Editing Velocity-addition formula

{{short description|Equation used in relativistic physics}}
[[File:Albert Einstein Head.jpg|thumb|upright|220px|The special theory of relativity, formulated in 1905 by [[Albert Einstein]], implies that addition of velocities does not behave in accordance with simple [[vector addition]].]]

In [[relativistic physics]], a '''velocity-addition formula''' is an equation that specifies how to combine the [[Velocity|velocities]] of objects in a way that is consistent with the requirement that no object's speed can exceed the [[speed of light]]. Such formulas apply to successive [[Lorentz transformations]], so they also relate different [[frame of reference|frames]]. Accompanying velocity addition is a kinematic effect known as [[Thomas precession]], whereby successive non-collinear [[Lorentz boost]]s become equivalent to the composition of a rotation of the coordinate system and a boost.

Standard applications of velocity-addition formulas include the [[Doppler shift]], [[Doppler navigation]], the [[aberration of light]], and the dragging of light in moving water observed in the 1851 [[Fizeau experiment]].<ref>{{harvnb|Kleppner|Kolenkow|1978|loc=Chapters 11–14}}</ref>

The notation employs {{math|'''u'''}} as velocity of a body within a Lorentz frame {{math|''S''}}, and {{math|'''v'''}} as velocity of a second frame {{math|''S''′}}, as measured in {{math|''S''}}, and {{math|'''u'''′}} as the transformed velocity of the body within the second frame.

== History ==
The speed of light in a fluid is slower than the speed of light in vacuum, and it changes if the fluid is moving along with the light. In 1851, [[Hippolyte Fizeau|Fizeau]] [[Fizeau experiment|measured]] the speed of light in a fluid moving parallel to the light using an [[Interferometry|interferometer]]. Fizeau's results were not in accord with the then-prevalent theories. Fizeau experimentally correctly determined the zeroth term of an expansion of the relativistically correct addition law in terms of {{math|{{sfrac|''V''|''c''}}}} as is described below. Fizeau's result led physicists to accept the empirical validity of the rather unsatisfactory theory by [[Augustin-Jean Fresnel|Fresnel]] that a fluid moving with respect to the stationary [[Luminiferous aether|aether]] ''partially'' drags light with it, i.e. the speed is {{math|{{sfrac|''c''|''n''}}  + (1 − {{sfrac|1|''n''<sup>2</sup>}})''V''}} instead of {{math|{{sfrac|''c''|''n''}} + ''V''}}, where {{math|''c''}} is the speed of light in the aether, {{math|''n''}} is the [[refractive index]] of the fluid, and {{math|''V''}} is the speed of the fluid with respect to the aether.

The aberration of light, of which the easiest explanation is the relativistic velocity addition formula, together with Fizeau's result, triggered the development of theories like [[Lorentz aether theory]] of electromagnetism in 1892. In 1905 [[Albert Einstein]], with the advent of [[special relativity]], derived the standard configuration formula ({{math|''V''}} in the {{nowrap|{{math|''x''}}-direction}}) for the addition of relativistic velocities.<ref name=Einstein_1905>{{harvnb|Einstein|1905|loc=See section 5, "The composition of velocities"}}</ref> The issues involving aether were, gradually over the years, settled in favor of special relativity.

== Galilean relativity ==

It was observed by [[Galileo's ship|Galileo]] that a person on a uniformly moving ship has the impression of being at rest and sees a heavy body falling vertically downward.<ref>{{harvnb|Galilei|2001}}</ref> This observation is now regarded as the first clear statement of the principle of mechanical relativity. Galileo saw that from the point of view of a person standing on the shore, the motion of falling downwards on the ship would be combined with, or added to, the forward motion of the ship.<ref>{{harvnb|Galilei|1954}} Galileo used this insight to show that the path of the weight when seen from the shore would be a parabola.</ref> In terms of velocities, it can be said that the velocity of the falling body relative to the shore equals the velocity of that body relative to ship plus the velocity of the ship relative to the shore.

In general for three objects A (e.g. Galileo on the shore), B (e.g. ship), C (e.g. falling body on ship) the velocity vector <math>\mathbf{u}</math> of C relative to A (velocity of falling object as Galileo sees it) is the sum of the velocity <math>\mathbf{u'}</math> of C relative to B (velocity of falling object relative to ship) plus the velocity {{math|'''v'''}} of B relative to A (ship's velocity away from the shore). The addition here is the vector addition of vector algebra and the resulting velocity is usually represented in the form

<math display="block"> \mathbf{u} = \mathbf{v} + \mathbf{u'}.</math>

The cosmos of Galileo consists of [[absolute space and time]] and the addition of velocities corresponds to composition of [[Galilean transformation]]s. The relativity principle is called [[Galilean relativity]]. It is obeyed by [[Newtonian mechanics]].

== Special relativity ==

According to the theory of [[special relativity]], the frame of the ship has a different clock rate and distance measure, and the notion of simultaneity in the direction of motion is altered, so the addition law for velocities is changed. This change is not noticeable at low velocities but as the velocity increases towards the speed of light it becomes important. The addition law is also called a '''composition law for velocities'''. For collinear motions, the speed of the object, <math>u'</math>, e.g. a cannonball fired horizontally out to sea, as measured from the ship, moving at speed <math>v</math>, would be measured by someone standing on the shore and watching the whole scene through a telescope as<ref>{{cite book |title=University Physics |first1=George |last1=Arfken |publisher=Academic Press |year=2012 |isbn=978-0-323-14202-1 |page=367 |url=https://books.google.com/books?id=PfadZy35Wh0C}} [https://books.google.com/books?id=PfadZy35Wh0C&pg=PA367 Extract of page 367]</ref>
<math display="block"> u = {v+u'\over 1+(vu'/c^2)} . </math>
The composition formula can take an algebraically equivalent form, which can be easily derived by using only the principle of constancy of the speed of light,<ref>{{harvnb|Mermin|2005|p=37}}</ref>
<math display="block"> {c-u \over c+u} = \left({c-u' \over c+u'}\right)\left({c-v \over c+v}\right).</math>
The cosmos of special relativity consists of [[Minkowski spacetime]] and the addition of velocities corresponds to composition of [[Lorentz transformation]]s. In the special theory of relativity Newtonian mechanics is modified into [[relativistic mechanics]].

== Standard configuration ==
The formulas for boosts in the [[Derivations of the Lorentz transformations#Standard configuration|standard configuration]] follow most straightforwardly from taking differentials of the [[Lorentz transformation#Velocity parametrization|inverse Lorentz boost]] in standard configuration.<ref name=LL>{{harvnb|Landau|Lifshitz|2002|p=13}}</ref><ref>{{harvnb|Kleppner|Kolenkow|1978|p=457}}</ref> If the primed frame is travelling with speed <math>v</math> with [[Lorentz factor]] <math display="inline">\gamma_{_v} = 1/\sqrt{1-v^2/c^2}</math> in the positive {{nowrap|{{math|''x''}}-direction}} relative to the unprimed frame, then the differentials are

<math display="block">dx = \gamma_{_v}(dx' + vdt'), \quad dy = dy', \quad dz = dz', \quad dt = \gamma_{_v}\left(dt' + \frac{v}{c^2}dx'\right).</math>

Divide the first three equations by the fourth,

<math display="block">\frac{dx}{dt} = \frac{\gamma_{_v}(dx' + vdt')}{\gamma_{_v}(dt' + \frac{v}{c^2}dx')},
 \quad \frac{dy}{dt} = \frac{dy'}{\gamma_{_v}(dt' + \frac{v}{c^2}dx')}, \quad \frac{dz}{dt} = \frac{dz'}{\gamma_{_v}(dt' + \frac{v}{c^2}dx')},</math>

or

<math display="block">u_x = \frac{dx}{dt} = \frac{\frac{dx'}{dt'} + v}{(1 + \frac{v}{c^2}\frac{dx'}{dt'})},
\quad u_y = \frac{dy}{dt} = \frac{\frac{dy'}{dt'}}{\gamma_{_v} \ (1 + \frac{v}{c^2}\frac{dx'}{dt'})},
\quad u_z = \frac{dz}{dt} = \frac{\frac{dz'}{dt'}}{\gamma_{_v} \ (1 + \frac{v}{c^2}\frac{dx'}{dt'})},</math>

which is

{{Equation box 1
|title='''Transformation of velocity''' (''Cartesian components'')
|indent =:
|equation =
<math display="block">u_x = \frac{u_x' + v}{1 + \frac{v}{c^2}u_x'}, \quad u_x' = \frac{u_x - v}{1 - \frac{v}{c^2}u_x},</math>
<math display="block">u_y = \frac{u_y'\sqrt{1-\frac{v^2}{c^2}}}{1 + \frac{v}{c^2}u_x'}, \quad u_y' = \frac{u_y\sqrt{1-\frac{v^2}{c^2}}}{1 - \frac{v}{c^2}u_x},</math>
<math display="block">u_z = \frac{u_z'\sqrt{1-\frac{v^2}{c^2}}}{1 + \frac{v}{c^2}u_x'}, \quad u_z' = \frac{u_z\sqrt{1-\frac{v^2}{c^2}}}{1 - \frac{v}{c^2}u_x},</math>
|cellpadding= 6
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|bgcolor=#F9FFF7}}

in which expressions for the primed velocities were obtained using the standard recipe by replacing {{math|'''v'''}} by {{math|–'''v'''}} and swapping primed and unprimed coordinates. If coordinates are chosen so that all velocities lie in a (common) {{math|''x''–''y''}} plane, then velocities may be expressed as
<math display="block">u_x = u\cos \theta, u_y = u\sin \theta,\quad u_x' = u'\cos \theta', \quad u_y' = u'\sin \theta',</math>
(see [[polar coordinates]]) and one finds<ref name=Einstein_1905 /><ref>{{harvnb|Jackson|1999|p=531}}</ref>
{{Equation box 1
|title='''Transformation of velocity''' (''Plane polar components'')
|indent =:
|equation =
<math display="block">u = \frac{\sqrt{u'^2 +v^2+2vu'\cos \theta' - \left(\frac{vu'\sin\theta'}{c}\right)^2}}{1 + \frac{v}{c^2}u'\cos \theta'},</math>
<math>\tan \theta = \frac{u_y}{u_x} = \frac{\sqrt{1-\frac{v^2}{c^2}}u_y'}{u_x' + v} = \frac{\sqrt{1-\frac{v^2}{c^2}}u'\sin \theta'}{u'\cos \theta' + v}.</math>
|cellpadding= 6
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|bgcolor=#F9FFF7}}

{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=Details for u}}
<math display="block">\begin{align}
u &= \sqrt{u_x^2 + u_y^2} = \frac{\sqrt{(u_x'+v)^2 + (1-\frac{v^2}{c^2})u_y'^2}}{1 + \frac{v}{c^2}u_x'}
= \frac{\sqrt{u_x'^2+v^2+2u_x'v + (1-\frac{v^2}{c^2})u_y'^2}}{1 + \frac{v}{c^2}u_x'}\\
&=\frac{\sqrt{u'^2 \cos^2 \theta'+v^2+2vu'\cos \theta' + u'^2\sin^2\theta' - \frac{v^2}{c^2}u'^2\sin^2\theta'}}{1 + \frac{v}{c^2}u_x'}\\
&=\frac{\sqrt{u'^2 +v^2+2vu'\cos \theta' - (\frac{vu'\sin\theta'}{c})^2}}{1 + \frac{v}{c^2}u'\cos \theta'}
\end{align}</math>
----
{{Hidden end}}

The proof as given is highly formal. There are other more involved proofs that may be more enlightening, such as the one below.
{{math proof | title=A proof using {{math|4}}-vectors and Lorentz transformation matrices | proof =
Since a relativistic transformation rotates space and time into each other much as geometric rotations in the plane rotate the {{math|''x''}}- and {{math|''y''}}-axes, it is convenient to use the same units for space and time, otherwise a unit conversion factor appears throughout relativistic formulae, being the [[speed of light]]. In a system where lengths and times are measured in the same units, the speed of light is dimensionless and equal to {{math|1}}. A velocity is then expressed as fraction of the speed of light.

To find the relativistic transformation law, it is useful to introduce the four-velocities {{math|1=''V'' = (''V''<sub>0</sub>, ''V''<sub>1</sub>, 0, 0)}}, which is the motion of the ship away from the shore, as measured from the shore, and {{math|1=''U′'' = (''U′''<sub>0</sub>, ''U′''<sub>1</sub>, ''U′''<sub>2</sub>,  ''U′''<sub>3</sub>)}} which is the motion of the fly away from the ship, as measured from the ship.  The [[Four vector#Four-velocity|four-velocity]] is defined to be a [[four-vector]] with [[Minkowski metric|relativistic length]] equal to {{math|1}}, future-directed and tangent to the [[world line]] of the object in spacetime. Here, {{math|''V''<sub>0</sub>}} corresponds to the time component and {{math|''V''<sub>1</sub>}} to the {{math|''x''}} component of the ship's velocity as seen from the shore. It is convenient to take the {{math|''x''}}-axis to be the direction of motion of the ship away from the shore, and the {{math|''y''}}-axis so that the {{math|''x''–''y''}} plane is the plane spanned by the motion of the ship and the fly. This results in several components of the velocities being zero:
{{math|1=''V''<sub>2</sub> = ''V''<sub>3</sub> = ''U′''<sub>3</sub> = 0}}

The ordinary velocity is the ratio of the rate at which the space coordinates are increasing to the rate at which the time coordinate is increasing:

<math display="block">\begin{align}
\mathbf{v}  &= (v_1, v_2, v_3) = (V_1/V_0, 0, 0),\\
\mathbf{u}' &= (u'_1, u'_2, u'_3) = (U'_1/U'_0, U'_2/U'_0, 0)
\end{align}</math>

Since the relativistic length of {{math|''V''}} is {{math|1}},
<math display="block"> V_0^2 - V_1^2 = 1,</math>
so
<math display="block"> V_0 = 1/\sqrt{1-v_1^2} \ = \gamma, \quad V_1 = v_1/\sqrt{1-v_1^2} =  v_1 \gamma .</math>

The Lorentz transformation matrix that converts velocities measured in the ship frame to the shore frame is the ''inverse'' of the transformation described on the [[Lorentz transformation]] page, so the minus signs that appear there must be inverted here:

<math display="block">
\begin{pmatrix} \gamma &  v_1 \gamma & 0 & 0 \\  v_1 \gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
</math>

This matrix rotates the pure time-axis vector {{math|(1, 0, 0, 0)}} to {{math|(''V''<sub>0</sub>, ''V''<sub>1</sub>, 0, 0)}}, and all its columns are relativistically orthogonal to one another, so it defines a Lorentz transformation.

If a fly is moving with four-velocity {{math|''U′''}} in the ship frame, and it is boosted by multiplying by the matrix above, the new four-velocity in the shore frame is {{math|1=''U'' = (''U''<sub>0</sub>, ''U''<sub>1</sub>, ''U''<sub>2</sub>, ''U''<sub>3</sub>)}},
<math display="block">\begin{align} U_0 &= V_0 U'_0 + V_1 U'_1,\\
U_1 &= V_1 U'_0 + V_0 U'_1,\\
U_2 &= U'_2,\\
U_3 &= U'_3.\end{align} </math>

Dividing by the time component {{math|''U''<sub>0</sub>}} and substituting for the components of the four-vectors {{math|''U′''}} and {{math|''V''}} in terms of the components of the three-vectors {{math|'''u′'''}} and {{math|'''v'''}} gives the relativistic composition law as

<math display="block">\begin{align} u_1 &= { v_1 + u'_1 \over 1 + v_1 u'_1 },\\
u_2 &= { u'_2 \over (1 + v_1 u'_1) }{ 1 \over V_0 } = { u'_2 \over 1 + v_1 u'_1 } \sqrt{1 - v_1^2},\\
u_3 &= 0\end{align} </math>

The form of the relativistic composition law can be understood as an effect of the failure of simultaneity at a distance. For the parallel component, the time dilation decreases the speed, the length contraction increases it, and the two effects cancel out. The failure of simultaneity means that the fly is changing slices of simultaneity as the projection of {{math|'''u′'''}} onto {{math|'''v'''}}. Since this effect is entirely due to the time slicing, the same factor multiplies the perpendicular component, but for the perpendicular component there is no length contraction, so the time dilation multiplies by a factor of {{math|1={{sfrac|1|''V''<sub>0</sub>}} = {{sqrt|(1 − ''v''<sub>1</sub><sup>2</sup>)}}}}.
}}

== General configuration ==
[[File:Velocity decomposition sr resymbolised.svg|thumb|200px|Decomposition of 3-velocity {{math|'''u'''}} into parallel and perpendicular components, and calculation of the components. The procedure for {{math|'''u'''′}} is identical.]]

Starting from the expression in coordinates for {{math|''v''}} parallel to the {{nowrap|{{math|''x''}}-axis}}, expressions for the perpendicular and parallel components can be cast in vector form as follows, a trick which also works for Lorentz transformations of other 3d physical quantities originally in set up standard configuration. Introduce the velocity vector {{math|'''u'''}} in the unprimed frame and {{math|'''u'''′}} in the primed frame, and split them into components parallel (∥) and perpendicular (&perp;) to the relative velocity vector {{math|'''v'''}} (see hide box below) thus
<math display="block">\mathbf{u} = \mathbf{u}_\parallel + \mathbf{u}_\perp,\quad \mathbf{u}' = \mathbf{u}'_\parallel + \mathbf{u}'_\perp ,</math>
then with the usual [[Cartesian coordinates#Representing a vector in the standard basis|Cartesian standard basis vectors]] {{math|'''e'''<sub>''x''</sub>, '''e'''<sub>''y''</sub>, '''e'''<sub>''z''</sub>}}, set the velocity in the unprimed frame to be
<math display="block">\mathbf{u}_\parallel = u_x \mathbf{e}_x,\quad \mathbf{u}_\perp = u_y \mathbf{e}_y + u_z \mathbf{e}_z ,\quad \mathbf{v} = v\mathbf{e}_x,</math>
which gives, using the results for the standard configuration,
<math display="block">\mathbf u_\parallel = \frac{\mathbf u_\parallel' + \mathbf v}{1 + \frac{\mathbf v \cdot \mathbf u_\parallel'}{c^2}},
\quad \mathbf u_\perp = \frac{\sqrt{1-\frac{v^2}{c^2}}\mathbf u_\perp'}{1 + \frac{\mathbf v\cdot \mathbf u_\parallel'}{c^2}}.
</math>
where {{math|·}} is the [[dot product]]. Since these are vector equations, they still have the same form for {{math|'''v'''}} in ''any'' direction. The only difference from the coordinate expressions is that the above expressions refers to ''vectors'', not components.

One obtains
<math display="block">\mathbf{u} = \mathbf u_\parallel + \mathbf u_\perp = \frac{1}{1+\frac{\mathbf{v}\cdot\mathbf{u}'}{c^{2}}}\left[\alpha_v\mathbf{u}'+ \mathbf{v} + (1-\alpha_v)\frac{(\mathbf{v}\cdot\mathbf{u}')}{v^{2}}\mathbf{v}\right] \equiv \mathbf v \oplus \mathbf u',</math>
where {{math|1=''α''<sub>''v''</sub> = 1/''γ''<sub>''v''</sub>}} is the reciprocal of the [[Lorentz factor]]. The ordering of operands in the definition is chosen to coincide with that of the standard configuration from which the formula is derived.
{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=The algebra}}
<math display="block">\begin{align}
\frac{\mathbf u'_\parallel + \mathbf v}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}} + \frac{\alpha_v \mathbf u'_\perp}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}} &= \frac{\mathbf v + \frac{\mathbf v \cdot \mathbf u'}{v^2}\mathbf v}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}}
+ \frac{\alpha_v \mathbf u' - \alpha_v\frac{\mathbf v \cdot \mathbf u'}{v^2}\mathbf v}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}}\\

&=\frac{1 + \frac{\mathbf v \cdot \mathbf u'}{v^2}(1 - \alpha_v)}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}}\mathbf v +
\alpha_v\frac{1}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}}\mathbf u'\\

&=\frac{1}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}}\mathbf v +
\alpha_v\frac{1}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}}\mathbf u' +
\frac{1}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}}\frac{\mathbf v \cdot \mathbf u'}{v^2}(1 - \alpha_v)\mathbf v\\

&=\frac{1}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}}\mathbf v +
\alpha_v\frac{1}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}}\mathbf u' +
\frac{1}{c^2}\frac{1}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}}\frac{\mathbf v \cdot \mathbf u'}{v^2/c^2}(1 - \alpha_v)\mathbf v\\

&=\frac{1}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}}\mathbf v +
\alpha_v\frac{1}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}}\mathbf u' +
\frac{1}{c^2}\frac{1}{1 + \frac{\mathbf v \cdot \mathbf u'}{c^2}}\frac{\mathbf v \cdot \mathbf u'}{(1-\alpha_v)(1+\alpha_v)}(1 - \alpha_v)\mathbf v\\

&= \frac{1}{1+\frac{\mathbf{v}\cdot\mathbf{u}'}{c^{2}}}\left[\alpha_v\mathbf{u}'+ \mathbf{v} + (1-\alpha_v)\frac{(\mathbf{v}\cdot\mathbf{u}')}{v^{2}}\mathbf{v}\right].

\end{align}
</math>
----
{{Hidden end}}
{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=Decomposition into parallel and perpendicular components in terms of {{math|''V''}}}}
Either the parallel or the perpendicular component for each vector needs to be found, since the other component will be eliminated by substitution of the full vectors.

The parallel component of {{math|'''u'''′}} can be found by [[Vector projection#Vector projection|projecting the full vector]] into the direction of the relative motion
<math display="block">\mathbf{u}'_\parallel = \frac{\mathbf{v} \cdot \mathbf{u}'}{v^2}\mathbf v,</math>
and the perpendicular component of {{math|''u''′}} can be found by the geometric properties of the [[cross product]] (see figure above right),
<math display="block">\mathbf{u}'_\perp = - \frac{\mathbf{v} \times (\mathbf{v} \times \mathbf{u}')}{v^2}.</math>

In each case, {{math|'''v'''/''v''}} is a [[unit vector]] in the direction of relative motion.

The expressions for {{math|'''u'''<sub>∥</sub>}} and {{math|'''u'''<sub>&perp;</sub>}} can be found in the same way. Substituting the parallel component into
<math display="block">\mathbf u = \frac{\mathbf u_\parallel' + \mathbf v}{1 + \frac{\mathbf v \cdot \mathbf u_\parallel'}{c^2}}
+\frac{\sqrt{1-\frac{v^2}{c^2}}(\mathbf u' - \mathbf u_\parallel')}{1 + \frac{\mathbf v\cdot \mathbf u_\parallel'}{c^2}},
</math>

results in the above equation.<ref>{{harvnb|Lerner|Trigg|1991|p=1053}}</ref>
----
{{Hidden end}}

Using an identity in <math>\alpha_v</math> and <math>\gamma_v</math>,<ref>{{harvnb|Friedman|2002|pp=1–21}}</ref><ref group=nb>These formulae follow from inverting {{math|''α<sub>v</sub>''}} for {{math|''v''<sup>2</sup>}} and applying the [[difference of two squares]] to obtain
{{block indent | em = 1.5 | text = {{math|1=''v''<sup>2</sup> = ''c''<sup>2</sup>(1 − ''α''<sub>''v''</sub><sup>2</sup>) = ''c''<sup>2</sup>(1 − ''α''<sub>''v''</sub>)(1 + ''α''<sub>''v''</sub>)}}}}
so that
{{block indent | em = 1.5 | text = {{math|1={{sfrac|(1 − ''α''<sub>''v''</sub>)|''v''<sup>2</sup>}} = {{sfrac|1|''c''<sup>2</sup>(1 + ''α''<sub>''v''</sub>)}} = {{sfrac|''γ''<sub>''v''</sub>|''c''<sup>2</sup>(1 + ''γ''<sub>''v''</sub>)}}}}.}}</ref>

{{Equation box 1
|title=
|indent =:
|equation =
<math display="block">\begin{align}
\mathbf v \oplus \mathbf u' \equiv \mathbf u &=\frac{1}{1 + \frac{\mathbf u' \cdot \mathbf v}{c^2}}\left[\mathbf v +
\frac{\mathbf u'}{\gamma_v} + \frac{1}{c^2}\frac{\gamma_v}{1+\gamma_v}(\mathbf u' \cdot \mathbf v)\mathbf v\right]\\
&= \frac{1}{1 + \frac{\mathbf u' \cdot \mathbf v}{c^2}}\left[\mathbf v + \mathbf u' +
\frac{1}{c^2}\frac{\gamma_v}{1+\gamma_v} \mathbf v \times(\mathbf v \times \mathbf u')\right],
\end{align}</math>
and in the forwards (v positive, S → S') direction
<math display="block">\begin{align}
\mathbf v \oplus \mathbf u \equiv \mathbf u' &=\frac{1}{1 - \frac{\mathbf u \cdot \mathbf v}{c^2}}\left[\frac{\mathbf u}{\gamma_v} - \mathbf v + \frac{1}{c^2}\frac{\gamma_v}{1+\gamma_v}(\mathbf u \cdot \mathbf v)\mathbf v\right]\\
&= \frac{1}{1 - \frac{\mathbf u \cdot \mathbf v}{c^2}}\left[ \mathbf u - \mathbf v +
\frac{1}{c^2}\frac{\gamma_v}{1+\gamma_v} \mathbf v \times(\mathbf v \times \mathbf u)\right]
\end{align}</math>
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where the last expression is by the standard [[vector algebra relations|vector analysis formula]] {{math|1='''v''' × ('''v''' × '''u''') = ('''v''' ⋅ '''u''')'''v''' − ('''v''' ⋅ '''v''')'''u'''}}. The first expression extends to any number of spatial dimensions, but the [[cross product]] is defined in three dimensions only. The objects {{math|''A'', ''B'', ''C''}} with {{math|''B''}} having velocity {{math|'''v'''}} relative to {{math|''A''}} and {{math|''C''}} having velocity {{math|'''u'''}} relative to {{math|''A''}} can be anything. In particular, they can be three frames, or they could be the laboratory, a decaying particle and one of the decay products of the decaying particle.

=== Properties ===

The relativistic addition of 3-velocities is [[non-linear]], so in general
<math display="block">(\lambda \mathbf{v}) \oplus (\lambda \mathbf{u}) \neq \lambda (\mathbf{v} \oplus \mathbf{u}) , </math>
for [[real number]] {{math|''λ''}}, although it is true that
<math display="block">(-\mathbf{v}) \oplus (-\mathbf{u}) = - (\mathbf{v} \oplus \mathbf{u}) , </math>

Also, due to the last terms, is in general neither [[commutative]]
<math display="block">\mathbf v \oplus \mathbf u  \ne \mathbf u \oplus \mathbf v, </math>
nor [[associative]]
<math display="block">\mathbf v \oplus (\mathbf u \oplus \mathbf w) \ne (\mathbf v \oplus \mathbf u) \oplus \mathbf w. </math>

It deserves special mention that if {{math|'''u'''}} and {{math|'''v′'''}} refer to velocities of pairwise parallel frames (primed parallel to unprimed and doubly primed parallel to primed), then, according to Einstein's velocity reciprocity principle, the unprimed frame moves with velocity {{math|−'''u'''}} relative to the primed frame, and the primed frame moves with velocity {{math|−'''v′'''}} relative to the doubly primed frame hence {{math|(−'''v′''' ⊕ −'''u''')}} is the velocity of the unprimed frame relative to the doubly primed frame, and one  might expect to have {{math|1='''u''' ⊕ '''v′''' = −(−'''v′''' ⊕ −'''u''')}} by naive application of the reciprocity principle. This does not hold, though the magnitudes are equal. The unprimed and doubly primed frames are ''not'' parallel, but related through a rotation. This is related to the phenomenon of [[Thomas precession]], and is not dealt with further here.

The norms are given by<ref>{{harvnb|Landau|Lifshitz|2002|p=37}} Equation (12.6) This is derived quite differently by consideration of invariant cross sections.</ref>
<math display="block">| \mathbf u |^2 \equiv |\mathbf v \oplus \mathbf u'|^2 = \frac{1}{\left(1+\frac{\mathbf v \cdot \mathbf u'}{c^2}\right)^2}\left[\left(\mathbf v + \mathbf u' \right)^2 - \frac{1}{c^2}\left(\mathbf v \times \mathbf u'\right)^2 \right] = |\mathbf u' \oplus \mathbf v|^2.</math>
and
<math display="block">| \mathbf u' |^2 \equiv |\mathbf v \oplus \mathbf u|^2 = \frac{1}{\left(1-\frac{\mathbf v \cdot \mathbf u}{c^2}\right)^2}\left[\left(\mathbf u - \mathbf v \right)^2 - \frac{1}{c^2}\left(\mathbf v \times \mathbf u\right)^2 \right] = |\mathbf u \oplus \mathbf v|^2.</math>

{{math proof | proof = <math display="block">\begin{align}
&\left(1+\frac{\mathbf v \cdot \mathbf u'}{c^2}\right)^2|\mathbf v \oplus \mathbf u'|^2\\
&= \left[ \mathbf v + \mathbf u' + \frac{1}{c^2}\frac{\gamma_v}{1+\gamma_v}\mathbf v \times(\mathbf v \times \mathbf u')\right]^2\\

&=(\mathbf v + \mathbf u')^2 +2\frac{1}{c^2} \frac{\gamma_v}{\gamma_v + 1}  \left[(\mathbf v \cdot \mathbf u')^2-(\mathbf v\cdot\mathbf v)(\mathbf u'\cdot\mathbf u')\right] + \frac{1}{c^4}\left(\frac{\gamma_v}{\gamma_v + 1}\right)^2\left[(\mathbf v\cdot\mathbf v)^2(\mathbf u' \cdot \mathbf u') - (\mathbf v\cdot \mathbf u')^2(\mathbf v\cdot \mathbf v)\right]\\

&=(\mathbf v + \mathbf u')^2 +2\frac{1}{c^2} \frac{\gamma_v}{\gamma_v + 1}  \left[(\mathbf v \cdot \mathbf u')^2-(\mathbf v\cdot\mathbf v)(\mathbf u'\cdot\mathbf u')\right]
+ \frac{v^2}{c^4}\left(\frac{\gamma_v}{\gamma_v + 1}\right)^2\left[(\mathbf v\cdot\mathbf v)(\mathbf u' \cdot \mathbf u') - (\mathbf v\cdot \mathbf u')^2\right]\\

&=(\mathbf v + \mathbf u')^2 +2\frac{1}{c^2} \frac{\gamma_v}{\gamma_v + 1}  \left[(\mathbf v \cdot \mathbf u')^2-(\mathbf v\cdot\mathbf v)(\mathbf u'\cdot\mathbf u')\right]
+ \frac{(1-\alpha_v)(1+\alpha_v)}{c^2}\left(\frac{\gamma_v}{\gamma_v + 1}\right)^2\left[(\mathbf v\cdot\mathbf v)(\mathbf u' \cdot \mathbf u') - (\mathbf v\cdot \mathbf u')^2\right]\\

&=(\mathbf v + \mathbf u')^2 +2\frac{1}{c^2} \frac{\gamma_v}{\gamma_v + 1}  \left[(\mathbf v \cdot \mathbf u')^2-(\mathbf v\cdot\mathbf v)(\mathbf u'\cdot\mathbf u')\right]
+ \frac{(\gamma_v-1)}{c^2(\gamma_v + 1)}\left[(\mathbf v\cdot\mathbf v)(\mathbf u' \cdot \mathbf u') - (\mathbf v\cdot \mathbf u')^2\right]\\

&=(\mathbf v + \mathbf u')^2 +2\frac{1}{c^2} \frac{\gamma_v}{\gamma_v + 1}  \left[(\mathbf v \cdot \mathbf u')^2-(\mathbf v\cdot\mathbf v)(\mathbf u'\cdot\mathbf u')\right]
+ \frac{(1-\gamma_v)}{c^2(\gamma_v + 1)}\left[(\mathbf v\cdot \mathbf u')^2 - (\mathbf v\cdot\mathbf v)(\mathbf u' \cdot \mathbf u')\right] \\

&=(\mathbf v + \mathbf u')^2 +\frac{1}{c^2} \frac{\gamma_v+1}{\gamma_v + 1}  \left[(\mathbf v \cdot \mathbf u')^2-(\mathbf v\cdot\mathbf v)(\mathbf u'\cdot\mathbf u')\right]\\

&=(\mathbf v + \mathbf u')^2 -\frac{1}{c^2}  |\mathbf v \times \mathbf u'|^2

\end{align}</math>
Reverse formula found by using [[Lorentz transformation#Transformation of other quantities|standard procedure]] of swapping {{math|'''v'''}} for {{math|−'''v'''}} and {{math|'''u'''}} for {{math|'''u'''′}}.
}}
It is clear that the non-commutativity manifests itself as an additional ''rotation'' of the coordinate frame when two boosts are involved, since the norm squared is the same for both orders of boosts.

The gamma factors for the combined velocities are computed as
<math display="block">\gamma_u = \gamma_{\mathbf v \oplus \mathbf u'} =\left[ 1 -   \frac{1}{c^2}\frac{1}{(1+\frac{\mathbf v \cdot \mathbf u'}{c^2})^2}
\left( (\mathbf v + \mathbf u')^2 - \frac{1}{c^2}(v^2u'^2 - (\mathbf v \cdot \mathbf u')^2)\right)\right]^{-\frac{1}{2}}=\gamma_v\gamma_u'\left(1+\frac{\mathbf v \cdot \mathbf u'}{c^2}\right), \quad \quad \gamma_u' = \gamma_v\gamma_u\left(1-\frac{\mathbf v \cdot \mathbf u}{c^2}\right)</math>

{{math proof | title = Detailed proof | proof =
<math display="block">\begin{align}
\gamma_{\mathbf v \oplus \mathbf u'} &= \left[ \frac{c^3(1+\frac{\mathbf v \cdot \mathbf u'}{c^2})^2}{c^2(1+\frac{\mathbf v \cdot \mathbf u'}{c^2})^2}
 -   \frac{1}{c^2}\frac{ (\mathbf v + \mathbf u')^2 - \frac{1}{c^2}(v^2u'^2 - (\mathbf v \cdot \mathbf u')^2)}{(1+\frac{\mathbf v \cdot \mathbf u'}{c^2})^2}\right]^{-\frac{1}{2}}\\

&=\left[ \frac{c^2(1+\frac{\mathbf v \cdot \mathbf u'}{c^2})^2 - (\mathbf v + \mathbf u')^2 + \frac{1}{c^2}(v^2u'^2 - (\mathbf v \cdot \mathbf u')^2)}{c^2(1+\frac{\mathbf v \cdot \mathbf u'}{c^2})^2}
\right]^{-\frac{1}{2}}\\

&=\left[ \frac{c^2(1+2\frac{\mathbf v \cdot \mathbf u'}{c^2} + \frac{(\mathbf v \cdot \mathbf u')^2}{c^4}) - v^2 - u'^2 - 2(\mathbf v \cdot \mathbf u') + \frac{1}{c^2}(v^2u'^2 - (\mathbf v \cdot \mathbf u')^2)}{c^2(1+\frac{\mathbf v \cdot \mathbf u'}{c^2})^2}
\right]^{-\frac{1}{2}}\\

&=\left[ \frac{1+2\frac{\mathbf v \cdot \mathbf u'}{c^2} + \frac{(\mathbf v \cdot \mathbf u')^2}{c^4} - \frac{v^2}{c^2} - \frac{u'^2}{c^2} - \frac{2}{c^2}(\mathbf v \cdot \mathbf u') + \frac{1}{c^4}(v^2u'^2 - (\mathbf v \cdot \mathbf u')^2)}{(1+\frac{\mathbf v \cdot \mathbf u'}{c^2})^2}
\right]^{-\frac{1}{2}}\\

&=\left[ \frac{1 + \frac{(\mathbf v \cdot \mathbf u')^2}{c^4} - \frac{v^2}{c^2} - \frac{u'^2}{c^2} + \frac{1}{c^4}(v^2u'^2 - (\mathbf v \cdot \mathbf u')^2)}{(1+\frac{\mathbf v \cdot \mathbf u'}{c^2})^2}
\right]^{-\frac{1}{2}}\\

&=\left[ \frac{\left(1-\frac{v^2}{c^2}\right)\left(1-\frac{u'^2}{c^2}\right)}{\left(1+\frac{\mathbf v \cdot \mathbf u'}{c^2}\right)^2}
\right]^{-\frac{1}{2}}
=\left[ \frac{1}{\gamma_v^2\gamma_u'^2\left(1+\frac{\mathbf v \cdot \mathbf u'}{c^2}\right)^2}
\right]^{-\frac{1}{2}}\\

&=\gamma_v\gamma_u' \left(1+\frac{\mathbf v \cdot \mathbf u'}{c^2}\right)
\end{align}</math>

Reverse formula found by using [[Lorentz transformation#Transformation of other quantities|standard procedure]] of swapping {{math|'''v'''}} for {{math|−'''v'''}} and {{math|'''u'''}} for {{math|'''u'''′}}.
}}

=== Notational conventions ===
Notations and conventions for the velocity addition vary from author to author. Different symbols may be used for the operation, or for the velocities involved, and the operands may be switched for the same expression, or the symbols may be switched for the same velocity. A completely separate symbol may also be used for the transformed velocity, rather than the prime used here. Since the velocity addition is non-commutative, one cannot switch the operands or symbols without changing the result.

Examples of alternative notation include:

; No specific operand : {{harvtxt|Landau|Lifshitz|2002}} (using units where ''c'' = 1) <math display="block">| \mathbf{v_{rel}} |^2 =\frac{1}{(1 - \mathbf{v_1}\cdot\mathbf{v_2})^2}\left[(\mathbf{v_1}-\mathbf{v_2})^2 - (\mathbf{v_1} \times \mathbf{v_2})^2\right] </math>
; Left-to-right ordering of operands : {{harvtxt|Mocanu|1992}} <math display="block">\mathbf{u}\oplus\mathbf{v} = \frac{1}{1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^2}}\left[\mathbf{v}+\mathbf{u}+\frac{1}{c^2}\frac{\gamma_\mathbf{u}}{\gamma_\mathbf{u}+1}\mathbf{u}\times(\mathbf{u}\times\mathbf{v})\right] </math> {{br}}{{harvtxt|Ungar|1988}} <math display="block">\mathbf{u}*\mathbf{v}=\frac{1}{1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^2}}\left[\mathbf{v}+\mathbf{u}+\frac{1}{c^2}\frac{\gamma_\mathbf{u}}{\gamma_\mathbf{u}+1}\mathbf{u}\times(\mathbf{u}\times\mathbf{v})\right] </math>
; Right-to-left ordering of operands : {{harvtxt|Sexl|Urbantke|2001}} <math display="block">\mathbf{w}\circ\mathbf{v}=\frac{1}{1+\frac{\mathbf{v}\cdot\mathbf{w}}{c^{2}}}\left[\frac{\mathbf{w}}{\gamma_\mathbf{v}}+\mathbf{v}+\frac{1}{c^{2}}\frac{\gamma_\mathbf{v}}{\gamma_\mathbf{v}+1}(\mathbf{w}\cdot\mathbf{v})\mathbf{v}\right]</math>

== Applications ==
Some classical applications of velocity-addition formulas, to the Doppler shift, to the aberration of light, and to the dragging of light in moving water, yielding relativistically valid expressions for these phenomena are detailed below. It is also possible to use the velocity addition formula, assuming conservation of momentum (by appeal to ordinary rotational invariance), the correct form of the {{math|3}}-vector part of the [[momentum four-vector]], without resort to electromagnetism, or a priori not known to be valid, relativistic versions of the [[Lagrangian mechanics|Lagrangian formalism]]. This involves experimentalist bouncing off relativistic billiard balls from each other. This is not detailed here, but see for reference {{harvtxt|Lewis|Tolman|1909}} [[s:The Principle of Relativity, and Non-Newtonian Mechanics|Wikisource version]] (primary source) and {{harvtxt|Sard|1970|loc=Section 3.2}}.

=== Fizeau experiment ===
[[File:Hippolyte Fizeau.jpg|right|thumb|[[Hippolyte Fizeau]] (1819–1896), a French physicist, was in 1851 the first to measure the speed of light in flowing water.]]
{{main|Fizeau experiment}}
When light propagates in a medium, its speed is reduced, in the rest frame of the medium, to {{math|1=''c''<sub>''m''</sub> = {{sfrac|''c''|''n''<sub>''m''</sub>}}}}, where {{math|''n''<sub>''m''</sub>}} is the [[index of refraction]] of the medium {{math|''m''}}. The speed of light in a medium uniformly moving with speed {{math|''V''}} in the positive {{math|''x''}}-direction as measured in the lab frame is given directly by the velocity addition formulas. For the forward direction (standard configuration, drop index {{mvar|m}} on {{math|''n''}}) one gets,<ref>{{harvnb|Kleppner|Kolenkow|1978|p=474}}</ref>
<math display="block">\begin{align}
c_m &= \frac{V + c_m'}{1 + \frac{Vc_m'}{c^2}} = \frac{V + \frac{c}{n}}{1 + \frac{Vc}{nc^2}} = \frac{c}{n} \frac{1 + \frac{nV}{c}}{1 + \frac{V}{nc}}\\
& =    \frac{c}{n} \left(1 + \frac{nV}{c}\right) \frac{1}{1 + \frac{V}{nc}} =
\left(\frac{c}{n} + V\right) \left(1 - \frac{V}{nc} + \left(\frac{V}{nc}\right)^2 - \cdots\right).
\end{align}</math>

Collecting the largest contributions explicitly,
<math display="block">c_m = \frac{c}{n} + V\left(1 - \frac{1}{n^2} - \frac{V}{nc} + \cdots\right).</math>
Fizeau found the first three terms.<ref>{{harvnb|Fizeau|1851e}}</ref><ref>{{harvnb|Fizeau|1860}}</ref> The classical result is the first two terms.

=== Aberration of light ===
{{main|Aberration of light}}

Another basic application is to consider the deviation of light, i.e. change of its direction, when transforming to a new reference frame with parallel axes, called [[aberration of light]]. In this case, {{math|1=''v''′ = ''v'' = ''c''}}, and insertion in the formula for {{math|tan ''θ''}} yields
<math display="block">\tan \theta = \frac{\sqrt{1-\frac{V^2}{c^2}}c\sin \theta'}{c\cos \theta' + V}
= \frac{\sqrt{1-\frac{V^2}{c^2}}\sin \theta'}{\cos \theta' + \frac{V}{c}}.</math>

For this case one may also compute {{math|sin ''θ''}} and {{math|cos ''θ''}} from the standard formulae,<ref>{{harvnb|Landau|Lifshitz|2002|p=14}}</ref>
<math display="block">\sin \theta =\frac{\sqrt{1-\frac{V^2}{c^2}}\sin \theta'}{1+\frac{V}{c}\cos \theta'},</math>

{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=Trigonometry}}
<math display="block">\begin{align}\frac{v_y}{v}
&= \frac{\frac{\sqrt{1-\frac{V^2}{c^2}}v_y'}{1 + \frac{V}{c^2}v_x'}}{\frac{\sqrt{v'^2 +V^2+2Vv'\cos \theta' - (\frac{Vv'\sin\theta'}{c})^2}}{1 + \frac{V}{c^2}v'\cos \theta'}}\\

&= \frac{c\sqrt{1-\frac{V^2}{c^2}}\sin \theta'}{\sqrt{c^2 +V^2+2Vc\cos \theta' - V^2\sin^2\theta'}}\\

&= \frac{c\sqrt{1-\frac{V^2}{c^2}}\sin \theta'}{\sqrt{c^2 +V^2+2Vc\cos \theta' - V^2(1 - \cos^2\theta')}}
= \frac{c\sqrt{1-\frac{V^2}{c^2}}\sin \theta'}{\sqrt{c^2 +2Vc\cos \theta' + V^2\cos^2\theta'}}\\

&= \frac{\sqrt{1-\frac{V^2}{c^2}}\sin \theta'}{1+\frac{V}{c}\cos \theta'},
\end{align}</math>
----
{{Hidden end}}
<math display="block">\cos \theta = \frac{\frac{V}{c} + \cos \theta'}{1+\frac{V}{c}\cos \theta'},</math>
[[File:James Bradley by Thomas Hudson.jpg|left|thumb|[[James Bradley]] (1693–1762) [[Fellow of the Royal Society|FRS]] provided an explanation of aberration of light correct at the classical level,<ref>{{harvnb|Bradley|1727–1728}}</ref> at odds with the later theories prevailing in the nineteenth century based on the existence of [[aether (classical element)|aether]].]]

the trigonometric manipulations essentially being identical in the {{math|cos}} case to the manipulations in the {{math|sin}} case. Consider the difference,

<math display="block">\begin{align}\sin \theta - \sin \theta' &= \sin \theta'\left(\frac{\sqrt{1 - \frac{V^2}{c^2}}}{1 + \frac{V}{c} \cos \theta'} - 1\right)\\
&\approx \sin \theta'\left(1 -\frac{V}{c} \cos \theta' - 1\right) = -\frac{V}{c}\sin\theta'\cos\theta',\end{align}</math>
correct to order {{math|{{sfrac|''v''|''c''}}}}. Employ in order to make small angle approximations a trigonometric formula,
<math display="block">\sin \theta' - \sin \theta = 2\sin \frac{1}{2}(\theta'-\theta)\cos\frac{1}{2}(\theta + \theta') \approx (\theta' - \theta)\cos\theta', </math>
where {{math|cos{{sfrac|1|2}}(''θ'' + ''θ''′) ≈ cos ''θ''′, sin{{sfrac|1|2}}(''θ'' − ''θ''′) ≈ {{sfrac|1|2}}(''θ'' − ''θ''′)}} were used.

Thus the quantity
<math display="block">\Delta \theta \equiv \theta' - \theta = \frac{V}{c}\sin \theta',</math>
the '''classical aberration angle''', is obtained in the limit {{math|{{sfrac|''V''|''c''}} → 0}}.

=== Relativistic Doppler shift ===
[[File:Christian Doppler.jpg|thumb|right|[[Christian Doppler]] (1803–1853) was an Austrian mathematician and physicist who discovered that the observed frequency of a wave depends on the relative speed of the source and the observer.]]
{{main|Relativistic Doppler effect}}

Here ''velocity components'' will be used as opposed to ''speed'' for greater generality, and in order to avoid perhaps seemingly [[ad hoc]] introductions of minus signs. Minus signs occurring here will instead serve to illuminate features when speeds less than that of light are considered.

For light waves in vacuum, [[time dilation]] together with a simple geometrical observation alone suffices to calculate the Doppler shift in standard configuration (collinear relative velocity of emitter and observer as well of observed light wave).

All velocities in what follows are parallel to the common positive {{nowrap|{{math|''x''}}-direction}}, so subscripts on velocity components are dropped. In the observers frame, introduce the geometrical observation
<math display="block">\lambda = -sT + VT = (-s + V)T</math>
as the spatial distance, or [[wavelength]], between two pulses (wave crests), where {{math|''T''}} is the time elapsed between the emission of two pulses. The time elapsed between the passage of two pulses ''at the same point in space'' is the ''time period'' {{mvar|τ}}, and its inverse {{math|1=''ν'' = {{sfrac|1|''τ''}}}} is the observed (temporal) [[frequency]]. The corresponding quantities in the emitters frame are endowed with primes.<ref>{{harvnb|Kleppner|Kolenkow|1978|p=477}} In the reference, the speed of an ''approaching'' emitter is taken as ''positive''. Hence the sign difference.</ref>

For light waves <math display="block">s = s' = -c,</math>
and the observed frequency is<ref name=Einstein_1905 /><ref>{{harvnb|Tipler|Mosca|2008|pp=1328–1329}}</ref><ref>{{harvnb|Mansfield|O'Sullivan|2011|pp=491–492}}</ref>
<math display="block">\nu = {-s \over \lambda} = {-s \over (V-s)T} = {c \over (V+c)\gamma_{_V} T'}
= \nu'\frac{c\sqrt{1 - {V^2 \over c^2}}}{c+V} = \nu'\sqrt{\frac{1-\beta}{1+\beta}}\,.</math>
where {{math|1=''T'' = ''γ''<sub>''V''</sub>''T''′}} is standard [[time dilation]] formula.

Suppose instead that the wave is not composed of light waves with speed {{mvar|''c''}}, but instead, for easy visualization, bullets fired from a relativistic machine gun, with velocity {{math|''s''′}} in the frame of the emitter. Then, in general, the geometrical observation is ''precisely the same''. But now, {{math|''s''′ ≠ ''s''}}, and {{math|''s''}} is given by velocity addition,
<math display="block">s = \frac{s' + V}{1+{s'V\over c^2}}.</math>

The calculation is then essentially the same, except that here it is easier carried out upside down with {{math|1=''τ'' = {{sfrac|1|''ν''}}}} instead of {{mvar|ν}}. One finds

{{Equation box 1|indent =:|equation =
<math>\tau= {1 \over \gamma_{_V}\nu'}\left(\frac{1}{1+{V\over s'}}\right), \quad \nu = \gamma_{_V}\nu'\left(1+{V\over s'}\right)</math>
|cellpadding=6|border|border colour = #0073CF|bgcolor=#F9FFF7}}

{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=Details in derivation}}
<math display="block">\begin{align}{L\over -s} &= \frac{\left(\frac{-s'-V}{1+{s'V\over c^2}} + V\right) T} {\frac{-s'-V}{1+{s'V\over c^2}}}\\

&={\gamma_{_V} \over \nu'}\frac{-s'-V + V(1+{s'V\over c^2})}{-s'-V}\\
&={\gamma_{_V} \over \nu'}\left(\frac{s'\left(1-{V^2\over c^2}\right)}{s'+V}\right)\\
&={\gamma_{_V} \over \nu'}\left(\frac{s'\gamma^{-2}}{s'+V}\right)\\
&={1 \over \gamma_{_V}\nu'}\left(\frac{1}{1+{V\over s'}}\right).\\
\end{align}</math>
----
{{Hidden end}}

Observe that in the typical case, the {{math|''s''′}} that enters is ''negative''. The formula has general validity though.<ref group=nb>Note that {{math|''s''′}} is negative in the sense for which that the problem is set up, i.e. emitter with ''positive'' velocity fires ''fast'' bullets  ''towards'' observer in unprimed system. The convention is that {{math|−''s'' > ''V''}} should yield ''positive'' frequency in accordance with the result for the ultimate velocity, {{math|1=''s'' = −''c''}}. Hence the minus sign is a convention, but a very natural convention, to the point of being canonical.{{pb}}The formula may also result in negative frequencies. The interpretation then is that the bullets are approaching from the negative {{math|''x''}}-axis. This may have two causes. The emitter can have large positive velocity and be firing slow bullets. It can also be the case that the emitter has small negative velocity and is firing fast bullets. But if the emitter has a large negative velocity and is firing slow bullets, the frequency is again positive.{{pb}}For some of these combination to make sense, it must be required that the emitter has been firing bullets for sufficiently long time, in the limit that the {{math|''x''}}-axis at any instant has equally spaced bullets everywhere.</ref> When {{math|1=''s''′ = −''c''}}, the formula reduces to the formula calculated directly for light waves above,

<math display="block">\nu = \nu'\gamma_{_V}(1-\beta) = \nu'\frac{1-\beta}{\sqrt{1-\beta}\sqrt{1+\beta}}=\nu'\sqrt{\frac{1-\beta}{1+\beta}}\,.</math>

If the emitter is not firing bullets in empty space, but emitting waves in a medium, then the ''formula still applies'', but now, it may be necessary to first calculate {{math|''s''′}} from the velocity of the emitter relative to the medium.

Returning to the case of a light emitter, in the case the observer and emitter are not collinear, the result has little modification,<ref name=Einstein_1905 /><ref>{{harvnb|Lerner|Trigg|1991|p=259}}</ref><ref>{{harvnb|Parker|1993|p=312}}</ref>
<math display="block">\nu = \gamma_{_V}\nu' \left(1+\frac{V}{s'}\cos\theta\right),</math>
where {{mvar|θ}} is the angle between the light emitter and the observer. This reduces to the previous result for collinear motion when {{math|1=''θ'' = 0}}, but for transverse motion corresponding to {{math|1=''θ'' = ''π''/2}}, the frequency is shifted by the [[Lorentz factor]]. This does not happen in the classical optical Doppler effect.

== Hyperbolic geometry ==
[[File:sinh cosh tanh.svg|thumb|The functions <span style="color:#b30000;">sinh</span>, <span style="color:#00b300;">cosh</span> and <span style="color:#0000b3;">tanh</span>. The function [[Hyperbolic function|tanh]] relates the rapidity {{math|−∞ < ''ς'' < +∞}} to relativistic velocity {{math|−1 < ''β'' < +1}}.]]

Associated to the relativistic velocity <math>\boldsymbol \beta</math> of an object is a quantity <math>\boldsymbol{\zeta}</math> whose norm is called [[rapidity]]. These are related through
<math display="block">\mathfrak{so}(3,1) \supset \mathrm{span}\{K_1, K_2, K_3\} \approx \mathbb{R}^3 \ni \boldsymbol{\zeta} = \boldsymbol{\hat{\beta}} \tanh^{-1}\beta, \quad \boldsymbol{\beta} \in \mathbb{B}^3,</math>
where the vector <math>\boldsymbol \zeta</math> is thought of as being [[Cartesian coordinates]] on a 3-dimensional subspace of the [[Lie algebra]] <math>\mathfrak{so}(3, 1)</math> of the Lorentz group spanned by the [[Representation theory of the Lorentz group#Conventions and Lie algebra bases|boost generators]] <math>K_1, K_2, K_3</math>. This space, call it ''rapidity space'', is [[isomorphic]] to {{math|ℝ<sup>3</sup>}} as a vector space, and is mapped to the open unit ball,
<math> \mathbb B^3</math>, ''velocity space'', via the above relation.<ref>{{harvnb|Jackson|1999|p=547}}</ref> The addition law on collinear form coincides with the law of addition of [[hyperbolic function|hyperbolic tangents]]
<math display="block">\tanh(\zeta_v + \zeta_{u'}) = {\tanh \zeta_v + \tanh \zeta_{u'} \over 1+ \tanh \zeta_v \tanh \zeta_{u'}}</math>
with
<math display="block">\frac{v}{c} = \tanh \zeta_v \ , \quad \frac{u'}{c} = \tanh \zeta_{u'} \ , \quad\, \frac{u}{c} = \tanh(\zeta_v + \zeta_{u'}).</math>

The [[line element]] in velocity space <math>\mathbb B^3</math> follows from the expression for ''relativistic relative velocity'' in any frame,<ref>{{harvnb|Landau|Lifshitz|2002|loc=Equation 12.6}}</ref>
<math display="block">v_{r} = \frac{\sqrt{(\mathbf{v_1}-\mathbf{v_2})^2 - (\mathbf{v_1} \times \mathbf{v_2})^2}}{1 - \mathbf{v_1}\cdot\mathbf{v_2}},</math>
where the speed of light is set to unity so that <math>v_i</math> and <math>\beta_i</math> agree. It this expression, <math>\mathbf{v}_1</math> and <math>\mathbf{v}_2</math> are velocities of two objects in any one given frame. The quantity <math>v_r</math> is the speed of one or the other object ''relative'' to the other object as seen ''in the given frame''. The expression is Lorentz invariant, i.e. independent of which frame is the given frame, but the quantity it calculates is ''not''. For instance, if the given frame is the rest frame of object one, then <math>v_r = v_2</math>.

The line element is found by putting <math>\mathbf v_2 = \mathbf v_1 + d\mathbf v</math> or equivalently <math>\boldsymbol \beta_2 = \boldsymbol \beta_1 + d\boldsymbol \beta</math>,<ref>{{harvnb|Landau|Lifshitz|2002|loc=Problem p. 38}}</ref>
<math display="block">dl_\boldsymbol{\beta}^2 = \frac{d\boldsymbol \beta^2 - (\boldsymbol \beta \times d\boldsymbol \beta)^2}{(1-\beta^2)^2} = \frac{d\beta^2}{(1-\beta^2)^2} + \frac{\beta^2}{1-\beta^2}(d\theta^2 + \sin^2\theta d\varphi^2),</math>
with {{math|''θ''}} and {{mvar|φ}} the usual spherical angle coordinates for <math>\boldsymbol \beta</math> taken in the {{math|''z''}}-direction. Now introduce {{mvar|ζ}} through
<math display="block">\zeta = |\boldsymbol \zeta| = \tanh^{-1}\beta,</math>
and the line element on rapidity space <math>\mathbb R^3</math> becomes
<math display="block">dl_{\boldsymbol \zeta}^2 = d\zeta^2 + \sinh^2\zeta(d\theta^2 + \sin^2\theta d\varphi^2).</math>

=== Relativistic particle collisions ===
In scattering experiments the primary objective is to measure the [[Scattering cross section|invariant scattering cross section]]. This enters the formula for scattering of two particle types into a final state <math>f</math> assumed to have two or more particles,<ref>{{harvnb|Cannoni|2017|p=1}}</ref>
<math display="block">dN_f = R_f \, dV \, dt = \sigma F \, dV \, dt</math>
or, in most textbooks,
<math display="block">dN_f = \sigma n_1 n_2 v_r \, dV \, dt</math>
where
* <math>dVdt</math> is spacetime volume. It is an invariant under Lorentz transformations.
* <math>dN_f</math> is the total number of reactions resulting in final state <math>f</math> in  spacetime volume <math>dVdt</math>. Being a number, it is invariant when the ''same'' spacetime volume is considered.
* <math>R_f = F\sigma</math> is the number of reactions resulting in final state <math>f</math> per unit spacetime, or ''reaction rate''. This is invariant.
* <math>F = n_1n_2v_{r}</math> is called the ''incident flux''. This is required to be invariant, but isn't in the most general setting.
* <math>\sigma</math> is the scattering cross section. It is required to be invariant.
* <math>n_1, n_2 </math> are the particle densities in the incident beams. These are not invariant as is clear due to [[length contraction]].
* <math>v_{r} = |\mathbf v_2 - \mathbf v_1|</math> is the ''relative speed'' of the two incident beams. This ''cannot'' be invariant since <math>F = n_1n_2v_{r}</math> is required to be so.

The objective is to find a correct expression for ''relativistic relative speed'' <math>v_\text{rel}</math> and an invariant expression for the incident flux.

Non-relativistically, one has for relative speed <math>v_r = |\mathbf v_2 - \mathbf v_1|</math>. If the system in which velocities are measured is the rest frame of particle type <math>1</math>, it is required that <math>v_\text{rel} = v_r = |\mathbf v_2|.</math> Setting the speed of light <math>c = 1</math>, the expression for <math>v_\text{rel}</math> follows immediately from the formula for the norm (second formula) in the ''general configuration'' as<ref name="Cannoni 2017 4">{{harvnb|Cannoni|2017|p=4}}</ref><ref>{{harvnb|Landau|Lifshitz|2002}}</ref>
<math display="block">v_\text{rel} =\frac{\sqrt{(\mathbf{v_1}-\mathbf{v_2})^2 - (\mathbf{v_1} \times \mathbf{v_2})^2}}{1 - \mathbf{v_1}\cdot\mathbf{v_2}}.</math>

The formula reduces in the classical limit to <math>v_r = |\mathbf v_1 - \mathbf v_2|</math> as it should, and gives the correct result in the rest frames of the particles. The relative velocity is ''incorrectly given'' in most, perhaps ''all'' books on particle physics and quantum field theory.<ref name="Cannoni 2017 4" /> This is mostly harmless, since if either one particle type is stationary or the relative motion is collinear, then the right result is obtained from the incorrect formulas. The formula is invariant, but not manifestly so. It can be rewritten in terms of four-velocities as
<math display="block">v_\text{rel} = \frac{\sqrt{(u_1 \cdot u_2)^2 - 1}}{u_1 \cdot u_2}.</math>

The correct expression for the flux, published by [[Christian Møller]]<ref>{{harvnb|Møller|1945}}</ref> in 1945, is given by<ref>{{harvnb|Cannoni|2017|p=8}}</ref>
<math display="block">F = n_1n_2\sqrt{(\mathbf v_1 - \mathbf v_2)^2 - (\mathbf v_1 \times \mathbf v_2)^2} \equiv n_1n_2\bar v.</math>

One notes that for collinear velocities, <math>F = n_1n_2|\mathbf v_2 - \mathbf v_1| = n_1n_2v_r</math>. In order to get a ''manifestly'' Lorentz invariant expression one writes <math>J_i = (n_i, n_i\mathbf v_i)</math> with <math>n_i = \gamma_i n_i^0</math>, where <math>n_i^0</math> is the density in the rest frame, for the individual particle fluxes and arrives at<ref name="Cannoni 2017 13">{{harvnb|Cannoni|2017|p=13}}</ref>
<math display="block">F = (J_1 \cdot J_2) v_\text{rel}.</math>

In the literature the quantity <math>\bar v</math> as well as <math>v_r</math> are both referred to as the relative velocity. In some cases (statistical physics and dark matter literature), <math>\bar v</math> is referred to as the ''Møller velocity'', in which case <math>v_r</math> means relative velocity. The true relative velocity is at any rate <math>v_\text{rel}</math>.<ref name="Cannoni 2017 13"/> The discrepancy between <math>v_\text{rel}</math> and <math>v_r</math> is relevant though in most cases velocities are collinear. At [[LHC]] the crossing angle is small, around {{val|300|u=urad}}, but at
the old Intersecting Storage Ring at [[CERN]], it was about 18°.<ref>{{harvnb|Cannoni|2017|p=15}}</ref>

==With rapidity==

[[File:Minkowski2.png|thumb|right|300px|[[Light cone]] and [[conjugate hyperbola]]s in Minkowski (1908)]]
{{main|Rapidity}}

For collinear velocities interpreted as rapidity, the sum formula is simple addition. According to Minkowski, the time-like vector (''zeitartiger Vektor'') for a given duration lies on a hyperbola. Since the hyperbola is traced by a [[hyperbolic angle]], and velocity determines a point on the hyperbola, there is a hyperbolic angle called ''rapidity'' associated with a particular velocity. For a unit of duration, the [[unit hyperbola]] provides a reference, where hyperbolic angle forms a [[one-parameter group]] isomorphic to the real number line under addition. The form of the formula for velocity addition can then be accounted in terms of the [[hyperbolic tangent]] function tanh which takes hyperbolic angle (rapidity) as an argument. In fact, the hyperbolic tangent of rapidity is the ratio of velocity to the speed of light in vacuum.<ref>{{Cite book |last=Silberstein |first=Ludwik |author-link=Ludwik Silberstein |year=1914 |title=The Theory of Relativity |location=London |publisher=Macmillan & Co. | page =179|url=https://archive.org/details/theoryofrelativi00silbrich/page/178/mode/2up?q=rapidity}}</ref>

== See also ==
* [[Hyperbolic law of cosines]]
* [[Biquaternion]]
* [[Relative velocity]]

== Remarks ==
{{reflist|group=nb}}

== Notes ==
{{reflist|30em}}

== References ==
{{refbegin}}
* {{cite journal|last=Cannoni|first=Mirco|title=Lorentz invariant relative velocity and relativistic binary collisions|journal=International Journal of Modern Physics A|year=2017|volume=32|issue=2n03|pages=1730002|doi=10.1142/S0217751X17300022|via=[[World Scientific]]|arxiv=1605.00569|bibcode=2017IJMPA..3230002C|s2cid=119223742}}
* {{cite journal|last=Einstein|first=A.|author-link=Albert Einstein|year=1905|trans-title=Zur Elektrodynamik bewegter Körper|title=On the Electrodynamics of moving bodies|journal=Annalen der Physik|issue=322|volume=10|pages=891–921|url=http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1905_relativity.pdf|doi=10.1002/andp.19053221004|bibcode=1905AnP...322..891E|doi-access=free}}
* {{cite book |doi=10.1016/C2013-0-05319-4 |title=The Theory of Space, Time and Gravitation |date=1964 |isbn=978-0-08-010061-6 |last1=Fock |first1=V }}
* {{cite book|title=Special Relativity|first=A.P.|last=French|publisher=[[W.W. Norton & Company]]|series = MIT Introductory Physics Series|year=1968|isbn=978-0-393-09793-1}}
* {{cite book|title=Physical applications of homogeneous balls|first1=Yaakov|last1=Friedman|first2=Tzvi|last2=Scarr|publisher=Birkhäuser|year=2005|isbn=978-0-8176-3339-4|pages=1–21|url=https://books.google.com/books?id=D_zYVBu0KAIC}}
* {{cite book|first=J. D.|last=Jackson|author-link=John David Jackson (physicist)|title=Classical Electrodynamics|edition=3d|year=1999|orig-year=1962|isbn=978-0-471-30932-1|publisher=[[John Wiley & Sons]]|chapter=Chapter 11}} (graduate level)
* {{cite book|first1=D.|last1=Kleppner|author-link1=Daniel Kleppner|first2=R. J.|last2=Kolenkow|author-link2=Robert J. Kolenkow|title=An Introduction to Mechanics|year=1978|orig-year=1973|isbn=978-0-07-035048-9|publisher=[[McGraw-Hill]]|location=London|url=https://archive.org/details/introductiontome00dani}} (introductory level)
* {{cite book|last1=Landau|first1=L.D.|author-link1=Lev Landau|last2=Lifshitz|first2=E.M.|author-link2=Evgeny Lifshitz|title=The Classical Theory of Fields|series=Course of Theoretical Physics|volume=2|edition=4th|publisher=[[Butterworth–Heinemann]]|isbn=0-7506-2768-9|year=2002|orig-year=1939}} (graduate level)
* {{cite book|first1=R.G.|last1=Lerner|author1-link=Rita G. Lerner|first2=G.L.|last2=Trigg|title=Encyclopaedia of Physics|publisher=VHC Publishers, Springer|edition=2nd|year=1991|isbn=978-0-07-025734-4}}
* {{cite book|last=Mermin|first=N. D.|year=2005|title=It's About Time: Understanding Einstein's Relativity|publisher=Princeton University Press|isbn=978-0-691-12201-4|url-access=registration|url=https://archive.org/details/itsabouttimeunde0000merm}}
* {{cite journal|last=Mocanu|first=C.I.|title=On the relativistic velocity composition paradox and the Thomas rotation|journal=Found. Phys. Lett.|volume=5|issue=5|year=1992|pages=443–456|doi=10.1007/BF00690425|issn=0894-9875|bibcode=1992FoPhL...5..443M|s2cid=122472788}}
* {{cite journal|first=C.|last=Møller|journal=D. KGL Danske Vidensk. Selsk. Mat.-Fys. Medd.|volume=23|issue=1|year=1945|title=General properties of the characteristic matrix in the theory of elementary particles I|author-link=Christian Møller|url=http://gymarkiv.sdu.dk/MFM/kdvs/mfm%2020-29/mfm-23-1.pdf}}
* {{cite book|first1=S. P.|last1=Parker|title=McGraw Hill Encyclopaedia of Physics|publisher=McGraw Hill|edition=2nd|year=1993|isbn=978-0-07-051400-3|url=https://archive.org/details/mcgrawhillencycl1993park}}
* {{cite book|last=Sard|first=R. D.|title=Relativistic Mechanics – Special Relativity and Classical Particle Dynamics|year=1970|publisher=W. A. Benjamin|location=New York|isbn=978-0-8053-8491-8|url-access=registration|url=https://archive.org/details/relativisticmech0000sard}}
* {{cite book|first1=R. U.|last1=Sexl|first2=H. K.|last2=Urbantke|title=Relativity, Groups Particles. Special Relativity and Relativistic Symmetry in Field and Particle Physics|year=2001|orig-year=1992|pages=38–43|publisher=Springer|isbn=978-3-211-83443-5|url=https://books.google.com/books?id=iyj0CAAAQBAJ&q=sexl+relativity}}
* {{cite book|first1=P. |last1=Tipler|first2=G. |last2=Mosca|title=Physics for Scientists and Engineers|edition=6th|year=2008|pages=1328–1329|publisher=Freeman|isbn=978-1-4292-0265-7|url=https://books.google.com/books?id=AttDBYgLeZkC&q=tipler+physics}}
* {{cite journal|last=Ungar|first=A. A.|title=Thomas rotation and parameterization of the Lorentz group|journal=Foundations of Physics Letters|volume=1|issue=1|pages=57–81|year=1988|doi=10.1007/BF00661317|issn=0894-9875|bibcode = 1988FoPhL...1...57U |s2cid=121240925}}

; Historical :
* {{cite journal|last=Bradley|first=James|author-link=James Bradley|title=A Letter from the Reverend Mr. James Bradley Savilian Professor of Astronomy at Oxford, and F.R.S. to Dr.Edmond Halley Astronom. Reg. &c. Giving an Account of a New Discovered Motion of the Fix'd Stars|year=1727–1728|doi=10.1098/rstl.1727.0064|journal=Phil. Trans. R. Soc.|volume=35|issue=399–406|pages=637–661|bibcode=1727RSPT...35..637B|type=PDF|doi-access=free}}
* {{citation|last=Doppler|first=C.|author-link=Christian Doppler|year=1903|orig-year=1842|title=Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels|trans-title=About the coloured light of the binary stars and some other stars of the heavens|publisher=Abhandlungen der Königl. Böhm. Gesellschaft der Wissenschaften|volume=2|issue=V|pages=465–482|location=Prague|language=de|url=https://archive.org/stream/ueberdasfarbigel00doppuoft#page/n5/mode/2up}}
* {{cite journal|last=Fizeau|first=H.|author-link=Hippolyte Fizeau|title=Sur les hypothèses relatives à l'éther lumineux|journal=Comptes Rendus|trans-title=The Hypotheses Relating to the Luminous Aether|volume=33|year=1851f|pages=349–355|url=http://gallica.bnf.fr/ark:/12148/bpt6k29901/f351.chemindefer|language=fr}}
* {{cite journal|author=Fizeau |first=H.|title=The Hypotheses Relating to the Luminous Aether|journal=Philosophical Magazine|volume=2|year=1851e|pages=568–573|title-link=s:The Hypotheses Relating to the Luminous Aether}}
* {{cite journal|author=Fizeau |first=H.|title=Sur les hypothèses relatives à l'éther lumineux|trans-title=The Hypotheses Relating to the Luminous Aether|journal=Ann. Chim. Phys.|volume=57|year=1859|pages=385–404 |url=http://gallica.bnf.fr/ark:/12148/bpt6k347981/f381.table|language=fr}}
* {{cite journal|author=Fizeau |first=H.|title=On the Effect of the Motion of a Body upon the Velocity with which it is traversed by Light|journal=Philosophical Magazine|volume=19|year=1860|pages=245–260|title-link=s:On the Effect of the Motion of a Body upon the Velocity with which it is traversed by Light}}
* {{cite book|last=Galilei|first=G.|author-link=Galileo Galilei|title=Dialogue Concerning the Two Chief World Systems|trans-title=Dialogo sopra i due massimi sistemi del mondo|others=Stillman Drake (Editor, Translator), Stephen Jay Gould (Editor), J. L. Heilbron (Introduction), Albert Einstein (Foreword)|year=2001|orig-year=1632|isbn=978-0-375-75766-2|publisher=Modern Library}}
* {{cite book|last=Galilei|first=G.|year=1954|orig-year=1638|title=Dialogues Concerning Two New Sciences|trans-title= Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze|isbn=978-1-4209-3815-9|publisher=Digiread.com|others=Henry Crew, Alfonso de Salvio (Translators)}}
* {{cite journal|first1=G. N.|last1=Lewis|author-link1=Gilbert N. Lewis|first2=R. C.|last2=Tolman|author-link2=Richard C. Tolman|title=The Principle of Relativity, and Non-Newtonian Mechanics|journal=Phil. Mag.|series=6|volume=18|issue=106|doi=10.1080/14786441008636725|pages=510–523|year=1909|url=https://zenodo.org/record/1430872}} [[s:The Principle of Relativity, and Non-Newtonian Mechanics|Wikisource version]]
{{refend}}

== External links ==
{{refbegin}}
* {{cite journal|last=Sommerfeld|first=A.|author-link=Arnold Sommerfeld|year=1909|url=https://en.wikisource.org/?curid=707458|title=On the Composition of Velocities in the Theory of Relativity|trans-title=Über die Zusammensetzung der Geschwindigkeiten in der Relativtheorie|journal=Verh. Dtsch. Phys. Ges.|volume=21|pages=577–582}}
{{refend}}

[[Category:Special relativity]]
[[Category:Equations]]
[[Category:Velocity|Addition formula]]
[[Category:Kinematics]]