Editing Special relativity (section)

== Lorentz invariance as the essential core of special relativity <span class="anchor" id="Lorentz transformation"></span> ==
{{Main|Lorentz transformation}}

=== Two- vs one- postulate approaches ===
{{main|Derivations of the Lorentz transformations}}

In Einstein's own view, the two postulates of relativity and the invariance of the speed of light lead to a single postulate, the Lorentz transformation:
{{blockquote|The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events&nbsp;... The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws&nbsp;...<ref name="autogenerated1" group=p/>}}

Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations,<ref>For a survey of such derivations, see Lucas and Hodgson, Spacetime and Electromagnetism, 1990</ref> but Einstein stuck to his approach throughout work.<ref group=p>Einstein, On the Relativity Principle and the Conclusions Drawn from It, 1907; "The Principle of Relativity and Its Consequences in Modern Physics", 1910; "The Theory of Relativity", 1911; Manuscript on the Special Theory of Relativity, 1912; Theory of Relativity, 1913; Einstein, Relativity, the Special and General Theory, 1916; The Principal Ideas of the Theory of Relativity, 1916; What Is The Theory of Relativity?, 1919; The Principle of Relativity (Princeton Lectures), 1921; Physics and Reality, 1936; The Theory of Relativity, 1949.</ref>

[[Henri Poincaré]] provided the mathematical framework for relativity theory by proving that [[Lorentz transformations]] are a subset of his [[Poincaré group]] of symmetry transformations. Einstein later derived these transformations from his axioms.

While the traditional two-postulate approach to special relativity is presented in innumerable college textbooks and popular presentations,<ref name="Miller2009">{{cite journal |last1=Miller |first1=D. J. |title=A constructive approach to the special theory of relativity |journal=American Journal of Physics |volume=78 |issue=6 |pages=633–638 |arxiv=0907.0902 |doi=10.1119/1.3298908 |year=2010 |bibcode=2010AmJPh..78..633M |s2cid=20444859 }}</ref> other treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of [[Minkowski spacetime]].<ref group=p>Das, A. (1993) ''The Special Theory of Relativity, A Mathematical Exposition'', Springer, {{isbn|0-387-94042-1}}.</ref><ref group=p>Schutz, J. (1997) Independent Axioms for Minkowski Spacetime, Addison Wesley Longman Limited, {{isbn|0-582-31760-6}}.</ref> Textbooks starting with the single postulate of Minkowski spacetime include those by Taylor and Wheeler<ref name="Taylor1992"/> and by Callahan.<ref name="Callahan">{{cite book |last1=Callahan |first1=James J. |title=The Geometry of Spacetime: An Introduction to Special and General Relativity |date=2011 |publisher=Springer |location=New York |isbn=9781441931429}}</ref>

=== Lorentz transformation and its inverse ===

Define an [[Spacetime#Basic concepts|event]] to have spacetime coordinates {{nowrap|(''t'', ''x'', ''y'', ''z'')}} in system ''S'' and {{nowrap|({{prime|''t''}}, {{prime|''x''}}, {{prime|''y''}}, {{prime|''z''}})}} in a reference frame moving at a velocity ''v'' on the ''x''-axis with respect to that frame, {{prime|''S''}}. Then the [[Lorentz transformation]] specifies that these coordinates are related in the following way:
<math display="block">\begin{align}
t' &= \gamma \ (t - vx/c^2) \\
x' &= \gamma \ (x - v t) \\
y' &= y \\
z' &= z ,
\end{align}</math>
where <math display="block">\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}</math> is the [[Lorentz factor]] and ''c'' is the [[speed of light]] in vacuum, and the velocity ''v'' of {{prime|''S''}}, relative to ''S'', is parallel to the ''x''-axis. For simplicity, the ''y'' and ''z'' coordinates are unaffected; only the ''x'' and ''t'' coordinates are transformed. These Lorentz transformations form a [[one-parameter group]] of [[linear mapping]]s, that parameter being called [[rapidity]].

Solving the four transformation equations above for the unprimed coordinates yields the inverse Lorentz transformation:
<math display="block">\begin{align}
t &= \gamma ( t' + v x'/c^2) \\
x &= \gamma ( x' + v t') \\
y &= y' \\
z &= z'.
\end{align}</math>

This shows that the unprimed frame is moving with the velocity −''v'', as measured in the primed frame.<ref>P. G. Bergmann (1976) ''Introduction to the Theory of Relativity'', Dover edition, Chapter IV, page 36 {{isbn|0-486-63282-2}}.</ref>

There is nothing special about the ''x''-axis. The transformation can apply to the ''y''- or ''z''-axis, or indeed in any direction parallel to the motion (which are warped by the ''γ'' factor) and perpendicular; see the article [[Lorentz transformation]] for details.

A quantity that is invariant under [[Lorentz transformations]] is known as a [[Lorentz scalar]].

Writing the Lorentz transformation and its inverse in terms of coordinate differences, where one event has coordinates {{nowrap|(''x''<sub>1</sub>, ''t''<sub>1</sub>)}} and {{nowrap|({{prime|''x''}}<sub>1</sub>, {{prime|''t''}}<sub>1</sub>)}}, another event has coordinates {{nowrap|(''x''<sub>2</sub>, ''t''<sub>2</sub>)}} and {{nowrap|({{prime|''x''}}<sub>2</sub>, {{prime|''t''}}<sub>2</sub>)}}, and the differences are defined as
* {{EquationRef|1|Eq. 1:}} &nbsp;&nbsp; <math>\Delta x' = x'_2-x'_1 \ , \ \Delta t' = t'_2-t'_1 \ .</math>
* {{EquationRef|2|Eq. 2:}} &nbsp;&nbsp; <math>\Delta x = x_2-x_1 \ , \ \ \Delta t = t_2-t_1 \ .</math>
we get
* {{EquationRef|3|Eq. 3:}} &nbsp;&nbsp; <math>\Delta x' = \gamma \ (\Delta x - v \,\Delta t) \ ,\ \ </math> <math>\Delta t' = \gamma \ \left(\Delta t - v \ \Delta x / c^{2} \right) \ . </math>
* {{EquationRef|4|Eq. 4:}} &nbsp;&nbsp; <math>\Delta x = \gamma \ (\Delta x' + v \,\Delta t') \ , \ </math> <math>\Delta t = \gamma \ \left(\Delta t' + v \ \Delta x' / c^{2} \right) \ . </math>

If we take differentials instead of taking differences, we get
* {{EquationRef|5|Eq. 5:}} &nbsp;&nbsp; <math>dx' = \gamma \ (dx - v\,dt) \ ,\ \ </math> <math>dt' = \gamma \ \left( dt - v \ dx / c^{2} \right) \ . </math>
* {{EquationRef|6|Eq. 6:}} &nbsp;&nbsp; <math>dx = \gamma \ (dx' + v\,dt') \ , \ </math> <math>dt = \gamma \ \left(dt' + v \ dx' / c^{2} \right) \ . </math>

=== Graphical representation of the Lorentz transformation ===
{{multiple image|perrow = 2|total_width=400
| image1 = Spacetime diagram development A.svg |width1=535|height1=535
| image2 = Spacetime diagram development B.svg |width2=535|height2=535
| image3 = Spacetime diagram development C.svg |width3=535|height3=535
| image4 = Spacetime diagram development D.svg|width4=535|height4=535
| footer = Figure 3-1. Drawing a Minkowski spacetime diagram to illustrate a Lorentz transformation.}}

Spacetime diagrams ([[Minkowski diagram]]s) are an extremely useful aid to visualizing how coordinates transform between different reference frames. Although it is not as easy to perform exact computations using them as directly invoking the Lorentz transformations, their main power is their ability to provide an intuitive grasp of the results of a relativistic scenario.<ref name=Morin2007/>

To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S′, in standard configuration, as shown in Fig.&nbsp;2-1.<ref name=Morin2007/><ref name="Mermin1968">{{cite book |last1=Mermin |first1=N. David |title=Space and Time in Special Relativity |url=https://archive.org/details/spacetimeinspeci0000merm |url-access=registration |date=1968 |publisher=McGraw-Hill |isbn=978-0881334203}}</ref>{{rp|155–199}}

'''Fig.&nbsp;3-1a'''. Draw the <math>x</math> and <math>t</math> axes of frame S. The <math>x</math> axis is horizontal and the <math>t</math> (actually <math>ct</math>) axis is vertical, which is the opposite of the usual convention in kinematics. The <math>ct</math> axis is scaled by a factor of <math>c</math> so that both axes have common units of length. In the diagram shown, the gridlines are spaced one unit distance apart. The 45° diagonal lines represent the ''worldlines'' of two photons passing through the origin at time <math>t = 0.</math> The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events, <math>\text{A}</math> and <math>\text{B},</math> have been plotted on this graph so that their coordinates may be compared in the S and S' frames.

'''Fig.&nbsp;3-1b'''. Draw the <math>x'</math> and <math>ct'</math> axes of frame S'. The <math>ct'</math> axis represents the worldline of the origin of the S' coordinate system as measured in frame S. In this figure, <math>v = c/2.</math> Both the <math>ct'</math> and <math>x'</math> axes are tilted from the unprimed axes by an angle <math>\alpha = \tan^{-1}(\beta),</math> where <math>\beta = v/c.</math> The primed and unprimed axes share a common origin because frames S and S' had been set up in standard configuration, so that <math>t=0</math> when <math>t'=0.</math>

'''Fig.&nbsp;3-1c'''. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that <math>(x', ct')</math> coordinates of <math>(0, 1)</math> in the primed coordinate system transform to <math> (\beta \gamma, \gamma)</math> in the unprimed coordinate system. Likewise, <math>(x', ct')</math> coordinates of <math>(1, 0)</math> in the primed coordinate system transform to <math>(\gamma, \beta \gamma)</math> in the unprimed system. Draw gridlines parallel with the <math>ct'</math> axis through points <math>(k \gamma, k \beta \gamma)</math> as measured in the unprimed frame, where <math> k </math> is an integer. Likewise, draw gridlines parallel with the <math>x'</math> axis through <math>(k \beta \gamma, k \gamma)</math> as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between <math>ct'</math> units equals <math display=inline>\sqrt{(1 + \beta ^2)/(1 - \beta ^2)}</math> times the spacing between <math>ct</math> units, as measured in frame S. This ratio is always greater than 1, and ultimately it approaches infinity as <math>\beta \to 1.</math>

'''Fig.&nbsp;3-1d'''. Since the speed of light is an invariant, the ''worldlines'' of two photons passing through the origin at time <math>t' = 0</math> still plot as 45° diagonal lines. The primed coordinates of <math>\text{A}</math> and <math>\text{B}</math> are related to the unprimed coordinates through the Lorentz transformations and ''could'' be approximately measured from the graph (assuming that it has been plotted accurately enough), but the real merit of a Minkowski diagram is its granting us a geometric view of the scenario. For example, in this figure, we observe that the two timelike-separated events that had different x-coordinates in the unprimed frame are now at the same position in space.

While the unprimed frame is drawn with space and time axes that meet at right angles, the primed frame is drawn with axes that meet at acute or obtuse angles. This asymmetry is due to unavoidable distortions in how spacetime coordinates map onto a [[Cartesian plane]], but the frames are actually equivalent.