Editing Lorentz transformation (section)

==Physical formulation of Lorentz boosts==
{{Further|Derivations of the Lorentz transformations}}

===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
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|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.

===Physical implications===

A critical requirement of the Lorentz transformations is the invariance of the speed of light, a fact used in their derivation, and contained in the transformations themselves. If in {{mvar|F}} the equation for a pulse of light along the {{mvar|x}} direction is {{math|1=''x'' = ''ct''}}, then in {{mvar|F′}} the Lorentz transformations give {{math|1=''x′'' = ''ct′''}}, and vice versa, for any {{math|−''c'' < ''v'' < ''c''}}.

For relative speeds much less than the speed of light, the Lorentz transformations reduce to the [[Galilean transformation]]:<ref>{{cite book |title=International Edition University Physics |author1=George Arfken |edition= |publisher=Elsevier |year=2012 |isbn=978-0-323-14203-8 |page=367 |url=https://books.google.com/books?id=XzQK42x6uaEC}} [https://books.google.com/books?id=XzQK42x6uaEC&pg=PA367 Extract of page 367]</ref><ref>{{cite book |title=Basic Electromagnetism |author1=E.R. Dobbs |edition=illustrated |publisher=Springer Science & Business Media |year=2013 |isbn=978-94-011-2112-5 |page=113 |url=https://books.google.com/books?id=v3fsCAAAQBAJ}} [https://books.google.com/books?id=v3fsCAAAQBAJ&pg=PA113 Extract of page 113]</ref>
<math display="block">\begin{align}
t' &\approx t \\
x' &\approx x - vt
\end{align}</math>
in accordance with the [[correspondence principle]]. It is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance".<ref>{{harvnb|Einstein|1916}}</ref>


Three counterintuitive, but correct, predictions of the transformations are:
;[[Relativity of simultaneity]]
: Suppose two events occur along the x axis simultaneously ({{math|1=Δ''t'' = 0}}) in {{mvar|F}}, but separated by a nonzero displacement {{math|Δ''x''}}.  Then in {{mvar|F′}}, we find that <math>\Delta t' = \gamma \frac{-v\,\Delta x}{c^2} </math>, so the events are no longer simultaneous according to a moving observer. 
;[[Time dilation]]
: Suppose there is a clock at rest in {{mvar|F}}. If a time interval is measured at the same point in that frame, so that {{math|1=Δ''x'' = 0}}, then the transformations give this interval in {{mvar|F′}} by {{math|1=Δ''t′'' = ''γ''Δ''t''}}. Conversely, suppose there is a clock at rest in {{mvar|F′}}. If an interval is measured at the same point in that frame, so that {{math|1=Δ''x′'' = 0}}, then the transformations give this interval in {{mvar|F}} by {{math|1=Δ''t'' = ''γ''Δ''t′''}}. Either way, each observer measures the time interval between ticks of a moving clock to be longer by a factor {{mvar|γ}} than the time interval between ticks of his own clock.
;[[Length contraction]]
: Suppose there is a rod at rest in {{mvar|F}} aligned along the {{mvar|x}} axis, with length {{math|Δ''x''}}.  In {{mvar|F′}}, the rod moves with velocity {{math|-''v''}}, so its length must be measured by taking two simultaneous ({{math|1=Δ''t′'' = 0}}) measurements at opposite ends.  Under these conditions, the inverse Lorentz transform shows that {{math|1=Δ''x'' = ''γ''Δ''x′''}}. In {{mvar|F}} the two measurements are no longer simultaneous, but this does not matter because the rod is at rest in {{mvar|F}}. So each observer measures the distance between the end points of a moving rod to be shorter by a factor {{math|1/''γ''}} than the end points of an identical rod at rest in his own frame. Length contraction affects any geometric quantity related to lengths, so from the perspective of a moving observer, areas and volumes will also appear to shrink along the direction of motion.

===Vector transformations===

{{Further|Euclidean vector|vector projection}}

[[File:Lorentz boost any direction standard configuration.svg|upright=1.75|thumb|An observer in frame {{mvar|F}} observes {{mvar|F′}} to move with velocity {{math|'''v'''}}, while {{mvar|F′}} observes {{mvar|F}} to move with velocity {{math|−'''v'''}}. {{According to whom|The coordinate axes of each frame are still parallel|date=November 2020}} and orthogonal. The position vector as measured in each frame is split into components parallel and perpendicular to the relative velocity vector {{math|'''v'''}}.<br />'''Left:''' Standard configuration. '''Right:''' Inverse configuration.]]

The use of vectors allows positions and velocities to be expressed in arbitrary directions compactly. A single boost in any direction depends on the full relative [[velocity vector]] {{math|'''v'''}} with a magnitude {{math|1={{abs|'''v'''}} = ''v''}} that cannot equal or exceed {{mvar|c}}, so that {{math|0 ≤ ''v'' < ''c''}}.

Only time and the coordinates parallel to the direction of relative motion change, while those coordinates perpendicular do not. With this in mind, split the spatial [[position vector]] {{math|'''r'''}} as measured in {{mvar|F}}, and {{math|'''r′'''}} as measured in {{mvar|F′}}, each into components perpendicular (⊥) and parallel ( ‖ ) to {{math|'''v'''}},
<math display="block">\mathbf{r}=\mathbf{r}_\perp+\mathbf{r}_\|\,,\quad \mathbf{r}' = \mathbf{r}_\perp' + \mathbf{r}_\|' \,, </math>
then the transformations are
<math display="block">\begin{align}
              t'  &= \gamma \left(t - \frac{\mathbf{r}_\parallel \cdot \mathbf{v}}{c^2} \right) \\
   \mathbf{r}_\|' &= \gamma (\mathbf{r}_\| - \mathbf{v} t) \\
\mathbf{r}_\perp' &= \mathbf{r}_\perp
\end{align}</math>
where {{math|·}} is the [[dot product]]. The Lorentz factor {{mvar|γ}} retains its definition for a boost in any direction, since it depends only on the magnitude of the relative velocity. The definition {{math|1='''β''' = '''v'''/''c''}} with magnitude {{math|0 ≤ ''β'' < 1}} is also used by some authors.

Introducing a [[unit vector]] {{math|1='''n''' = '''v'''/''v'' = '''β'''/''β''}} in the direction of relative motion, the relative velocity is {{math|1='''v''' = ''v'''''n'''}} with magnitude {{mvar|v}} and direction {{math|'''n'''}}, and [[vector projection]] and rejection give respectively
<math display="block">\mathbf{r}_\parallel = (\mathbf{r}\cdot\mathbf{n})\mathbf{n}\,,\quad \mathbf{r}_\perp = \mathbf{r} - (\mathbf{r}\cdot\mathbf{n})\mathbf{n}</math>

Accumulating the results gives the full transformations,

{{Equation box 1
|title='''Lorentz boost''' (''in direction '' {{math|'''n'''}} '' with magnitude '' {{mvar|v}})
|indent =:
|equation =
<math>\begin{align}
          t'  &= \gamma \left(t - \frac{v\mathbf{n}\cdot \mathbf{r}}{c^2} \right) \,, \\
  \mathbf{r}' &= \mathbf{r} + (\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n} - \gamma t v\mathbf{n} \,.
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

The projection and rejection also applies to {{math|'''r′'''}}. For the inverse transformations, exchange {{math|'''r'''}} and {{math|'''r′'''}} to switch observed coordinates, and negate the relative velocity {{math|'''v''' → −'''v'''}} (or simply the unit vector {{math|'''n''' → −'''n'''}} since the magnitude {{mvar|v}} is always positive) to obtain

{{Equation box 1
|title='''Inverse Lorentz boost''' (''in direction '' {{math|'''n'''}} '' with magnitude '' {{mvar|v}})
|indent =:
|equation =
<math>\begin{align}
           t &= \gamma \left(t' + \frac{\mathbf{r}' \cdot v\mathbf{n}}{c^2} \right) \,, \\
  \mathbf{r} &= \mathbf{r}' + (\gamma-1)(\mathbf{r}'\cdot\mathbf{n})\mathbf{n} + \gamma t' v\mathbf{n} \,,
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

The unit vector has the advantage of simplifying equations for a single boost, allows either {{math|'''v'''}} or {{math|'''β'''}} to be reinstated when convenient, and the rapidity parametrization is immediately obtained by replacing {{mvar|β}} and {{math|''βγ''}}. It is not convenient for multiple boosts.

The vectorial relation between relative velocity and rapidity is<ref>{{harvnb|Barut|1964|page=18–19}}</ref>
<math display="block"> \boldsymbol{\beta} = \beta \mathbf{n} = \mathbf{n} \tanh\zeta \,,</math>
and the "rapidity vector" can be defined as
<math display="block"> \boldsymbol{\zeta} = \zeta\mathbf{n} = \mathbf{n}\tanh^{-1}\beta \,, </math>
each of which serves as a useful abbreviation in some contexts. The magnitude of {{math|'''ζ'''}} is the absolute value of the rapidity scalar confined to {{math|0 ≤ ''ζ'' < ∞}}, which agrees with the range {{math|0 ≤ ''β'' < 1}}.

===Transformation of velocities===

{{Further|differential of a function|velocity addition formula}}

[[File:Lorentz transformation of velocity including velocity addition.svg|upright=1.75|thumb|The transformation of velocities provides the definition [[velocity addition formula|relativistic velocity addition]] {{math|⊕}}, the ordering of vectors is chosen to reflect the ordering of the addition of velocities; first {{math|'''v'''}} (the velocity of {{mvar|F′}} relative to {{mvar|F}}) then {{math|'''u′'''}} (the velocity of {{mvar|X}} relative to {{mvar|F′}}) to obtain {{math|'''u''' {{=}} '''v''' ⊕ '''u′'''}} (the velocity of {{mvar|X}} relative to {{mvar|F}}).]]

Defining the coordinate velocities and Lorentz factor by

:<math>\mathbf{u} = \frac{d\mathbf{r}}{dt} \,,\quad \mathbf{u}' = \frac{d\mathbf{r}'}{dt'} \,,\quad \gamma_\mathbf{v} = \frac{1}{\sqrt{1-\dfrac{\mathbf{v}\cdot\mathbf{v}}{c^2}}}</math>

taking the differentials in the coordinates and time of the vector transformations, then dividing equations, leads to

:<math>\mathbf{u}' = \frac{1}{ 1 - \frac{\mathbf{v}\cdot\mathbf{u}}{c^2} }\left[\frac{\mathbf{u}}{\gamma_\mathbf{v}} - \mathbf{v} + \frac{1}{c^2}\frac{\gamma_\mathbf{v}}{\gamma_\mathbf{v} + 1}\left(\mathbf{u}\cdot\mathbf{v}\right)\mathbf{v}\right] </math>

The velocities {{math|'''u'''}} and {{math|'''u′'''}} are the velocity of some massive object. They can also be for a third inertial frame (say {{mvar|F′′}}), in which case they must be ''constant''. Denote either entity by {{mvar|X}}. Then {{mvar|X}} moves with velocity {{math|'''u'''}} relative to {{mvar|F}}, or equivalently with velocity {{math|'''u′'''}} relative to {{mvar|F′}}, in turn {{mvar|F′}} moves with velocity {{math|'''v'''}} relative to {{mvar|F}}. The inverse transformations can be obtained in a similar way, or as with position coordinates exchange {{math|'''u'''}} and {{math|'''u′'''}}, and change {{math|'''v'''}} to {{math|−'''v'''}}.

The transformation of velocity is useful in [[stellar aberration]], the [[Fizeau experiment]], and the [[relativistic Doppler effect]].

The [[Acceleration (special relativity)#Three-acceleration|Lorentz transformations of acceleration]] can be similarly obtained by taking differentials in the velocity vectors, and dividing these by the time differential.

===Transformation of other quantities===

In general, given four quantities {{mvar|A}} and {{math|1='''Z''' = (''Z''{{sub|''x''}}, ''Z''{{sub|''y''}}, ''Z''{{sub|''z''}})}} and their Lorentz-boosted counterparts {{mvar|A′}} and {{math|1='''Z′''' = (''Z′''{{sub|''x''}}, ''Z′''{{sub|''y''}}, ''Z′''{{sub|''z''}})}}, a relation of the form
<math display="block">A^2 - \mathbf{Z}\cdot\mathbf{Z} = {A'}^2 - \mathbf{Z}'\cdot\mathbf{Z}'</math>
implies the quantities transform under Lorentz transformations similar to the transformation of spacetime coordinates;
<math display="block">\begin{align}
           A' &= \gamma \left(A - \frac{v\mathbf{n}\cdot \mathbf{Z}}{c} \right) \,, \\
  \mathbf{Z}' &= \mathbf{Z} + (\gamma-1)(\mathbf{Z}\cdot\mathbf{n})\mathbf{n} - \frac{\gamma A v\mathbf{n}}{c} \,.
\end{align}</math>

The decomposition of {{math|'''Z'''}} (and {{math|'''Z′'''}}) into components perpendicular and parallel to {{math|'''v'''}} is exactly the same as for the position vector, as is the process of obtaining the inverse transformations (exchange {{math|(''A'', '''Z''')}} and {{math|(''A′'', '''Z′''')}} to switch observed quantities, and reverse the direction of relative motion by the substitution {{math|'''n''' ↦ −'''n'''}}).

The quantities {{math|(''A'', '''Z''')}} collectively make up a ''[[four-vector]]'', where {{mvar|A}} is the "timelike component", and {{math|'''Z'''}} the "spacelike component". Examples of {{mvar|A}} and {{math|'''Z'''}} are the following:

{| class="wikitable"
|-
! Four-vector
! {{mvar|A}}
! {{math|'''Z'''}}
|-
| Position [[four-vector]]
| [[Time]] (multiplied by {{mvar|c}}), {{math|''ct''}}
| [[Position vector]], {{math|'''r'''}}
|-
| [[Four-momentum]]
| [[Energy]] (divided by {{mvar|c}}), {{math|''E''/''c''}}
| [[Momentum]], {{math|'''p'''}}
|-
| [[Four-vector|Four-wave vector]]
| [[angular frequency]] (divided by {{mvar|c}}), {{math|''ω''/''c''}}
| [[wave vector]], {{math|'''k'''}}
|-
| [[Four-spin]]
| (No name), {{math|''s''{{sub|''t''}}}}
| [[Spin (physics)|Spin]], {{math|'''s'''}}
|-
| [[Four-current]]
| [[Charge density]] (multiplied by {{mvar|c}}), {{math|''ρc''}}
| [[Current density]], {{math|'''j'''}}
|-
| [[Electromagnetic four-potential]]
| [[Electric potential]] (divided by {{mvar|c}}), {{math|''φ''/''c''}}
| [[Magnetic vector potential]], {{math|'''A'''}}
|}

For a given object (e.g., particle, fluid, field, material), if {{mvar|A}} or {{math|'''Z'''}} correspond to properties specific to the object like its [[charge density]], [[mass density]], [[Spin (physics)|spin]], etc., its properties can be fixed in the rest frame of that object. Then the Lorentz transformations give the corresponding properties in a frame moving relative to the object with constant velocity. This breaks some notions taken for granted in non-relativistic physics. For example, the energy {{mvar|E}} of an object is a scalar in non-relativistic mechanics, but not in relativistic mechanics because energy changes under Lorentz transformations; its value is different for various inertial frames. In the rest frame of an object, it has a [[rest energy]] and zero momentum. In a boosted frame its energy is different and it appears to have a momentum. Similarly, in non-relativistic quantum mechanics the spin of a particle is a constant vector, but in [[relativistic quantum mechanics]] spin {{math|'''s'''}} depends on relative motion. In the rest frame of the particle, the spin pseudovector can be fixed to be its ordinary non-relativistic spin with a zero timelike quantity {{math|''s''{{sub|''t''}}}}, however a boosted observer will perceive a nonzero timelike component and an altered spin.<ref>{{harvnb|Chaichian|Hagedorn|1997|page=239}}</ref>

Not all quantities are invariant in the form as shown above, for example orbital [[angular momentum]] {{math|'''L'''}} does not have a timelike quantity, and neither does the [[electric field]] {{math|'''E'''}} nor the [[magnetic field]] {{math|'''B'''}}. The definition of angular momentum is {{math|1='''L''' = '''r''' × '''p'''}}, and in a boosted frame the altered angular momentum is {{math|1='''L′''' = '''r′''' × '''p′'''}}. Applying this definition using the transformations of coordinates and momentum leads to the transformation of angular momentum. It turns out {{math|'''L'''}} transforms with another vector quantity {{math|1='''N''' = (''E''/''c''{{sup|2}})'''r''' − ''t'''''p'''}} related to boosts, see [[relativistic angular momentum]] for details. For the case of the {{math|'''E'''}} and {{math|'''B'''}} fields, the transformations cannot be obtained as directly using vector algebra. The [[Lorentz force]] is the definition of these fields, and in {{mvar|F}} it is {{math|1='''F''' = ''q''('''E''' + '''v''' × '''B''')}} while in {{mvar|F′}} it is {{math|1='''F′''' = ''q''('''E′''' + '''v′''' × '''B′''')}}. A method of deriving the EM field transformations in an efficient way which also illustrates the unit of the electromagnetic field uses tensor algebra, [[Lorentz transformation#Transformation of the electromagnetic field|given below]].