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 a 3-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different frames. Accompanying velocity addition is a kinematic effect known as Thomas precession, whereby successive non-collinear Lorentz boosts 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.[1]

## History

The speed of light in the fluid is slower than the speed of light in vacuum, and it changes if the fluid is moving along with the light. In 1851, Fizeau measured the speed of light in a fluid moving parallel to the light using a Michelson 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 Vc as is described below. Fizeau's result led physicists to accept the empirical validity of the rather unsatisfactory theory by Fresnel that a fluid moving with respect to the stationary aether partially drags light with it, i.e. the speed is c + (1 − 1n2)V instead of c + V, where c is the speed of light in the aether, and 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 (V in the x-direction) for the addition of relativistic velocities.[2] The issues involving aether were, gradually over the years, settled in favor of special relativity.

## Galilean relativity

It was observed by Galilei that a person on a uniformly moving ship has the impression of being at rest and sees a heavy body falling vertically downward.[3] This observation is now regarded as the first clear statement of the principle of mechanical relativity. Galilei 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.[4] 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. Galilei), B (e.g. ship), C (e.g. falling body on ship) the velocity vector s of C relative to A (velocity of falling object as Galilei sees it) is the sum of the velocity v of C relative to B (velocity of falling object relative to ship) plus the velocity u of B relative to A (ships velocity relative to the shore). The addition here is the vector addition of vector algebra and the resulting velocity is usually represented in the form

${\displaystyle \mathbf {s} =\mathbf {v} +\mathbf {u} .}$

The cosmos of Galileo consists of absolute space and time and the addition of velocities corresponds to composition of Galilean transformations. 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 (e.g. a cannonball fired horizontally away from, or towards Galilei) on the ship relative to the shore is given by

${\displaystyle s={v+u \over 1+(vu/c^{2})}.}$

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,[5]

${\displaystyle {c-s \over c+s}=\left({c-u \over c+u}\right)\left({c-v \over c+v}\right).}$

The cosmos of special relativity consists of Minkowski spacetime and the addition of velocities corresponds to composition of Lorentz transformations. In the special theory of relativity Newtonian mechanics is modified into relativistic mechanics.

## Standard configuration

The formulas for boosts in the standard configuration follow most straightforwardly from taking differentials of the inverse Lorentz boost in standard configuration.[6][7] If the primed frame is travelling with speed V in the positive x-direction relative to the unprimed frame then the formulas are

${\displaystyle dx=\gamma _{_{V}}(dx'+Vdt'),\quad dy=dy',\quad dz=dz',\quad dt=\gamma _{_{V}}\left(dt'+{\frac {V}{c^{2}}}dx'\right),}$

where

${\displaystyle \gamma _{_{V}}={\frac {1}{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}}.}$

Divide the first three equations by the fourth,

${\displaystyle {\frac {dx}{dt}}={\frac {\gamma _{_{V}}(dx'+Vdt')}{\gamma _{_{V}}(dt'+{\frac {V}{c^{2}}}dx')}},\quad {\frac {dy}{dt}}={\frac {dy'}{\gamma _{_{V}}(dt'+{\frac {V}{c^{2}}}dx')}},\quad {\frac {dz}{dt}}={\frac {dz'}{\gamma _{_{V}}(dt'+{\frac {V}{c^{2}}}dx')}},}$

or

${\displaystyle {\frac {dx}{dt}}={\frac {dx'+Vdt'}{dt'(1+{\frac {V}{c^{2}}}{\frac {dx'}{dt'}})}},\quad {\frac {dy}{dt}}={\frac {dy'}{\gamma _{_{V}}\ dt'(1+{\frac {V}{c^{2}}}{\frac {dx'}{dt'}})}},\quad {\frac {dz}{dt}}={\frac {dz'}{\gamma _{_{V}}\ dt'(1+{\frac {V}{c^{2}}}{\frac {dx'}{dt'}})}},}$

which is

 Transformation of velocity (Cartesian components) ${\displaystyle v_{x}={\frac {v_{x}'+V}{1+{\frac {V}{c^{2}}}v_{x}'}},\quad v_{y}={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}v_{y}'}{1+{\frac {V}{c^{2}}}v_{x}'}},\quad v_{z}={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}v_{z}'}{1+{\frac {V}{c^{2}}}v_{x}'}}.}$

If coordinates are chosen so that all velocities lie in a (common) xy plane, then velocities may be expressed as

${\displaystyle v_{x}=v\cos \theta ,v_{y}=v\sin \theta ,\quad v_{x}'=v'\cos \theta ',\quad v_{y}'=v'\sin \theta ',}$

(see polar coordinates) and one finds[2][8]

 Transformation of velocity (Plane polar components) {\displaystyle {\begin{aligned}v&={\frac {\sqrt {v'^{2}+V^{2}+2Vv'\cos \theta '-({\frac {Vv'\sin \theta '}{c}})^{2}}}{1+{\frac {V}{c^{2}}}v'\cos \theta '}},\\\tan \theta &={\frac {v_{y}}{v_{x}}}={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}v_{y}'}{v_{x}'+V}}={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}v'\sin \theta '}{v'\cos \theta '+V}}.\end{aligned}}}

The proof as given is highly formal. There are other more involved proofs that may be more enlightening, such as the one below.

## General configuration

Decomposition of 3-velocity v into parallel and perpendicular components, and calculation of the components. The procedure for v is identical.

Starting from the expression in coordinates for V parallel to the 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 v in the unprimed frame and v in the primed frame, and split them into components parallel ( ∥ ) and perpendicular (⊥) to the relative velocity vector V (see hide box below) thus

${\displaystyle \mathbf {v} =\mathbf {v} _{\parallel }+\mathbf {v} _{\perp },\quad \mathbf {v} '=\mathbf {v} '_{\parallel }+\mathbf {v} '_{\perp },}$

then with the usual Cartesian unit basis vectors ex, ey, ez, set the velocity in the unprimed frame to be

${\displaystyle \mathbf {v} _{\parallel }=v_{x}\mathbf {e} _{x},\quad \mathbf {v} _{\perp }=v_{y}\mathbf {e} _{y}+v_{z}\mathbf {e} _{z},\quad \mathbf {V} =V\mathbf {e} _{x},}$

which gives, using the results for the standard configuration,

${\displaystyle \mathbf {v} _{\parallel }={\frac {\mathbf {v} _{\parallel }'+\mathbf {V} }{1+{\frac {\mathbf {V} \cdot \mathbf {v} _{\parallel }'}{c^{2}}}}},\quad \mathbf {v} _{\perp }={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}\mathbf {v} _{\perp }'}{1+{\frac {\mathbf {V} \cdot \mathbf {v} _{\parallel }'}{c^{2}}}}}.}$

where · is the dot product. Since these are vector equations, they still have the same form for V in any direction. The only difference from the coordinate expressions is that the above expressions refers to vectors, not components.

One obtains

${\displaystyle \mathbf {v} =\mathbf {v} _{\parallel }+\mathbf {v} _{\perp }={\frac {1}{1+{\frac {\mathbf {V} \cdot \mathbf {v} '}{c^{2}}}}}\left[\alpha _{V}\mathbf {v} '+\mathbf {V} +(1-\alpha _{V}){\frac {(\mathbf {V} \cdot \mathbf {v} ')}{V^{2}}}\mathbf {V} \right]\equiv \mathbf {V} \oplus \mathbf {v} ',}$

where αV = 1/γV 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.

In order to facilitate generalization and to avoid proliferation of primes, change notation of V to u, and v to v. Thus[10][nb 1]

 {\displaystyle {\begin{aligned}\mathbf {u} \oplus \mathbf {v} &={\frac {1}{1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}}}\left[\mathbf {u} +{\frac {\mathbf {v} }{\gamma _{u}}}+{\frac {1}{c^{2}}}{\frac {\gamma _{u}}{1+\gamma _{u}}}(\mathbf {u} \cdot \mathbf {v} )\mathbf {u} \right]\\&={\frac {1}{1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}}}\left[\mathbf {u} +\mathbf {v} +{\frac {1}{c^{2}}}{\frac {\gamma _{u}}{1+\gamma _{u}}}\mathbf {u} \times (\mathbf {u} \times \mathbf {v} )\right],\end{aligned}}}

where the last expression is by the standard vector analysis formula u × (u × v) = (uv)u − (uu)v. The first expression extends to any number of spatial dimensions, but the cross product is defined in three dimensions only. The initial requirement that the first operand refers to the velocity of a frame is nowhere used, and the objects A, B, C with B having velocity u relative to A and C having velocity v relative to B 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

${\displaystyle (\lambda \mathbf {u} )\oplus (\mu \mathbf {v} )\neq \lambda \mu (\mathbf {u} \oplus \mathbf {v} ),}$

for any real numbers λ and μ, although it is true that

${\displaystyle (-\mathbf {u} )\oplus (-\mathbf {v} )=-(\mathbf {u} \oplus \mathbf {v} ),}$

Also, due to the last terms, is in general neither commutative

${\displaystyle \mathbf {u} \oplus \mathbf {v} \neq \mathbf {v} \oplus \mathbf {u} ,}$

nor associative

${\displaystyle \mathbf {u} \oplus (\mathbf {v} \oplus \mathbf {w} )\neq (\mathbf {u} \oplus \mathbf {v} )\oplus \mathbf {w} .}$

It deserves special mention that if u and v refer to boost velocities, then both uv, and vu are correct expressions for the combined boost velocity. They are just given in different coordinate systems, the unprimed and the (would be) doubly primed respectively in the old notation, related by a rotation. This is the phenomenon of Thomas precession, and is not dealt with further here.

The norm is given by[11]

${\displaystyle |\mathbf {u} \oplus \mathbf {v} |^{2}={\frac {1}{\left(1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}\right)^{2}}}\left[\left(\mathbf {u} +\mathbf {v} \right)^{2}-{\frac {1}{c^{2}}}\left(\mathbf {u} \times \mathbf {v} \right)^{2}\right]=|\mathbf {v} \oplus \mathbf {u} |^{2}.}$

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 factor for the combined velocity is computed as

${\displaystyle \gamma _{\mathbf {u} \oplus \mathbf {v} }=\left[1-{\frac {1}{c^{2}}}{\frac {1}{(1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}})^{2}}}\left((\mathbf {u} +\mathbf {v} )^{2}-{\frac {1}{c^{2}}}(u^{2}v^{2}-(\mathbf {u} \cdot \mathbf {v} )^{2})\right)\right]^{-{\frac {1}{2}}}=\gamma _{u}\gamma _{v}\left(1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}\right).}$

### Notational conventions

Notations and conventions for the velocity addition vary from author to author. Different symbols may be used for the operation, and the operands may be switched for the same expression. Since the velocity addition is non-commutative, we cannot naïvely switch the operands without changing the result.

Left-to-right ordering of operands

${\displaystyle \mathbf {u} \oplus \mathbf {v} ={\frac {1}{1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}}}\left[\mathbf {v} +\mathbf {u} +{\frac {1}{c^{2}}}{\frac {\gamma _{\mathbf {u} }}{\gamma _{\mathbf {u} }+1}}\mathbf {u} \times (\mathbf {u} \times \mathbf {v} )\right]}$

Ungar (1988, 1989)

${\displaystyle \mathbf {u} *\mathbf {v} ={\frac {1}{1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}}}\left[\mathbf {v} +\mathbf {u} +{\frac {1}{c^{2}}}{\frac {\gamma _{\mathbf {u} }}{\gamma _{\mathbf {u} }+1}}\mathbf {u} \times (\mathbf {u} \times \mathbf {v} )\right]}$
Right-to-left ordering of operands

Sexl & Urbantke (1992)

${\displaystyle \mathbf {v} \circ \mathbf {u} ={\frac {1}{1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}}}\left[{\frac {\mathbf {v} }{\gamma _{\mathbf {u} }}}+\mathbf {u} +{\frac {1}{c^{2}}}{\frac {\gamma _{\mathbf {u} }}{\gamma _{\mathbf {u} }+1}}(\mathbf {v} \cdot \mathbf {u} )\mathbf {u} \right]}$

## 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 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 formalism. This involves experimentalist bouncing off relativistic billiard balls from each other. This is not detailed here, but see for reference Lewis & Tolman (1909) Wikisource version (primary source) and Sard (1970, Section 3.2).

### Fizeau experiment

Hippolyte Fizeau (1819 – 1896), a French physicist, was in 1851 the first to measure the speed of light in flowing water.
Main article: Fizeau experiment

When light propagates in a medium, its speed is reduced, in the rest frame of the medium, to cm = cnm, where nm is the index of refraction of the medium m. The speed of light in a medium uniformly moving with speed V in the positive x-direction as measured in the lab frame is given directly by the velocity addition formulas. For the forward direction (standard configuration, drop index m on n) one gets,[12]

{\displaystyle {\begin{aligned}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}}(1+{\frac {nV}{c}}){\frac {1}{1+{\frac {V}{nc}}}}=({\frac {c}{n}}+V)\left(1-{\frac {V}{nc}}+\left({\frac {V}{nc}}\right)^{2}-\cdots \right).\end{aligned}}}

Collecting the largest contributions explicitly,

${\displaystyle c_{m}={\frac {c}{n}}+V(1-{\frac {1}{n^{2}}}-{\frac {V}{nc}}+\cdots ).}$

Fizeau found the first three terms.[13][14] The classical result is the first two terms.

### Aberration of light

Main article: 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, v′ = v = c, and insertion in the formula for tan θ yields

${\displaystyle \tan \theta ={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}c\sin \theta '}{c\cos \theta '+V}}={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}\sin \theta '}{\cos \theta '+{\frac {V}{c}}}}.}$

For this case one may also compute sin θ and cos θ from the standard formulae,[15]

{\displaystyle {\begin{aligned}\sin \theta &={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}\sin \theta '}{1+{\frac {V}{c}}\cos \theta '}},\end{aligned}}}
${\displaystyle \cos \theta ={\frac {{\frac {V}{c}}+\cos \theta '}{1+{\frac {V}{c}}\cos \theta '}},}$
James Bradley (1693 – 1762) FRS, provided an explanation of aberration of light correct at the classical level,[16] at odds with the later theories prevailing in the nineteenth century based on the existence of aether.

the trigonometric manipulations essentially being identical in the cos case to the manipulations in the sin case. Consider the difference,

{\displaystyle {\begin{aligned}\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{aligned}}}

correct to order vc. Employ in order to make small angle approximations a trigonometric formula,

{\displaystyle {\begin{aligned}\sin \theta '-\sin \theta &=2\sin {\frac {1}{2}}(\theta '-\theta )\cos {\frac {1}{2}}(\theta +\theta ')\approx (\theta '-\theta )\cos \theta ',\end{aligned}}}

where cos1/2(θ + θ′) ≈ cos θ′, sin1/2(θθ′) ≈ 1/2(θθ′) were used.

Thus the quantity

${\displaystyle \Delta \theta \equiv \theta '-\theta ={\frac {V}{c}}\sin \theta ',}$

the classical aberration angle, is obtained in the limit Vc → 0.

### Relativistic Doppler shift

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.

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 x-direction, so subscripts on velocity components are dropped. In the observers frame, introduce the geometrical observation

${\displaystyle \lambda =-sT+VT=(-s+V)T}$

as the spatial distance, or wavelength, between two pulses (wave crests), where 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 τ, and its inverse ν = 1τ is the observed (temporal) frequency. The corresponding quantities in the emitters frame are endowed with primes.[17]

For light waves

${\displaystyle s=s'=-c,}$

and the observed frequency is[2][18][19]

${\displaystyle \nu ={-s \over \lambda }={-s \over (V-s)T}={c \over (V+c)\gamma _{_{V}}T'}=\nu '{\frac {c{\sqrt {1-{V^{2} \over c^{2}}}}}{c+V}}=\nu '{\sqrt {\frac {1-\beta }{1+\beta }}}\,.}$

where T = γVT is standard time dilation formula.

Suppose instead that the wave is not composed of light waves with speed c, but instead, for easy visualization, bullets fired from a relativistic machine gun, with velocity s in the frame of the emitter. Then, in general, the geometrical observation is precisely the same. But now, s′ ≠ s, and s is given by velocity addition,

${\displaystyle s={\frac {s'+V}{1+{s'V \over c^{2}}}}.}$

The calculation is then essentially the same, except that here it is easier carried out upside down with τ = 1ν instead of ν. One finds

 ${\displaystyle \tau ={1 \over \gamma _{_{V}}\nu '}\left({\frac {1}{1+{V \over s'}}}\right),\quad \nu =\gamma _{_{V}}\nu '\left(1+{V \over s'}\right)}$

Observe that in the typical case, the s that enters is negative. The formula has general validity though.[nb 2] When s′ = −c, the formula reduces to the formula calculated directly for light waves above,

${\displaystyle \nu =\nu '\gamma _{_{V}}(1-\beta )=\nu '{\frac {1-\beta }{{\sqrt {1-\beta }}{\sqrt {1+\beta }}}}=\nu '{\sqrt {\frac {1-\beta }{1+\beta }}}\,.}$

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 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,[2][20][21]

${\displaystyle \nu =\gamma _{_{V}}\nu '\left(1+{\frac {V}{s'}}\cos \theta \right),}$

where θ is the angle between the light emitter and the observer. This reduces to the previous result for collinear motion when θ = 0, but for transverse motion corresponding to θ = π/2, the frequency is shifted by the Lorentz factor. This does not happen in the classical optical Doppler effect.

## Hyperbolic geometry

The addition law on collinear form is also the law of addition of hyperbolic tangents

${\displaystyle \tanh(a+b)={\tanh a+\tanh b \over 1+\tanh a\tanh b}}$

where

${\displaystyle {v \over c}=\tanh a\ ,\quad {u \over c}=\tanh b\ ,\quad \,{s \over c}=\tanh(a+b),}$

which shows that the composition of collinear velocities is associative and commutative.

The quantities a and b (equal to the artanh of the velocities divided by c) are known as rapidities. The reason that the velocities are hyperbolic tangents is because the Lorentz transformation can be thought of as the application of a hyperbolic rotation through a hyperbolic angle which is the rapidity. Suppose the velocity of a line in space-time is the slope of the line, which is the hyperbolic tangent of the rapidity, just as the slope of the x-axis after a rotation is given by the tangent of the rotation angle. When a plane is successively rotated by two angles, the final rotation is by the sum of the two angles. So the final slope of the x-axis is the tangent of the sum of the two angles. In the same way, the slope of the time axis after two boosts is the hyperbolic tangent of the sum of the two rapidities.

The a and b of above can be thought of as being the radial coordinate on a 3-dimensional subspace with spherical coordinates, or the norm of a cartesian vector, of the Lie algebra of the Lorentz group spanned by the boost generators. This space is homeomorphic with 3, and is mapped to the open unit ball B3 via[22]

${\displaystyle {\mathfrak {so}}(3,1)\supset \mathrm {span} \{K_{1},K_{2},K_{3}\}\approx \mathbb {R} ^{3}\ni {\boldsymbol {\zeta }}={\boldsymbol {\hat {\beta }}}\tanh ^{-1}\beta ,\quad {\boldsymbol {\beta }}\in \mathbb {B} ^{3},}$

where ζ is a 3-vector in the boost subspace, expressed in cartesian coordinates, called the boost parameter and its norm the rapidity. The line element in 3 is given by[23]

${\displaystyle dl_{\boldsymbol {\zeta }}^{2}={\frac {d\mathbf {\zeta } ^{2}-({\boldsymbol {\zeta }}\times d{\boldsymbol {\zeta }})^{2}}{(1-\zeta ^{2})^{2}}}={\frac {d\zeta ^{2}}{1-\zeta ^{2}}}(d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}).}$

Now introduce β through

${\displaystyle \zeta =|{\boldsymbol {\zeta }}|=\tanh ^{-1}\beta ,}$

with θ and φ the usual spherical angle coordinates, and the line element on the open unit ball becomes

${\displaystyle dl_{\boldsymbol {\zeta }}^{2}=d\beta ^{2}+\sinh ^{2}\beta (d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}).}$

## Remarks

1. ^ These formulae follow from inverting αu for u2 and applying the difference of two squares to obtain
u2 = c2(1 − αu2) = c2(1 − αu)(1 + αu)
so that
(1 − αu)/u2 = 1/c2(1 + αu) = γu/c2(1 + γu).
2. ^ Note that 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 s > V should yield positive frequency in accordance with the result for the ultimate velocity, s = −c. Hence the minus sign is a convention, but a very natural convention, to the point of being canonical.

The formula may also result in negative frequencies. The interpretation then is that the bullets are approaching from the negative 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.

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 x-axis at any instant has equally spaced bullets everywhere.

## Notes

1. ^ Kleppner & Kolenkow 1978, Chapters 11–14
2. ^ a b c d Einstein 1905, See section 5, "The composition of velocities".
3. ^ Galilei 2001
4. ^ Galilei 1954 Galilei used this insight to show that the path of the weight when seen from the shore would be a parabola.
5. ^ Mermin 2005, p. 37
6. ^ Landau & Lifshitz 2002, p. 13
7. ^ Kleppner & Kolenkow 1978, p. 457
8. ^ Jackson 1999, p. 531
9. ^ Lerner & Trigg 1991, p. 1053
10. ^ Friedman 2002, pp. 1–21
11. ^ Landau & Lifshitz 2002, p. 37 Equation (12.6) This is derived quite differently by consideration of invariant cross sections.
12. ^ Kleppner & Kolenkow 1978, p. 474
13. ^ Fizeau 1851E
14. ^ Fizeau 1860
15. ^ Landau & Lifshitz 2002, p. 14