Editing Special relativity (section)

=== Rapidity ===
{{Main|Rapidity}}
{{multiple image
 | direction         = horizontal
 | width1            = 135
 | image1            = Trig functions (sine and cosine).svg
 | caption1          = Figure 7-1a.  A ray through the [[unit circle]] {{nowrap|1=''x''<sup>2</sup> + ''y''<sup>2</sup> = 1}} in the point {{nowrap|1=(cos ''a'', sin ''a'')}}, where ''a'' is twice the area between the ray, the circle, and the ''x''-axis.
 | width2            = 190
 | image2            = Hyperbolic functions-2.svg
 | caption2          = Figure 7-1b. A ray through the [[unit hyperbola]] {{nowrap|1=''x''<sup>2</sup> − ''y''<sup>2</sup> = 1}} in the point {{nowrap|1=(cosh ''a'', sinh ''a'')}}, where ''a'' is twice the area between the ray, the hyperbola, and the ''x''-axis.
}}
[[File:Sinh+cosh+tanh.svg|thumb|180px|Figure 7–2. Plot of the three basic [[Hyperbolic function]]s: hyperbolic sine ([[:File:Hyperbolic Sine.svg|sinh]]), hyperbolic cosine ([[:File:Hyperbolic Cosine.svg|cosh]]) and hyperbolic tangent ([[:File:Hyperbolic Tangent.svg|tanh]]). Sinh is red, cosh is blue and tanh is green.]]
Lorentz transformations relate coordinates of events in one reference frame to those of another frame. Relativistic composition of velocities is used to add two velocities together. The formulas to perform the latter computations are nonlinear, making them more complex than the corresponding Galilean formulas.

This nonlinearity is an artifact of our choice of parameters.<ref name="Taylor1992" />{{rp|47–59}} We have previously noted that in an {{nowrap|1=''x''–''ct''}} spacetime diagram, the points at some constant spacetime interval from the origin form an invariant hyperbola. We have also noted that the coordinate systems of two spacetime reference frames in standard configuration are hyperbolically rotated with respect to each other.

The natural functions for expressing these relationships are the [[Hyperbolic functions|hyperbolic analogs of the trigonometric functions]]. Fig.&nbsp;7-1a shows a [[unit circle]] with sin(''a'') and cos(''a''), the only difference between this diagram and the familiar unit circle of elementary trigonometry being that ''a'' is interpreted, not as the angle between the ray and the {{nowrap|1=''x''-axis}}, but as twice the area of the sector swept out by the ray from the {{nowrap|1=''x''-axis}}. Numerically, the angle and {{nowrap|1=2 × area}} measures for the unit circle are identical. Fig.&nbsp;7-1b shows a [[unit hyperbola]] with sinh(''a'') and cosh(''a''), where ''a'' is likewise interpreted as twice the tinted area.<ref>{{cite book|last1=Thomas|first1=George B.|last2=Weir|first2=Maurice D.|last3=Hass|first3=Joel|last4=Giordano|first4=Frank R.|title=Thomas' Calculus: Early Transcendentals|date=2008|publisher=Pearson Education, Inc.|location=Boston|isbn=978-0-321-49575-4|page=533|edition=Eleventh}}</ref> Fig.&nbsp;7-2 presents plots of the sinh, cosh, and tanh functions.

For the unit circle, the slope of the ray is given by
: <math>\text{slope} = \tan a  = \frac{\sin a }{\cos a }.</math>

In the Cartesian plane, rotation of point {{nowrap|1=(''x'', ''y'')}} into point {{nowrap|1=(''x''{{'}}, ''y''{{'}})}} by angle ''&theta;'' is given by
: <math>
\begin{pmatrix}
x' \\
y' \\
\end{pmatrix} = \begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \\
\end{pmatrix}\begin{pmatrix}
x \\
y \\
\end{pmatrix}.</math>

In a spacetime diagram, the velocity parameter <math>\beta</math> is the analog of slope. The ''rapidity'', ''&phi;'', is defined by<ref name="Morin2007" />{{rp|96–99}}
: <math>\beta \equiv \tanh \phi \equiv \frac{v}{c},</math>
where
: <math>\tanh \phi = \frac{\sinh \phi}{\cosh \phi} = \frac{e^\phi-e^{-\phi}}{e^\phi+e^{-\phi}}.</math>

The rapidity defined above is very useful in special relativity because many expressions take on a considerably simpler form when expressed in terms of it. For example, rapidity is simply additive in the collinear velocity-addition formula;<ref name="Taylor1992" />{{rp|47–59}}
: <math>\beta = \frac{\beta_1 + \beta_2}{1 + \beta_1 \beta_2} = </math> <math>\frac{\tanh \phi_1 + \tanh \phi_2}{1 + \tanh \phi_1 \tanh \phi_2} =</math> <math>\tanh(\phi_1 + \phi_2),</math>
or in other words, {{tmath|1= \phi = \phi_1 + \phi_2 }}.

The Lorentz transformations take a simple form when expressed in terms of rapidity. The ''&gamma;'' factor can be written as
: <math>\gamma = \frac{1}{\sqrt{1 - \beta^2}} = \frac{1}{\sqrt{1 - \tanh^2 \phi}}</math> <math>= \cosh \phi,</math>
: <math>\gamma \beta = \frac{\beta}{\sqrt{1 - \beta^2}} = \frac{\tanh \phi}{\sqrt{1 - \tanh^2 \phi}}</math> <math>= \sinh \phi.</math>

Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called ''boosts''.

Substituting ''&gamma;'' and ''&gamma;β'' into the transformations as previously presented and rewriting in matrix form, the Lorentz boost in the {{nowrap|1=''x''-direction}} may be written as
: <math>
  \begin{pmatrix}
    c t' \\
    x'
  \end{pmatrix}
  =
  \begin{pmatrix}
    \cosh \phi & -\sinh \phi \\
    -\sinh \phi & \cosh \phi
  \end{pmatrix}
  \begin{pmatrix}
    ct \\
    x
  \end{pmatrix},</math>
and the inverse Lorentz boost in the {{nowrap|1=''x''-direction}} may be written as
: <math>
  \begin{pmatrix}
    c t \\
    x
  \end{pmatrix}
  =
  \begin{pmatrix}
    \cosh \phi & \sinh \phi \\
    \sinh \phi & \cosh \phi
  \end{pmatrix}
  \begin{pmatrix}
    c t' \\
    x'
  \end{pmatrix}.</math>

In other words, Lorentz boosts represent [[hyperbolic rotation]]s in Minkowski spacetime.<ref name="Morin2007" />{{rp|96–99}}

The advantages of using hyperbolic functions are such that some textbooks such as the classic ones by Taylor and Wheeler introduce their use at a very early stage.<ref name="Taylor1992">{{cite book|url=https://archive.org/details/spacetime_physics/|title=Spacetime Physics|last1=Taylor|first1=Edwin F.|last2=Wheeler|first2=John Archibald|date=1992|publisher=W. H. Freeman|isbn=0-7167-2327-1|edition=2nd}}</ref><ref group=note>Rapidity arises naturally as a coordinates on the pure [[Representation theory of the Lorentz group#Conventions and Lie algebra bases|boost generators]] inside the [[Lie algebra]] algebra of the Lorentz group. Likewise, rotation angles arise naturally as coordinates (modulo {{nowrap|2{{pi}}}}) on the pure [[Representation theory of the Lorentz group#Conventions and Lie algebra bases|rotation generators]] in the Lie algebra. (Together they coordinatize the whole Lie algebra.) A notable difference is that the resulting rotations are periodic in the rotation angle, while the resulting boosts are not periodic in rapidity (but rather one-to-one). The similarity between boosts and rotations is formal resemblance.</ref>
{{anchor|4‑vectors}}