space with respect to the origin.]]
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| The '''Minkowski diagram''', also known as a '''spacetime diagram''', was developed in 1908 by [[Hermann Minkowski]] and provides an illustration of the properties of space and time in the [[special theory of relativity]]. It allows a quantitative understanding of the corresponding phenomena like [[time dilation]] and [[length contraction]] without mathematical equations.
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| The term Minkowski diagram is used in both a generic and particular sense. In general, a Minkowski diagram is a graphic depiction of a portion of [[Minkowski space]], often where space has been curtailed to a single dimension. These two-dimensional diagrams portray [[worldline]]s as curves in a plane that correspond to motion along the spatial axis. The vertical axis is usually temporal, and the [[units of measurement]] are taken such that the [[light cone]] at an event consists of the lines of [[slope]] plus or minus one through that event.<ref>Mermin (1968) Chapter 17</ref>
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| A particular Minkowski diagram illustrates the result of a [[Lorentz transformation]]. The origin corresponds to an event where a change of [[velocity]] takes place. The new worldline forms an angle α with the vertical, with α < π/4. The Lorentz transformation that moves the vertical to α also moves the horizontal by α. The horizontal corresponds to the usual notion of ''simultaneous events'', for a stationary observer at the origin. After the Lorentz transformation the new simultaneous events lie on the α-inclined line. Whatever the magnitude of α, the line ''t = x'' forms the universal<ref>See [[Vladimir Karapetoff]]</ref> [[bisector]].
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| == Basics ==
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| [[Image:Minkowski diagram - photon.svg|thumb|right|256px|Choosing ct instead of t on the time axis the world line of a photon becomes a straight line with a slope of 45°.]]
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| For simplification in Minkowski diagrams, usually only events in a one dimensional world are considered. Unlike common distance-time diagrams, the distance will be displayed on the ''x''-axis ([[abscissa]]) and the time on the ''y''-axis ([[ordinate]]). In this manner the events happening on a horizontal path in reality can be transferred easily to a horizontal line in the diagram. Objects plotted on the diagram can be thought of as moving from bottom to top as time passes. In this way each object, like an observer or a vehicle, follows in the diagram a certain curve which is called its [[world line]].
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| Each point in the diagram represents a certain position in space and time. Such a position is called an event whether or not anything happens at that position.
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| For convenience, the (vertical) time axis represents, not ''t'', but the corresponding quantity ''ct'', where ''c'' =299,792,458 m/s is the speed of light. In this way, one [[second]] on the ordinate corresponds to a distance of 299,792,458 m on the abscissa. Due to ''x''=''ct'' for a [[photon]] passing through the [[Origin (mathematics)|origin]] to the right, its world line is a straight line with a slope of 45°, if the scales on both axes are chosen to be identical.
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| == Path-time diagram in Newtonian physics ==
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| [[Image:Minkowski diagram - Newtonian physics.svg|thumb|right|256px|In Newtonian physics for both observers the event at A is assigned to the same point in time.]]
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| The black axes labelled ''x'' and ''ct'' on the adjoining diagram are the coordinate system of an observer which we will refer to as 'at rest', and who is positioned at ''x''=0. His world line is identical with the time axis. Each parallel line to this axis would correspond also to an object at rest but at another position. The blue line, however, describes an object moving with constant speed ''v'' to the right, such as a moving observer. | |
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| This blue line labelled ''ct''' may be interpreted as the time axis for the second observer. Together with the path axis (labeled ''x'', which is identical for both observers) it represents his coordinate system. Both observers agree on the location of the [[origin (mathematics)|origin]] of their coordinate systems. The axes for the moving observer are not [[perpendicular]] to each other and the scale on his time axis is stretched. To determine the coordinates of a certain event, two lines parallel to the two axes must be constructed passing through the event, and their intersections with the axes read off.
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| Determining position and time of the event A as an example in the diagram leads to the same time for both observers, as expected. Only for the position different values result, because the moving observer has approached the position of the event A since ''t''=0. Generally stated, all events on a line parallel to the path axis (x axis) happen simultaneously for both observers. There is only one universal time ''t''=''t<nowiki>'</nowiki>'' which corresponds with the existence of only one common path axis. On the other hand due to two different time axes the observers usually measure different path coordinates for the same event. This graphical translation from ''x'' and ''t'' to ''x<nowiki>'</nowiki>'' and ''t<nowiki>'</nowiki>'' and vice versa is described mathematically by the so called [[Galilean transformation]].
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| == Minkowski diagram in special relativity ==
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| [[Image:Minkowski diagram - asymmetric.svg|thumb|right|256px|In the theory of relativity each observer assigns the event at A to a different time and location.]]
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| [[Albert Einstein]] (1905) discovered that the description above is not correct,<ref>{{Citation
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| |doi=10.1002/andp.19053221004
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| |author=Einstein, Albert
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| |year=1905a
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| |title=Zur Elektrodynamik bewegter Körper
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| |journal=Annalen der Physik
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| |volume=322
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| |issue=10
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| |pages=891–921|bibcode = 1905AnP...322..891E
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| |url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_891-921.pdf}}. See also: [http://www.fourmilab.ch/etexts/einstein/specrel/ English translation].</ref> with [[Hermann Minkowski]] (1908) providing the graphical representation.<ref name=minko>{{Citation|author=Minkowski, Hermann|year=1909|title=[[s:de:Raum und Zeit (Minkowski)|Raum und Zeit]]|journal=Physikalische Zeitschrift|volume=10|pages=75–88}}
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| :*Various English translations on Wikisource: [[s:Space and Time|Space and Time]]</ref> Space and time have properties which lead to different rules for the translation of coordinates in case of moving observers. In particular, events which are estimated to happen simultaneously from the viewpoint of one observer, happen at different times for the other.
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| In the Minkowski diagram this [[relativity of simultaneity]] corresponds with the introduction of a separate path axis for the moving observer. Following the rule described above each observer interprets all events on a line parallel to his path axis as simultaneous. The sequence of events from the viewpoint of an observer can be illustrated graphically by shifting this line in the diagram from bottom to top.
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| If ''ct'' instead of ''t'' is assigned on the time axes, the angle ''α'' between both path axes will be identical with that between both time axes. This follows from the second postulate of the special relativity, saying that the speed of light is the same for all observers, regardless of their relative motion (see below). ''α'' is given by
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| :<math>\tan(\alpha)=\frac{v}{c}=\beta</math>.
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| [[File:MinkDiagrScale.svg|thumb|right|256px|Different scales on the axes.]]
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| The corresponding translation from ''x'' and ''t'' to ''x<nowiki>'</nowiki>'' and ''t<nowiki>'</nowiki>'' and vice versa is described mathematically by the so-called [[Lorentz transformation]]. Whatever space and time axes arise through such transformation, in a Minkowski diagram they correspond to [[conjugate diameters]] of a pair of [[hyperbola]]s. The scales on the axes are given as follows: If ''U'' is the unit length on the axes of ''ct'' and ''x'' respectively, the unit length on the axes of ''ct’'' and ''x’'' is:<ref>{{cite book |author=Jürgen Freund |title=Special Relativity for Beginners: A Textbook for Undergraduates |publisher=World Scientific| year=2008 |pages=49|isbn=981277159X}}</ref>
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| :<math>U'=U\cdot\sqrt{\frac{1+\beta^{2}}{1-\beta^{2}}}</math>.
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| ;History
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| In Minkowski’s 1908 paper there were three diagrams, first to illustrate the Lorentz transformation, then the partition of the plane by the light-cone, and finally illustration of worldlines.<ref name=minko /> The first diagram used a branch of the [[unit hyperbola]] <math>\scriptstyle t^2 - x^2 = 1 </math> to show the locus of a unit of [[proper time]] depending on velocity, thus illustrating time dilation. The second diagram showed the conjugate hyperbola to calibrate space, where a similar stretching leaves the impression of [[Fitzgerald contraction]]. In 1914 [[Ludwik Silberstein]]<ref>Silberstein (1914) The Theory of Relativity, page 131</ref> included a diagram of "Minkowski’s representation of the Lorentz transformation". This diagram included the unit hyperbola, its conjugate, and a pair of [[conjugate diameters]]. Since the 1960s a version of this more complete configuration has been referred to as The Minkowski Diagram, and used as a standard illustration of the [[transformation geometry]] of special relativity. [[E. T. Whittaker]] has pointed out that the [[Principle of relativity]] is tantamount to the arbitrariness of what hyperbola radius is selected for [[time]] in the Minkowski diagram. In 1912 [[Gilbert N. Lewis]] and [[Edwin B. Wilson]] applied the methods of [[synthetic geometry]] to develop the properties of the non-Euclidean plane that has Minkowski diagrams.
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| == Loedel diagram ==
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| [[Image:Minkowski diagram - simultaneity2.svg|thumb|right|256px|Symmetric Minkowski diagram (Loedel diagram). The two sets of parallel lines represent simultaneous events, one set for each moving frame.]]
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| While the rest frame has space and time axes at right angles, the moving frame has primed axes which form an acute angle. Since the frames are meant to be equivalent, the asymmetry may be disturbing. However, several authors showed that there is a frame of reference between the resting and moving ones where their symmetry would be apparent ("median frame").<ref name=mirimanoff /> In this frame, the two other frames are moving in opposite directions with equal speed. Using such coordinates makes the units of length and time the same for both axes. If <math>\beta=v/c</math> and <math>\gamma=\sqrt{1-\beta^{2}}</math> is given between <math>S</math> and <math>S'</math>, then these expressions are connected with the values in their median frame <math>S_0</math> as follows:<ref name=mirimanoff>{{cite journal|author=Mirimanoff, Dmitry|title=La transformation de Lorentz-Einstein et le temps universel de M. Ed. Guillaume|journal=Archives des sciences physiques et naturelles (supplement) |series=5|volume=3|pages=46-48|year=1921|url=http://gallica.bnf.fr/ark:/12148/bpt6k2991536/f682.image}} (Translation: [[s:en:Translation:The Lorentz-Einstein transformation and the universal time of Ed. Guillaume|The Lorentz-Einstein transformation and the universal time of Ed. Guillaume]])</ref><ref>{{cite book |author=Albert Shadowitz |title=The Electromagnetic Field |isbn=0486132013 |publisher=Courier Dover Publications |edition=Reprint of 1975 edition |year=2012|pages=460}} See {{Google books quote|id=31hjdifsTeQC|page=460|text=distortion|Google books, p. 460}}</ref>
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| :<math>\begin{align}
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| (1) & & \beta & =\frac{2\beta_{0}}{1+\beta_{0}^{2}},\\
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| (2) & & \beta_{0} & =\frac{\gamma-1}{\beta\gamma},\\
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| (3) & & \gamma_{0} & =\sqrt{\frac{\gamma+1}{2}}.
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| \end{align}</math>
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| For instance, if <math>\beta=0.5</math> between S and S', then by (2) they are moving in their median frame <math>S_0</math> with approximately ±0.268c each in opposite directions. On the other hand, if <math>\beta_0=0.5</math> in <math>S_0</math>, then by (1) the relative velocity between S and S' in their own rest frames is 0.8c. | |
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| However, it turns out that when it is about to design such a symmetric diagram, the axes of the median frame <math>S_0</math> are not required to be indicated, and some authors derived the diagram's relations even without mentioning the median frame and the magnitude <math>\beta_0</math> at all. Therefore it is common to design the coordinates and angles of S and S' directly on the basis of their relative velocity <math>\beta=v/c</math>.<ref name=sartori>{{cite book |author=Leo Sartori |title=Understanding Relativity: a simplified approach to Einstein's theories |isbn=0-520-20029-2 |publisher=University of California Press |year=1996|pages=151ff}}</ref>
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| [[File:Gruner1.svg|thumb|right|256px|Gruner and Sauter used symmetric Minkowski diagrams, in which the x'- and ct-axes are mutually perpendicular, as well as the x- and ct'-axes]]
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| The x'- and ct-axes have to be mutually perpendicular, as well as the x-axis and the ct'-axis. If <math>\varphi</math> is the angle between the ct'- and ct-axes (and between the x- and x'-axes) and <math>\theta</math> between the x'- and ct' axes:<ref name=gruner1 /><ref name=gruner2 /><ref name=shado /><ref name=sartori />
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| :<math>\begin{align}
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| \sin\varphi=\cos\theta & =\beta,\\
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| \cos\varphi=\sin\theta & =\sqrt{1-\beta^{2}}=1/\gamma,\\
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| \tan\varphi=\cot\theta & =\beta\cdot\gamma.
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| \end{align}</math>
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| In addition, the parallel projections <math>(x, t; x',t')</math> of vector <math>R</math> are its [[Covariance and contravariance of vectors|contravariant]] components, and <math>(\xi, \tau; \xi', \tau')</math> its covariant components (see image on the right).<ref name=gruner1 /><ref name=gruner2 />
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| ;History
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| *[[Max Born]] (1920) draw Minkowski diagrams by placing the ct'-axis almost perpendicular to the x-axis, as well as the ct-axis to the x'-axis, in order to demonstrate length contraction and time dilation in the symmetric case of two rods and two clocks moving in opposite direction.<ref>{{Cite book|author=Born, Max|title=Die Relativitätstheorie Einsteins|location=|pages=177-180|publisher=Springer|year=1920|edition=First|url=http://catalog.hathitrust.org/Record/006663730}} See also {{Google books quote|id=de_6AAAAQBAJ|page=187|Reprint (2013) of third edition (1922) at Google books, p. 187}}</ref>
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| * [[Dmitry Mirimanoff]] (1921) showed that there is always a median frame with respect to two relatively moving frames, and derived the relations between them from the Lorentz transformation. However, he didn't give a graphical representation in a diagram.<ref name=mirimanoff />
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| * Symmetric diagrams were systematically developed by [[Paul Gruner]] in collaboration with Josef Sauter in two papers in 1921. Relativistic effects such as length contraction and time dilation and some relations to covariant and contravariant vectors were demonstrated by them.<ref name=gruner1>{{cite journal|author=Gruner, Paul & Sauter, Josef|title=Représentation géométrique élémentaire des formules de la théorie de la relativité|journal=Archives des sciences physiques et naturelles|series=5|volume=3|pages=295-296|year=1921|url=http://gallica.bnf.fr/ark:/12148/bpt6k2991536/f295.image}} (Translation: [[s:en:Translation:Elementary geometric representation of the formulas of the special theory of relativity|Elementary geometric representation of the formulas of the special theory of relativity]])</ref><ref name=gruner2>{{cite journal|author=Gruner, Paul|title=Eine elementare geometrische Darstellung der Transformationsformeln der speziellen Relativitätstheorie|journal=Physikalische Zeitschrift|volume=22|pages=384-385|year=1921}} (Translation: [[s:Translation:An elementary geometrical representation of the transformation formulas of the special theory of relativity|An elementary geometrical representation of the transformation formulas of the special theory of relativity]])</ref> Gruner extended this method in subsequent papers (1922-1924), and gave credit to Mirimanoff's treatment as well.<ref name=gruner3>{{Cite book|author=Gruner, Paul|year=1922|title=Elemente der Relativitätstheorie|trans_title=Elements of the theory of relativity|location=Bern|publisher=P. Haupt}}</ref><ref name=gruner4>{{cite journal|author=Gruner, Paul|title=Graphische Darstellung der speziellen Relativitätstheorie in der vierdimensionalen Raum-Zeit-Welt I|trans_title=Graphical representation of the special theory of relativity in the four-dimensional spacetime-world I|journal=Zeitschrift für Physik|volume=10|issue=1|pages=22-37|year=1922|doi=10.1007/BF01332542}}</ref><ref name=gruner5>{{cite journal|author=Gruner, Paul|title=Graphische Darstellung der speziellen Relativitätstheorie in der vierdimensionalen Raum-Zeit-Welt II|trans_title=Graphical representation of the special theory of relativity in the four-dimensional spacetime-world II|journal=Zeitschrift für Physik|volume=10|issue=1|pages=227-235|year=1922|doi=10.1007/BF01332563}}</ref><ref name=gruner6>{{cite journal|author=Gruner, Paul|title=a) Représentation graphique de l’univers espace-temps à quatre dimensions. b) Représentation graphique du temps universel dans la théorie de la relativité|journal=Archives des sciences physiques et naturelles|series=5|volume=4|pages=234-236|year=1921|url=http://gallica.bnf.fr/ark:/12148/bpt6k299154k/f235.image}} (Translation: [[s:en:Translation:Graphical representation of the four-dimensional space-time universe|Graphical representation of the four-dimensional space-time universe]])</ref><ref name=gruner7>{{cite journal|author=Gruner, Paul|title=Die Bedeutung "reduzierter" orthogonaler Koordinatensysteme für die Tensoranalysis und die spezielle Relativitätstheorie|trans_title=The importance of "reduced" orthogonal coordinate-systems for tensor analysis and the special theory of relativity|journal=Zeitschrift für Physik|volume=10|issue=1|pages=236-242|year=1922|doi=10.1007/BF01332564}}</ref><ref name=gruner8>{{cite journal|author=Gruner, Paul|title=Geometrische Darstellungen der speziellen Relativitätstheorie, insbesondere des elektromagnetischen Feldes bewegter Körper|trans_title=Geometrich representations of the special theory of relativity, in particular the electromagnetic field of moving bodies|journal=Zeitschrift für Physik|volume=21|issue=1|pages=366-371|year=1924|doi=10.1007/BF01328285}}</ref>
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| *The construction of symmetric Minkowski diagrams was later independently rediscovered by several authors. For instance, starting in 1948, [[Enrique Loedel Palumbo]] published a series of papers in Spanish language, presenting the details of such an approach.<ref name=loed48>{{cite journal|author=Loedel, Enrique|title=Aberracion y Relatividad |journal=Anales soc. cient. argentina|volume=145|pages=3-13|year=1948}}</ref><ref>''Fisica relativista'', Kapelusz Editorial, Buenos Aires, Argentina (1955).</ref> In 1955, [[Henri Amar]] also published a paper presenting such relations, and gave credit to Loedel in a subsequent paper in 1957.<ref>{{cite journal|author=Amar, Henri|title=New Geometric Representation of the Lorentz Transformation |journal=American Journal of Physics|volume=23|issue=8|pages=487-489 |year=1955|doi=10.1119/1.1934074}}</ref><ref>{{cite journal|author=Amar, Henri & Loedel, Enrique|title=Geometric Representation of the Lorentz Transformation |journal=American Journal of Physics|volume=25|issue=5|pages=326-327 |year=1957|doi=10.1119/1.1934453}}</ref> Also some authors of modern textbooks use symmetric Minkowski diagrams of that kind, with denoting them as "Loedel diagrams".<ref name=shado>{{cite book |author=Albert Shadowitz |title=Special relativity |isbn=0-486-65743-4 |publisher=Courier Dover Publications |edition=Reprint of 1968 edition |year=1988|pages=20–22}}</ref><ref name=sartori />
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| == Time dilation ==
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| [[Image:Minkowski diagram - time dilation.svg|thumb|right|256px|Time dilation: Both observers consider the clock of the other as running slower.]]
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| Relativistic time dilation means that a clock moving relative to an observer is observed to run slower. In fact, time itself in the frame of the moving clock is observed to run slower. This can be read immediately from the adjoining Minkowski diagram. The observer whose reference frame is given by the black axes is assumed to move from the origin O towards A. The moving clock has the reference frame given by the blue axes and moves from O to B. For the black observer all events happening simultaneously with the event at A are located on a straight line parallel to its space axis. This line passes through A and B, so A and B are simultaneous from the reference frame of the observer with black axes. However, the clock that is moving relative to the black observer marks off time along the blue time axis. This is represented by the distance from O to B. Therefore, the observer at A with the black axes notices his or her clock as reading the distance from O to A while he or she observes the clock moving relative him or her to read the distance from O to B. Due to the distance from O to B being smaller than the distance from O to A, he or she concludes that the time passed on the clock moving relative to him or her is smaller than that passed on his own clock.
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| A second observer having moved together with the clock from O to B will argue that the other clock has reached only C until this moment and therefore this clock runs slower. The reason for these apparently paradoxical statements is the different determination of the events happening synchronously at different locations. Due to the principle of relativity the question of "who is right" has no answer and does not make sense.
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| == Length contraction ==
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| [[Image:Minkowski diagram - length contraction.svg|thumb|right|256px|Length contraction: Both observers consider objects moving with the other observer as being shorter.]]
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| Relativistic length contraction means that the length of an object moving relative to an observer is decreased and finally also the space itself is contracted in this system. The observer is assumed again to move along the ''ct''-axis. The world lines of the endpoints of an object moving relative to him are assumed to move along the ''ct<nowiki>'</nowiki>''-axis and the parallel line passing through A and B. For this observer the endpoints of the object at ''t''=0 are O and A. For a second observer moving together with the object, so that for him the object is at rest, it has the length OB at ''t<nowiki>'</nowiki>''=0. Due to OA<OB the object is contracted for the first observer.
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| The second observer will argue that the first observer has evaluated the endpoints of the object at O and A respectively and therefore at different times, leading to a wrong result due to his motion in the meantime. If the second observer investigates the length of another object with endpoints moving along the ''ct''-axis and a parallel line passing through C and D he concludes the same way this object to be contracted from OD to OC. Each observer estimates objects moving with the other observer to be contracted. This apparently paradoxical situation is again a consequence of the relativity of simultaneity as demonstrated by the analysis via Minkowski diagram.
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| For all these considerations it was assumed, that both observers take into account the speed of light and their distance to all events they see in order to determine the actual times at which these events happen from their point of view.
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| == Constancy of the speed of light ==
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| [[Image:Minkowski diagram - constancy of the speed of light.svg|thumb|right|256px|For the speed of a photon passing A both observers measure the same value even though they move relative to each other.]]
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| Another postulate of special relativity is the constancy of the speed of light. It says that any observer in an inertial reference frame measuring the vacuum speed of light relative to himself obtains the same value regardless of his own motion and that of the light source. This statement seems to be paradoxical, but it follows immediately from the differential equation yielding this, and the Minkowski diagram agrees. It explains also the result of the [[Michelson–Morley experiment]] which was considered to be a mystery before the theory of relativity was discovered, when photons were thought to be waves through an undetectable medium.
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| For world lines of photons passing the origin in different directions ''x''=''ct'' and ''x''=−''ct'' holds. That means any position on such a world line corresponds with steps on ''x''- and ''ct''-axis of equal absolute value. From the rule for reading off coordinates in coordinate system with tilted axes follows that the two world lines are the angle bisectors of the ''x''- and ''ct''-axis. The Minkowski diagram shows, that they are angle bisectors of the ''x'-'' and ''ct'-''axis as well. That means both observers measure the same speed ''c'' for both photons.
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| [[Image:Minkowski diagram - 3 systems.svg|thumb|right|256px|Minkowski diagram for 3 coordinate systems. For the speeds relative to the system in black ''v<nowiki>'</nowiki>'' = 0.4''c'' and ''v"'' = 0.8''c'' holds.]]
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| Further coordinate systems corresponding to observers with arbitrary velocities can be added to this Minkowski diagram. For all these systems both photon world lines represent the angle bisectors of the axes. The more the relative speed approaches the speed of light the more the axes approach the corresponding angle bisector. The path axis is always more flat and the time axis more steep than the photon world lines. The scales on both axes are always identical, but usually different from those of the other coordinate systems.
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| == Speed of light and causality ==
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| [[Image:Minkowski diagram - causality.svg|thumb|right|256px|Past and future relative to the origin. For the grey areas a corresponding temporal classification is not possible.]]
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| Straight lines passing the origin which are steeper than both photon world lines correspond with objects moving more slowly than the speed of light. If this applies to an object, then it applies from the viewpoint of all observers, because the world lines of these photons are the angle bisectors for any inertial reference frame. Therefore any point above the origin and between the world lines of both photons can be reached with a speed smaller than that of the light and can have a cause-effect-relationship with the origin. This area is the absolute future, because any event there happens later compared to the event represented by the origin regardless of the observer, which is obvious graphically from the Minkowski diagram.
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| Following the same argument the range below the origin and between the photon world lines is the absolute past relative to the origin. Any event there belongs definitely to the past and can be the cause of an effect at the origin.
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| The relationship between of such pairs of event is called ''timelike'', because they have a time distance greater than zero for all observers. A straight line connecting these two events is always the time axis of a possible observer for whom they happen at the same place. Two events which can be connected just with the speed of light are called ''lightlike''.
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| In principle a further dimension of space can be added to the Minkowski diagram leading to a three-dimensional representation. In this case the ranges of future and past become [[cone (geometry)|cone]]s with apexes touching each other at the origin. They are called [[light cones]].
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| == The speed of light as a limit ==
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| [[Image:Minkowski diagram - time travel.svg|thumb|right|256px|Sending a message at superluminal speed from O via A to B into the past. Both observers consider the temporal order of the pairs of events O and A as well as A and B different.]]
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| Following the same argument, all straight lines passing through the origin and which are more nearly horizontal than the photon world lines, would correspond to objects or signals moving [[Faster-than-light|faster than light]] regardless of the speed of the observer. Therefore no event outside the light cones can be reached from the origin, even by a light-signal, nor by any object or signal moving with less than the speed of light. Such pairs of events are called ''spacelike'' because they have a finite spatial distance different from zero for all observers. On the other hand a straight line connecting such events is always the space coordinate axis of a possible observer for whom they happen at the same time. By a slight variation of the velocity of this coordinate system in both directions it is always possible to find two inertial reference frames whose observers estimate the chronological order of these events to be different.
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| Therefore an object moving faster than light, say from O to A in the adjoining diagram, would imply that, for any observer watching the object moving from O to A, there can be found another observer (moving at less than the speed of light with respect to the first) for whom the object moves from A to O. The question of which observer is right has no unique answer, and therefore makes no physical sense. Any such moving object or signal would violate the principle of causality.
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| Also, any general technical means of sending signals faster than light would permit information to be sent into the originator's own past. In the diagram, an observer at O in the ''x''-''ct''-system sends a message moving faster than light to A. At A it is received by another observer, moving so as to be in the ''x'-ct'-''system, who sends it back, again faster than light by the same technology, arriving at B. But B is in the past relative to O. The absurdity of this process becomes obvious when both observers subsequently confirm that they received no message at all but all messages were directed towards the other observer as can be seen graphically in the Minkowski diagram. Indeed, if it was possible to accelerate an observer to the speed of light, the space and time axes would coincide with their angle bisector. The coordinate system would collapse.
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| These considerations show that the speed of light as a limit is a consequence of the properties of spacetime, and not of the properties of objects such as technologically imperfect space ships. The prohibition of faster-than-light motion actually has nothing in particular to do with electromagnetic waves or light, but depends on the structure of spacetime.
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| ==Eponym==
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| When Taylor and Wheeler composed ''Spacetime Physics'' (1966), they did ''not'' use the term "Minkowski diagram" for their spacetime geometry. Instead they included an acknowledgement of Minkowski’s contribution to philosophy by the totality of his innovation of 1908.<ref>Taylor/Wheeler (1966) page 37: "Minkowski's insight is central to the understanding of the physical world. It focuses attention on those quantities, such as interval, which are the same in all frames of reference. It brings out the relative character of quantities, such as velocity, energy, time, distance, which depend on the frame of reference."</ref>
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| As an [[eponym]], the term ''Minkowski diagram'' is subject to [[Stigler’s law of eponymy]], namely that Minkowski is wrongly designated as originator. The earlier works of [[Alexander Macfarlane]] contain algebra and diagrams that correspond well with the Minkowski diagram. See for instance the plate of figures in ''Proceedings of the Royal Society in Edinburgh'' for 1900. Macfarlane was building on what one sees in [[William Kingdon Clifford]]’s ''Elements of Dynamic'' (1878), page 90.
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| When abstracted to a line drawing, then any figure showing conjugate hyperbolas, with a selection of conjugate diameters, falls into this category. Students making drawings to accompany the exercises in [[George Salmon]]’s ''A Treatise on Conic Sections'' (1900) at pages 165–71 (on conjugate diameters) will be making Minkowski diagrams.
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| == See also ==
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| * [[Minkowski space]]
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| * [[Penrose diagram]]
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| * [[Rapidity]]
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| ==References==
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| {{reflist}}
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| * [[Anthony French]] (1968) ''Special Relativity'', pages 82 & 83, New York: [[W W Norton & Company]].
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| * E.N. Glass (1975) "Lorentz boosts and Minkowski diagrams" [[American Journal of Physics]] 43:1013,4.
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| * N. David Mermin (1968) ''Space and Time in Special Relativity'', Chapter 17 Minkowski diagrams: The Geometry of Spacetime, pages 155–99 [[McGraw-Hill]].
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| * {{cite book | author=Rindler, Wolfgang | title=Relativity: Special, General and Cosmological | publisher=Oxford University Press | year=2001 | isbn =0-19-850836-0 }}
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| * W.G.V. Rosser (1964) ''An Introduction to the Theory of Relativity'', page 256, Figure 6.4, London: [[Butterworths]].
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| * [[Edwin F. Taylor]] and [[John Archibald Wheeler]] (1963) ''Spacetime Physics'', pages 27 to 38, New York: [[W. H. Freeman and Company]], Second edition (1992).
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| *{{Citation|author=Walter, Scott|year=1999|contribution=The non-Euclidean style of Minkowskian relativity|editor=J. Gray|title=The Symbolic Universe: Geometry and Physics|pages=91–127|publisher=Oxford University Press|contribution-url=http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf}} (see page 10 of e-link)
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| ==External links==
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| {{Commons category inline|bullet=none|Minkowski diagrams}}
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| {{Relativity}}
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| [[Category:Special relativity]]
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| [[Category:Geometry]]
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| [[Category:Diagrams]]
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