Editing Special relativity (section)

=== 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>