Editing Lorentz transformation (section)

===Coordinate transformation===
{{anchor|Coordinate transformation}} <!-- "Spacetime" links here -->
[[File:Lorentz boost x direction standard configuration.svg|thumb|right|upright=1.75|The spacetime coordinates of an event, as measured by each observer in their inertial reference frame (in standard configuration) are shown in the speech bubbles.<br />'''Top:''' frame {{mvar|F′}} moves at velocity {{mvar|v}} along the {{mvar|x}}-axis of frame {{mvar|F}}.<br />'''Bottom:''' frame {{mvar|F}} moves at velocity −{{mvar|v}} along the {{mvar|x′}}-axis of frame {{mvar|F′}}.<ref>{{harvnb|Young|Freedman|2008}}</ref>]]

A "stationary" observer in frame {{mvar|F}} defines events with coordinates {{mvar|t}}, {{mvar|x}}, {{mvar|y}}, {{mvar|z}}. Another frame {{mvar|F′}} moves with velocity {{mvar|v}} relative to {{mvar|F}}, and an observer in this "moving" frame {{mvar|F′}} defines events using the coordinates {{mvar|t′}}, {{mvar|x′}}, {{mvar|y′}}, {{mvar|z′}}.

The coordinate axes in each frame are parallel (the {{mvar|x}} and {{mvar|x′}} axes are parallel, the {{mvar|y}} and {{mvar|y′}} axes are parallel, and the {{mvar|z}} and {{mvar|z′}} axes are parallel), remain mutually perpendicular, and relative motion is along the coincident {{math|''xx′''}} axes. At {{math|1=''t'' = ''t′'' = 0}}, the origins of both coordinate systems are the same, {{math|1=(''x'', ''y'', ''z'') = (''x′'', ''y′'', ''z′'') = (0, 0, 0)}}. In other words, the times and positions are coincident at this event. If all these hold, then the coordinate systems are said to be in '''standard configuration''', or '''synchronized'''.

If an observer in {{mvar|F}} records an event {{mvar|t}}, {{mvar|x}}, {{mvar|y}}, {{mvar|z}}, then an observer in {{mvar|F′}} records the ''same'' event with coordinates<ref>{{harvnb|Forshaw|Smith|2009}}</ref>

{{Equation box 1
|title='''Lorentz boost''' ({{mvar|x}} '' direction'')
|indent =:
|equation =
<math>\begin{align}
  t' &= \gamma \left( t - \frac{v x}{c^2} \right)  \\
  x' &= \gamma \left( x - v t \right)\\
  y' &= y \\
  z' &= z
\end{align}</math>
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|border = 1
|border colour = black
|background colour=white}}

where {{mvar|v}} is the relative velocity between frames in the {{mvar|x}}-direction, {{mvar|c}} is the [[speed of light]], and
<math display="block"> \gamma = \frac{1}{ \sqrt{1 - \frac{v^2}{c^2}}}</math>
(lowercase [[gamma]]) is the [[Lorentz factor]].

Here, {{mvar|v}} is the ''[[parameter]]'' of the transformation, for a given boost it is a constant number, but can take a continuous range of values. In the setup used here, positive relative velocity {{math|''v'' > 0}} is motion along the positive directions of the {{math|''xx′''}} axes, zero relative velocity {{math|''v'' {{=}} 0}} is no relative motion, while negative relative velocity {{math|''v'' < 0}} is relative motion along the negative directions of the {{math|''xx′''}} axes. The magnitude of relative velocity {{mvar|v}} cannot equal or exceed {{mvar|c}}, so only subluminal speeds {{math|−''c'' < ''v'' < ''c''}} are allowed. The corresponding range of {{mvar|γ}} is {{math|1 ≤ ''γ'' < ∞}}.

The transformations are not defined if {{mvar|v}} is outside these limits. At the speed of light ({{math|''v'' {{=}} ''c''}}) {{mvar|γ}} is infinite, and [[faster than light]] ({{math|''v'' > ''c''}}) {{mvar|γ}} is a [[complex number]], each of which make the transformations unphysical. The space and time coordinates are measurable quantities and numerically must be real numbers.

As an [[active transformation]], an observer in {{mvar|F′}} notices the coordinates of the event to be "boosted" in the negative directions of the {{math|''xx′''}} axes, because of the {{math|−''v''}} in the transformations. This has the equivalent effect of the ''coordinate system'' {{mvar|F′}} boosted in the positive directions of the {{math|''xx′''}} axes, while the event does not change and is simply represented in another coordinate system, a [[passive transformation]].

The inverse relations ({{mvar|t}}, {{mvar|x}}, {{mvar|y}}, {{mvar|z}} in terms of {{mvar|t′}}, {{mvar|x′}}, {{mvar|y′}}, {{mvar|z′}}) can be found by algebraically solving the original set of equations. A more efficient way is to use physical principles. Here {{mvar|F′}} is the "stationary" frame while {{mvar|F}} is the "moving" frame. According to the principle of relativity, there is no privileged frame of reference, so the transformations from {{mvar|F′}} to {{mvar|F}} must take exactly the same form as the transformations from {{mvar|F}} to {{mvar|F′}}. The only difference is {{mvar|F}} moves with velocity {{math|−''v''}} relative to {{mvar|F′}} (i.e., the relative velocity has the same magnitude but is oppositely directed). Thus if an observer in {{mvar|F′}} notes an event {{mvar|t′}}, {{mvar|x′}}, {{mvar|y′}}, {{mvar|z′}}, then an observer in {{mvar|F}} notes the ''same'' event with coordinates

{{Equation box 1
|title='''Inverse Lorentz boost''' ({{mvar|x}} '' direction'')
|indent =:
|equation =
<math>\begin{align}
  t &= \gamma \left( t' + \frac{v x'}{c^2} \right)  \\
  x &= \gamma \left( x' + v t' \right)\\
  y &= y' \\
  z &= z',
\end{align}</math>
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|background colour=white}}

and the value of {{mvar|γ}} remains unchanged. This "trick" of simply reversing the direction of relative velocity while preserving its magnitude, and exchanging primed and unprimed variables, always applies to finding the inverse transformation of every boost in any direction.<ref>{{cite book |title=Special Relativity and How it Works |author1=Moses Fayngold |edition=illustrated |publisher=John Wiley & Sons |year=2008 |isbn=978-3-527-40607-4 |page=102 |url=https://books.google.com/books?id=Q3egk8Ds6ogC}} [https://books.google.com/books?id=Q3egk8Ds6ogC&pg=PA102 Extract of page 102]</ref><ref>{{cite book |title=Advanced University Physics |author1=Mircea S. Rogalski |author2=Stuart B. Palmer |edition=2nd, revised |publisher=CRC Press |year=2018 |isbn=978-1-4200-5712-6 |page=70 |url=https://books.google.com/books?id=cmYIEQAAQBAJ}} [https://books.google.com/books?id=cmYIEQAAQBAJ&pg=PA70 Extract of page 70]</ref>

Sometimes it is more convenient to use {{math|''β'' {{=}} ''v''/''c''}} (lowercase [[beta]]) instead of {{mvar|v}}, so that
<math display="block">\begin{align}
  ct' &= \gamma \left( ct - \beta x \right) \,, \\
   x' &= \gamma \left( x - \beta ct \right) \,, \\
\end{align}</math>
which shows much more clearly the symmetry in the transformation. From the allowed ranges of {{mvar|v}} and the definition of {{mvar|β}}, it follows {{math|−1 < ''β'' < 1}}. The use of {{mvar|β}} and {{mvar|γ}} is standard throughout the literature. In the case of three spatial dimensions [''ct'',''x'',''y'',''z''], where the boost <math>\beta</math> is in the ''x'' direction, the [[Eigenvalues and eigenvectors|eigenstates]] of the  transformation are {{math|[1,1,0,0]}} with eigenvalue <math>\sqrt{(1-\beta)/(1+\beta)}</math>, {{math|[1, −1,0,0]}} with eigenvalue <math>\sqrt{(1+\beta)/(1-\beta)}</math>, and {{math|[0,0,1,0]}} and {{math|[0,0,0,1]}}, the latter two with eigenvalue 1. 

When the boost velocity <math>\boldsymbol{v}</math> is in an arbitrary vector direction with the boost vector <math>\boldsymbol{\beta}=\boldsymbol{v}/c</math>, then the transformation from an unprimed spacetime coordinate system to a primed coordinate system is given by<ref>{{cite book |title=Relativity Made Relatively Easy |last1=Steane |first1=Andrew M. |edition=illustrated |publisher=OUP Oxford |year=2012 |isbn=978-0-19-966286-9 |page=124 |url=https://books.google.com/books?id=75rCErZkh7EC}} [https://books.google.com/books?id=75rCErZkh7EC&pg=PA124 Extract of page 124]</ref><ref>{{cite web |last1=Steane |first1=Andrew |title=The Lorentz transformation |url=https://users.physics.ox.ac.uk/~Steane/teaching/rel_A.pdf}}</ref>

<math display="block">\begin{bmatrix} ct' \vphantom{-\gamma\beta _x} \\ x' \vphantom{1+\frac{\gamma^2}{1+\gamma}\beta _x^2} \\ y' \vphantom{ \frac{\gamma^2}{1+\gamma}\beta _x \beta _y } \\ z' \vphantom{\frac{\gamma^2}{1+\gamma}\beta _y\beta _z} \end{bmatrix} =
\begin{bmatrix}
  \gamma & -\gamma\beta _x  & -\gamma\beta _y  & -\gamma\beta _z \\
  -\gamma\beta _x   & 1+\frac{\gamma^2}{1+\gamma}\beta _x^2 & \frac{\gamma^2}{1+\gamma}\beta _x \beta _y& \frac{\gamma^2}{1+\gamma}\beta _x\beta _z \\
  -\gamma\beta _y &  \frac{\gamma^2}{1+\gamma}\beta _x \beta _y & 1+\frac{\gamma^2}{1+\gamma}\beta _y^2 & \frac{\gamma^2}{1+\gamma}\beta _y\beta _z \\
  -\gamma\beta _z &  \frac{\gamma^2}{1+\gamma}\beta _x\beta _z & \frac{\gamma^2}{1+\gamma}\beta _y\beta _z  & 1+\frac{\gamma^2}{1+\gamma}\beta _z^2 \\
\end{bmatrix}
\begin{bmatrix} ct \vphantom{-\gamma\beta _x} \\ x \vphantom{1+\frac{\gamma^2}{1+\gamma}\beta _x^2} \\ y \vphantom{ \frac{\gamma^2}{1+\gamma}\beta _x \beta _y } \\ z \vphantom{\frac{\gamma^2}{1+\gamma}\beta _y\beta _z} \end{bmatrix}, </math>

where the [[Lorentz factor]] is <math>\gamma =1/\sqrt{1 - \boldsymbol{\beta}^2} </math>. The [[determinant]] of the transformation matrix is +1 and its [[Trace (linear algebra)|trace]] is <math>2(1+\gamma)</math>. The inverse of the transformation is given by reversing the sign of <math>\boldsymbol{\beta}</math>. The quantity <math> c^2t^2-x^2-y^2-z^2</math> is invariant under the transformation: namely <math>(ct'^2-x'^2-y'^2-z'^2)=(ct^2-x^2-y^2-z^2)</math>.

The Lorentz transformations can also be derived in a way that resembles circular rotations in 3-dimensional space using the [[hyperbolic function]]s. For the boost in the {{mvar|x}} direction, the results are

{{Equation box 1
|title='''Lorentz boost''' ({{mvar|x}} '' direction with rapidity '' {{mvar|ζ}})
|indent =:
|equation =
<math>\begin{align}
  ct' &=  ct \cosh\zeta - x \sinh\zeta \\
   x' &= x \cosh\zeta - ct \sinh\zeta \\
   y' &= y \\
   z' &= z
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

where {{mvar|ζ}} (lowercase [[zeta]]) is a parameter called ''[[rapidity]]'' (many other symbols are used, including {{mvar|θ}}, {{mvar|ϕ}}, {{mvar|φ}}, {{mvar|η}}, {{mvar|ψ}}, {{mvar|ξ}}). Given the strong resemblance to rotations of spatial coordinates in 3-dimensional space in the Cartesian {{math|''xy''}}, {{math|''yz''}}, and {{math|''zx''}} planes, a Lorentz boost can be thought of as a [[hyperbolic rotation]] of spacetime coordinates in the xt, yt, and zt Cartesian-time planes of 4-dimensional [[Minkowski space]]. The parameter {{mvar|ζ}} is the [[hyperbolic angle]] of rotation, analogous to the ordinary angle for circular rotations. This transformation can be illustrated with a [[Minkowski diagram]].

The hyperbolic functions arise from the ''difference'' between the squares of the time and spatial coordinates in the spacetime interval, rather than a sum. The geometric significance of the hyperbolic functions can be visualized by taking {{math|1=''x'' = 0}} or {{math|1=''ct'' = 0}} in the transformations. Squaring and subtracting the results, one can derive hyperbolic curves of constant coordinate values but varying {{mvar|ζ}}, which parametrizes the curves according to the identity
<math display="block"> \cosh^2\zeta - \sinh^2\zeta = 1 \,. </math>

Conversely the {{math|''ct''}} and {{mvar|x}} axes can be constructed for varying coordinates but constant {{mvar|ζ}}. The definition
<math display="block"> \tanh\zeta = \frac{\sinh\zeta}{\cosh\zeta} \,, </math>
provides the link between a constant value of rapidity, and the [[slope]] of the {{math|''ct''}} axis in spacetime. A consequence these two hyperbolic formulae is an identity that matches the Lorentz factor
<math display="block"> \cosh\zeta = \frac{1}{\sqrt{1 - \tanh^2\zeta}} \,. </math>

Comparing the Lorentz transformations in terms of the relative velocity and rapidity, or using the above formulae, the connections between {{mvar|β}}, {{mvar|γ}}, and {{mvar|ζ}} are
<math display="block">\begin{align}
       \beta &= \tanh\zeta  \,, \\
      \gamma &= \cosh\zeta  \,, \\
\beta \gamma &= \sinh\zeta  \,.
\end{align}</math>

Taking the inverse hyperbolic tangent gives the rapidity <math display="block"> \zeta = \tanh^{-1}\beta  \,.</math>

Since {{math|−1 < ''β'' < 1}}, it follows {{math|−∞ < ''ζ'' < ∞}}. From the relation between {{mvar|ζ}} and {{mvar|β}}, positive rapidity {{math|''ζ'' > 0}} is motion along the positive directions of the {{math|''xx′''}} axes, zero rapidity {{math|1=''ζ'' = 0}} is no relative motion, while negative rapidity {{math|''ζ'' < 0}} is relative motion along the negative directions of the {{math|''xx′''}} axes.

The inverse transformations are obtained by exchanging primed and unprimed quantities to switch the coordinate frames, and negating rapidity {{math|''ζ'' → −''ζ''}} since this is equivalent to negating the relative velocity. Therefore,

{{Equation box 1
|title='''Inverse Lorentz boost''' ({{mvar|x}} '' direction with rapidity '' {{mvar|ζ}})
|indent =:
|equation =
<math>\begin{align}
  ct & = ct' \cosh\zeta + x' \sinh\zeta \\
   x &= x' \cosh\zeta + ct' \sinh\zeta \\
   y &= y' \\
   z &= z'
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

The inverse transformations can be similarly visualized by considering the cases when {{math|1=''x′'' = 0}} and {{math|1=''ct′'' = 0}}.

So far the Lorentz transformations have been applied to ''one event''. If there are two events, there is a spatial separation and time interval between them. It follows from the [[linear transformation|linearity]] of the Lorentz transformations that two values of space and time coordinates can be chosen, the Lorentz transformations can be applied to each, then subtracted to get the Lorentz transformations of the differences;

<math display="block">\begin{align}
\Delta t' &= \gamma \left( \Delta t - \frac{v \, \Delta x}{c^2} \right) \,, \\
\Delta x' &= \gamma \left( \Delta x - v \, \Delta t \right) \,,
\end{align}</math>
with inverse relations
<math display="block">\begin{align}
\Delta t &= \gamma \left( \Delta t' + \frac{v \, \Delta x'}{c^2} \right) \,, \\
\Delta x &= \gamma \left( \Delta x' + v \, \Delta t' \right) \,.
\end{align}</math>

where {{math|Δ}} (uppercase [[delta (letter)|delta]]) indicates a difference of quantities; e.g., {{math|1=Δ''x'' = ''x''{{sub|2}} − ''x''{{sub|1}}}} for two values of {{mvar|x}} coordinates, and so on.

These transformations on ''differences'' rather than spatial points or instants of time are useful for a number of reasons:
* in calculations and experiments, it is lengths between two points or time intervals that are measured or of interest (e.g., the length of a moving vehicle, or time duration it takes to travel from one place to another),
* the transformations of velocity can be readily derived by making the difference infinitesimally small and dividing the equations, and the process repeated for the transformation of acceleration,
* if the coordinate systems are never coincident (i.e., not in standard configuration), and if both observers can agree on an event {{math|''t''{{sub|0}}, ''x''{{sub|0}}, ''y''{{sub|0}}, ''z''{{sub|0}}}} in {{mvar|F}} and {{math|''t''{{sub|0}}′, ''x''{{sub|0}}′, ''y''{{sub|0}}′, ''z''{{sub|0}}′}} in {{mvar|F′}}, then they can use that event as the origin, and the spacetime coordinate differences are the differences between their coordinates and this origin, e.g., {{math|1=Δ''x'' = ''x'' − ''x''{{sub|0}}}}, {{math|1=Δ''x′'' = ''x′'' − ''x''{{sub|0}}′}}, etc.