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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 three-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]

The notation employs u as velocity of a body within a Lorentz frame S, and v as velocity of a second frame S′, as measured in S, and 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, Fizeau measured the speed of light in a fluid moving parallel to the light using a 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 ​^V⁄_c 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 − ​¹⁄_n²)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 on the shore), B (e.g. ship), C (e.g. falling body on ship) the velocity vector udisplaystyle mathbf u of C relative to A (velocity of falling object as Galilei sees it) is the sum of the velocity u′displaystyle mathbf u' of C relative to B (velocity of falling object relative to ship) plus the velocity 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

u=v+u′.displaystyle mathbf u =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 out to sea) as measured from the ship would be measured by someone standing on the shore and watching the whole scene through a telescope as^[5]

u=v+u′1+(vu′/c2).displaystyle u=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,^[6]

c−uc+u=(c−u′c+u′)(c−vc+v).displaystyle c-u over c+u=left(c-u' over c+u'right)left(c-v over c+vright).

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.^[7][8] If the primed frame is travelling with speed vdisplaystyle v with Lorentz factor γv=1/1−v2/c2displaystyle gamma __v=1/sqrt 1-v^2/c^2 in the positive x-direction relative to the unprimed frame, then the differentials are

dx=γv(dx′+vdt′),dy=dy′,dz=dz′,dt=γv(dt′+vc2dx′).displaystyle dx=gamma __v(dx'+vdt'),quad dy=dy',quad dz=dz',quad dt=gamma __vleft(dt'+frac vc^2dx'right).

Divide the first three equations by the fourth,

dxdt=γv(dx′+vdt′)γv(dt′+vc2dx′),dydt=dy′γv(dt′+vc2dx′),dzdt=dz′γv(dt′+vc2dx′),displaystyle frac dxdt=frac gamma __v(dx'+vdt')gamma __v(dt'+frac vc^2dx'),quad frac dydt=frac dy'gamma __v(dt'+frac vc^2dx'),quad frac dzdt=frac dz'gamma __v(dt'+frac vc^2dx'),

or

ux=dxdt=dx′dt′+v(1+vc2dx′dt′),uy=dydt=dy′dt′γv (1+vc2dx′dt′),uz=dzdt=dz′dt′γv (1+vc2dx′dt′),displaystyle u_x=frac dxdt=frac frac dx'dt'+v(1+frac vc^2frac dx'dt'),quad u_y=frac dydt=frac frac dy'dt'gamma __v (1+frac vc^2frac dx'dt'),quad u_z=frac dzdt=frac frac dz'dt'gamma __v (1+frac vc^2frac dx'dt'),

which is

Transformation of velocity (Cartesian components) ux=ux′+v1+vc2ux′,ux′=ux−v1−vc2ux,displaystyle u_x=frac u_x'+v1+frac vc^2u_x',quad u_x'=frac u_x-v1-frac vc^2u_x, uy=1−v2c2uy′1+vc2ux′,uy′=1−v2c2uy1−vc2ux,displaystyle u_y=frac sqrt 1-frac v^2c^2u_y'1+frac vc^2u_x',quad u_y'=frac sqrt 1-frac v^2c^2u_y1-frac vc^2u_x, uz=1−v2c2uz′1+vc2ux′,uz′=1−v2c2uz1−vc2ux,displaystyle u_z=frac sqrt 1-frac v^2c^2u_z'1+frac vc^2u_x',quad u_z'=frac sqrt 1-frac v^2c^2u_z1-frac vc^2u_x,

in which expressions for the primed velocities were obtained using the standard recipe by replacing v by –v and swapping primed and unprimed coordinates. If coordinates are chosen so that all velocities lie in a (common) x–y plane, then velocities may be expressed as

ux=ucos⁡θ,uy=usin⁡θ,ux′=u′cos⁡θ′,uy′=u′sin⁡θ′,displaystyle u_x=ucos theta ,u_y=usin theta ,quad u_x'=u'cos theta ',quad u_y'=u'sin theta ',

(see polar coordinates) and one finds^[2][9]

Transformation of velocity (Plane polar components) u=u′2+v2+2vu′cos⁡θ′−(vu′sin⁡θ′c)21+vc2u′cos⁡θ′,tan⁡θ=uyux=1−v2c2uy′ux′+v=1−v2c2u′sin⁡θ′u′cos⁡θ′+v.displaystyle beginalignedu&=frac sqrt u'^2+v^2+2vu'cos theta '-(frac vu'sin theta 'c)^21+frac vc^2u'cos theta ',\tan theta &=frac u_yu_x=frac sqrt 1-frac v^2c^2u_y'u_x'+v=frac sqrt 1-frac v^2c^2u'sin theta 'u'cos theta '+v.endaligned

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 u into parallel and perpendicular components, and calculation of the components. The procedure for u′ 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 u in the unprimed frame and u′ in the primed frame, and split them into components parallel ( ∥ ) and perpendicular (⊥) to the relative velocity vector v (see hide box below) thus

u=u∥+u⊥,u′=u∥′+u⊥′,displaystyle mathbf u =mathbf u _parallel +mathbf u _perp ,quad mathbf u '=mathbf u '_parallel +mathbf u '_perp ,

then with the usual Cartesian unit basis vectors e_x, e_y, e_z, set the velocity in the unprimed frame to be

u∥=uxex,u⊥=uyey+uzez,v=vex,displaystyle mathbf u _parallel =u_xmathbf e _x,quad mathbf u _perp =u_ymathbf e _y+u_zmathbf e _z,quad mathbf v =vmathbf e _x,

which gives, using the results for the standard configuration,

u∥=u∥′+v1+v⋅u∥′c2,u⊥=1−v2c2u⊥′1+v⋅u∥′c2.displaystyle 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^2c^2mathbf u _perp '1+frac mathbf v cdot mathbf u _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

u=u∥+u⊥=11+v⋅u′c2[αvu′+v+(1−αv)(v⋅u′)v2v]≡v⊕u′,displaystyle mathbf u =mathbf u _parallel +mathbf u _perp =frac 11+frac mathbf v cdot mathbf u 'c^2left[alpha _vmathbf u '+mathbf v +(1-alpha _v)frac (mathbf v cdot mathbf u ')v^2mathbf v right]equiv mathbf v oplus mathbf u ',

where α_v = 1/γ_v 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.

Using an identity in αvdisplaystyle alpha _v and γvdisplaystyle gamma _v,^{[11][nb 1]}

v⊕u′≡u=11+u′⋅vc2[v+u′γv+1c2γv1+γv(u′⋅v)v]=11+u′⋅vc2[v+u′+1c2γv1+γvv×(v×u′)],displaystyle beginalignedmathbf v oplus mathbf u 'equiv mathbf u &=frac 11+frac mathbf u 'cdot mathbf v c^2left[mathbf v +frac mathbf u 'gamma _v+frac 1c^2frac gamma _v1+gamma _v(mathbf u 'cdot mathbf v )mathbf v right]\&=frac 11+frac mathbf u 'cdot mathbf v c^2left[mathbf v +mathbf u '+frac 1c^2frac gamma _v1+gamma _vmathbf v times (mathbf v times mathbf u ')right],endaligned and in the forwards (v positive, S -> S') direction v⊕u≡u′=11−u⋅vc2[uγv−v+1c2γv1+γv(u⋅v)v]=11−u⋅vc2[u−v+1c2γv1+γvv×(v×u)]displaystyle beginalignedmathbf v oplus mathbf u equiv mathbf u '&=frac 11-frac mathbf u cdot mathbf v c^2left[frac mathbf u gamma _v-mathbf v +frac 1c^2frac gamma _v1+gamma _v(mathbf u cdot mathbf v )mathbf v right]\&=frac 11-frac mathbf u cdot mathbf v c^2left[mathbf u -mathbf v +frac 1c^2frac gamma _v1+gamma _vmathbf v times (mathbf v times mathbf u )right]endaligned

where the last expression is by the standard vector analysis formula 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 A, B, C with B having velocity v relative to A and C having velocity u relative to 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

(λv)⊕(μu)≠λμ(v⊕u),displaystyle (lambda mathbf v )oplus (mu mathbf u )neq lambda mu (mathbf v oplus mathbf u ),

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

(−v)⊕(−u)=−(v⊕u),displaystyle (-mathbf v )oplus (-mathbf u )=-(mathbf v oplus mathbf u ),

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

v⊕u≠u⊕v,displaystyle mathbf v oplus mathbf u neq mathbf u oplus mathbf v ,

nor associative

v⊕(u⊕w)≠(v⊕u)⊕w.displaystyle mathbf v oplus (mathbf u oplus mathbf w )neq (mathbf v oplus mathbf u )oplus mathbf w .

It deserves special mention that if u and 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 −u relative to the primed frame, and the primed frame moves with velocity −v′ relative to the doubly primed frame hence (−v′ ⊕ −u) is the velocity of the unprimed frame relative to the doubly primed frame, and one might expect to have 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 fares 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^[12]

|u|2≡|v⊕u′|2=1(1+v⋅u′c2)2[(v+u′)2−1c2(v×u′)2]=|u′⊕v|2.mathbf u

and

|u′|2≡|v⊕u|2=1(1−v⋅uc2)2[(u−v)2−1c2(v×u)2]=|u⊕v|2.displaystyle

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

γu=γv⊕u′=[1−1c21(1+v⋅u′c2)2((v+u′)2−1c2(v2u′2−(v⋅u′)2))]−12=γvγu′(1+v⋅u′c2),γu′=γvγu(1−v⋅uc2)displaystyle gamma _u=gamma _mathbf v oplus mathbf u '=left[1-frac 1c^2frac 1(1+frac mathbf v cdot mathbf u 'c^2)^2left((mathbf v +mathbf u ')^2-frac 1c^2(v^2u'^2-(mathbf v cdot mathbf u ')^2)right)right]^-frac 12=gamma _vgamma _u'left(1+frac mathbf v cdot mathbf u 'c^2right),quad quad gamma _u'=gamma _vgamma _uleft(1-frac mathbf v cdot mathbf u c^2right)

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

Landau & Lifshitz (2002) (using units where c = 1)

|vrel|2=1(1−v1⋅v2)2[(v1−v2)2−(v1×v2)2]^2=frac 1(1-mathbf v_1 cdot mathbf v_2 )^2left[(mathbf v_1 -mathbf v_2 )^2-(mathbf v_1 times mathbf v_2 )^2right]

Left-to-right ordering of operands

Mocanu (1992)

u⊕v=11+u⋅vc2[v+u+1c2γuγu+1u×(u×v)]displaystyle mathbf u oplus mathbf v =frac 11+frac mathbf u cdot mathbf v c^2left[mathbf v +mathbf u +frac 1c^2frac gamma _mathbf u gamma _mathbf u +1mathbf u times (mathbf u times mathbf v )right]

Ungar (1988)

u∗v=11+u⋅vc2[v+u+1c2γuγu+1u×(u×v)]displaystyle mathbf u *mathbf v =frac 11+frac mathbf u cdot mathbf v c^2left[mathbf v +mathbf u +frac 1c^2frac gamma _mathbf u gamma _mathbf u +1mathbf u times (mathbf u times mathbf v )right]

Right-to-left ordering of operands

Sexl & Urbantke (2001)

w∘v=11+v⋅wc2[wγv+v+1c2γvγv+1(w⋅v)v]displaystyle mathbf w circ mathbf v =frac 11+frac mathbf v cdot mathbf w c^2left[frac mathbf w gamma _mathbf v +mathbf v +frac 1c^2frac gamma _mathbf v gamma _mathbf v +1(mathbf w cdot mathbf v )mathbf v 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.

When light propagates in a medium, its speed is reduced, in the rest frame of the medium, to c_m = ​^c⁄_nm, where n_m 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,^[13]

cm=V+cm′1+Vcm′c2=V+cn1+Vcnc2=cn1+nVc1+Vnc=cn(1+nVc)11+Vnc=(cn+V)(1−Vnc+(Vnc)2−⋯).displaystyle beginalignedc_m&=frac V+c_m'1+frac Vc_m'c^2=frac V+frac cn1+frac Vcnc^2=frac cnfrac 1+frac nVc1+frac Vnc\&=frac cn(1+frac nVc)frac 11+frac Vnc=(frac cn+V)left(1-frac Vnc+left(frac Vncright)^2-cdots right).endaligned

Collecting the largest contributions explicitly,

cm=cn+V(1−1n2−Vnc+⋯).displaystyle c_m=frac cn+V(1-frac 1n^2-frac Vnc+cdots ).

Fizeau found the first three terms.^[14][15] The classical result is the first two terms.

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

tan⁡θ=1−V2c2csin⁡θ′ccos⁡θ′+V=1−V2c2sin⁡θ′cos⁡θ′+Vc.displaystyle tan theta =frac sqrt 1-frac V^2c^2csin theta 'ccos theta '+V=frac sqrt 1-frac V^2c^2sin theta 'cos theta '+frac Vc.

For this case one may also compute sin θ and cos θ from the standard formulae,^[16]

sin⁡θ=1−V2c2sin⁡θ′1+Vccos⁡θ′,displaystyle beginalignedsin theta &=frac sqrt 1-frac V^2c^2sin theta '1+frac Vccos theta ',endaligned

cos⁡θ=Vc+cos⁡θ′1+Vccos⁡θ′,displaystyle cos theta =frac frac Vc+cos theta '1+frac Vccos theta ',

James Bradley (1693–1762) FRS, provided an explanation of aberration of light correct at the classical level,^[17] 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,

sin⁡θ−sin⁡θ′=sin⁡θ′(1−V2c21+Vccos⁡θ′−1)≈sin⁡θ′(1−Vccos⁡θ′−1)=−Vcsin⁡θ′cos⁡θ′,displaystyle beginalignedsin theta -sin theta '&=sin theta 'left(frac sqrt 1-frac V^2c^21+frac Vccos theta '-1right)\&approx sin theta 'left(1-frac Vccos theta '-1right)=-frac Vcsin theta 'cos theta ',endaligned

correct to order ​^v⁄_c. Employ in order to make small angle approximations a trigonometric formula,

sin⁡θ′−sin⁡θ=2sin⁡12(θ′−θ)cos⁡12(θ+θ′)≈(θ′−θ)cos⁡θ′,displaystyle beginalignedsin theta '-sin theta &=2sin frac 12(theta '-theta )cos frac 12(theta +theta ')approx (theta '-theta )cos theta ',endaligned

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

Thus the quantity

Δθ≡θ′−θ=Vcsin⁡θ′,displaystyle Delta theta equiv theta '-theta =frac Vcsin theta ',

the classical aberration angle, is obtained in the limit ​^V⁄_c → 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

λ=−sT+VT=(−s+V)Tdisplaystyle 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 = ​¹⁄_τ is the observed (temporal) frequency. The corresponding quantities in the emitters frame are endowed with primes.^[18]

For light waves

s=s′=−c,displaystyle s=s'=-c,

and the observed frequency is^[2][19][20]

ν=−sλ=−s(V−s)T=c(V+c)γVT′=ν′c1−V2c2c+V=ν′1−β1+β.displaystyle nu =-s over lambda =-s over (V-s)T=c over (V+c)gamma __VT'=nu 'frac csqrt 1-V^2 over c^2c+V=nu 'sqrt frac 1-beta 1+beta ,.

where T = γ_VT′ 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,

s=s′+V1+s′Vc2.displaystyle s=frac s'+V1+s'V over c^2.

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

τ=1γVν′(11+Vs′),ν=γVν′(1+Vs′)displaystyle tau =1 over gamma __Vnu 'left(frac 11+V over s'right),quad nu =gamma __Vnu '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,

ν=ν′γV(1−β)=ν′1−β1−β1+β=ν′1−β1+β.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][21][22]

ν=γVν′(1+Vs′cos⁡θ),displaystyle nu =gamma __Vnu 'left(1+frac Vs'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 functions sinh, cosh and tanh. The function tanh relates the rapidity −∞ < ς < +∞ to relativistic velocity −1 < β < +1.

Associated to the relativistic velocity βdisplaystyle boldsymbol beta of an object is a quantity ζdisplaystyle boldsymbol zeta whose norm is called rapidity. These are related through

so(3,1)⊃spanK1,K2,K3≈R3∋ζ=β^tanh−1⁡β,β∈B3,displaystyle mathfrak so(3,1)supset mathrm span K_1,K_2,K_3approx mathbb R ^3ni boldsymbol zeta =boldsymbol hat beta tanh ^-1beta ,quad boldsymbol beta in mathbb B ^3,

where the vector ζdisplaystyle boldsymbol zeta is thought of as being Cartesian coordinates on a 3-dimensional subspace of the Lie algebra so(3,1)displaystyle mathfrak so(3,1) of the Lorentz group spanned by the boost generators K1,K2,K3displaystyle K_1,K_2,K_3. This space, call it rapidity space, is isomorphic to ℝ³ as a vector space, and is mapped to the open unit ball,
B3displaystyle mathbb B ^3, velocity space, via the above relation.^[23] The addition law on collinear form coincides with the law of addition of hyperbolic tangents

tanh⁡(ζv+ζu′)=tanh⁡ζv+tanh⁡ζu′1+tanh⁡ζvtanh⁡ζu′displaystyle tanh(zeta _v+zeta _u')=tanh zeta _v+tanh zeta _u' over 1+tanh zeta _vtanh zeta _u'

with

vc=tanh⁡ζv ,u′c=tanh⁡ζu′ ,uc=tanh⁡(ζv+ζu′).displaystyle v over c=tanh zeta _v ,quad u' over c=tanh zeta _u' ,quad ,u over c=tanh(zeta _v+zeta _u').

The line element in velocity space B3displaystyle mathbb B ^3 follows from the expression for relativistic relative velocity in any frame,^[24]

vr=(v1−v2)2−(v1×v2)21−v1⋅v2,displaystyle v_r=sqrt frac (mathbf v_1 -mathbf v_2 )^2-(mathbf v_1 times mathbf v_2 )^21-mathbf v_1 cdot mathbf v_2 ,

where the speed of light is set to unity so that vidisplaystyle v_i and βidisplaystyle beta _i agree. It this expression, v1displaystyle mathbf v _1 and v2displaystyle mathbf v _2 are velocities of two objects in any one given frame. The quantity vrdisplaystyle v_r 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 vr=v2displaystyle v_r=v_2.

The line element is found by putting v2=v1+dvdisplaystyle mathbf v _2=mathbf v _1+dmathbf v or equivalently β2=β1+dβdisplaystyle boldsymbol beta _2=boldsymbol beta _1+dboldsymbol beta ,^[25]

dlβ2=dβ2−(β×dβ)2(1−β2)2=dβ2(1−β2)2+β21−β2(dθ2+sin2⁡θdφ2),displaystyle dl_boldsymbol beta ^2=frac dboldsymbol beta ^2-(boldsymbol beta times dboldsymbol beta )^2(1-beta ^2)^2=frac dbeta ^2(1-beta ^2)^2+frac beta ^21-beta ^2(dtheta ^2+sin ^2theta dvarphi ^2),

with θ and φ the usual spherical angle coordinates for βdisplaystyle boldsymbol beta taken in the z-direction. Now introduce ζ through

ζ=|ζ|=tanh−1⁡β,=tanh ^-1beta ,

and the line element on rapidity space R3displaystyle mathbb R ^3 becomes

dlζ2=dζ2+sinh2⁡ζ(dθ2+sin2⁡θdφ2).displaystyle dl_boldsymbol zeta ^2=dzeta ^2+sinh ^2zeta (dtheta ^2+sin ^2theta dvarphi ^2).

Relativistic particle collisions

In scattering experiments the primary objective is to measure the invariant scattering cross section. This enters the formula for scattering of two particle types into a final state fdisplaystyle f assumed to have two or more particles,^[26]

dNf=RfdVdt=σFdVdt(given as σn1n2vrdVdt in most textbooks),displaystyle dN_f=R_fdVdt=sigma FdVdtquad (textgiven as sigma n_1n_2v_rdVdttext in most textbooks),

where

• 
  dVdtdisplaystyle dVdt is spacetime volume. It is an invariant under Lorentz transformations.


• 
  dNfdisplaystyle dN_f is the total number of reactions reactions resulting in final state fdisplaystyle f in spacetime volume dVdtdisplaystyle dVdt. Being a number, it is invariant when the same spacetime volume is considered.


• 
  Rf=Fσdisplaystyle R_f=Fsigma is the number of reactions resulting in final state fdisplaystyle f per unit spacetime, or reaction rate. This is invariant.


• 
  F=n1n2vrdisplaystyle F=n_1n_2v_r is called the incident flux. This is required to be invariant, but isn't in the most general setting.


• 
  σdisplaystyle sigma is the scattering cross section. It is required to be invariant.


• 
  n1,n2displaystyle n_1,n_2 are the particle densities in the incident beams. These are not invariant as is clear due to length contraction.


• 
  vr=|v2−v1|displaystyle v_r= is the relative speed of the two incident beams. This cannot be invariant since F=n1n2vrdisplaystyle F=n_1n_2v_r is required to be so.

The objective is to find a correct expression for relativistic relative speed vreldisplaystyle v_rel and an invariant expression for the incident flux.

Non-relativistically, one has for relative speed vr=|v2−v1|displaystyle v_r=. If the system in which velocities are measured is the rest frame of particle type 1displaystyle 1, it is required that vrel=vr=|v2|.. Setting the speed of light c=1displaystyle c=1, the expression for vreldisplaystyle v_rel follows immediately from the formula for the norm (second formula) in the general configuration as^[27][28]

vrel=(v1−v2)2−(v1×v2)21−v1⋅v2.displaystyle v_rel=sqrt frac (mathbf v_1 -mathbf v_2 )^2-(mathbf v_1 times mathbf v_2 )^21-mathbf v_1 cdot mathbf v_2 .

The formula reduces in the classical limit to vr=|v1−v2| 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.^[27] 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

vrel=(u1⋅u2)2−1u1⋅u2.displaystyle v_rel=frac sqrt (u_1cdot u_2)^2-1u_1cdot u_2.

The correct expression for the flux, published by Christian Møller^[29] in 1945, is given by^[30]

F=n1n1(v1−v2)2−(v1×v2)2≡n1n2v¯.displaystyle F=n_1n_1sqrt (mathbf v _1-mathbf v _2)^2-(mathbf v _1times mathbf v _2)^2equiv n_1n_2bar v.

One notes that for collinear velicities, F=n1n2|v2−v1|=n1n2vr=n_1n_2v_r. In order to get a manifestly Lorentz invariant expression one writes Ji=(ni,nivi)displaystyle J_i=(n_i,n_imathbf v _i) with ni=γini0displaystyle n_i=gamma _in_i^0, where ni0displaystyle n_i^0is the density in the rest frame, for the individual particle fluxes and arrives at^[31]

F=(J1⋅J2)vrel.displaystyle F=(J_1cdot J_2)v_rel.

In the literature the quantity v¯displaystyle bar v as well as vrdisplaystyle v_r are both referred to as the relative velocity. In some cases (statistical physics and dark matter literature), v¯displaystyle bar v is referred to as the Møller velocity, in which case vrdisplaystyle v_r means relative velocity. The true relative velocity is at any rate vreldisplaystyle v_rel.^[31] The discrepancy between vreldisplaystyle v_rel and vrdisplaystyle v_r is relevant though in most cases velocities are collinear. At LHC the crossing angle is small, around 300 μrad, but at
the old Intersecting Storage Ring at CERN, it was about 18^◦.^[32]

See also

• Hyperbolic law of cosines


• Biquaternion

Remarks

Notes

References

• 
  Cannoni, Mirco (2017). "Lorentz invariant relative velocity and relativistic binary collisions". International Journal of Modern Physics A. 32 (2n03): 1730002. arXiv:1605.00569. Bibcode:2017IJMPA..3230002C. doi:10.1142/S0217751X17300022 – via World Scientific. (Subscription required (help)).
  


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  Einstein, A. (1905). "On the Electrodynamics of moving bodies" [Zur Elektrodynamik bewegter Körper] (PDF). Annalen der Physik. 10 (322): 891–921. Bibcode:1905AnP...322..891E. doi:10.1002/andp.19053221004.
  


• 
  Fock, V.A. (1964). The theory of space, time, and gravitation (2nd ed.). ISBN 978-0-08-010061-6 – via ScienceDirect. (Subscription required (help)).
  


• 
  French, A.P. (1968). Special Relativity. MIT Introductory Physics Series. W.W. Norton & Company. ISBN 978-0-393-09793-1.
  


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  Friedman, Yaakov; Scarr, Tzvi (2005). Physical applications of homogeneous balls. Birkhäuser. pp. 1–21. ISBN 978-0-8176-3339-4.
  


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  Jackson, J. D. (1999) [1962]. "Chapter 11". Classical Electrodynamics (3d ed.). John Wiley & Sons. ISBN 978-0-471-30932-1. (graduate level)


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  Kleppner, D.; Kolenkow, R. J. (1978) [1973]. An Introduction to Mechanics. London: McGraw-Hill. ISBN 978-0-07-035048-9. (introductory level)


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  Landau, L.D.; Lifshitz, E.M. (2002) [1939]. The Classical Theory of Fields. Course of Theoretical Physics. 2 (4th ed.). Butterworth–Heinemann. ISBN 0 7506 2768 9. (graduate level)


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  Lerner, R.G.; Trigg, G.L. (1991). Encyclopaedia of Physics (2nd ed.). VHC Publishers, Springer. ISBN 978-0-07-025734-4.
  


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  Mermin, N. D. (2005). It's About Time: Understanding Einstein's Relativity. Princeton University Press. ISBN 978-0-691-12201-4.
  


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  Mocanu, C.I. (1992). "On the relativistic velocity composition paradox and the Thomas rotation". Found. Phys. Lett. 5 (5): 443–456. Bibcode:1992FoPhL...5..443M. doi:10.1007/BF00690425. ISSN 0894-9875.
  


• 
  Møller, C. (1945). "General properties of the characteristic matrix in the theory of elementary particles I" (PDF). D. Kgl Danske Vidensk. Selsk. Mat.-Fys. Medd. 23 (1).
  


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  Parker, S. P. (1993). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. ISBN 978-0-07-051400-3.
  


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  Sard, R. D. (1970). Relativistic Mechanics – Special Relativity and Classical Particle Dynamics. New York: W. A. Benjamin. ISBN 978-0-8053-8491-8.
  


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  Sexl, R. U.; Urbantke, H. K. (2001) [1992]. Relativity, Groups Particles. Special Relativity and Relativistic Symmetry in Field and Particle Physics. Springer. pp. 38–43. ISBN 978-3-211-83443-5.
  


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  Tipler, P.; Mosca, G. (2008). Physics for Scientists and Engineers (6th ed.). Freeman. pp. 1328–1329. ISBN 978-1-4292-0265-7.
  


• 
  Ungar, A. A. (1988). "Thomas rotation and parameterization of the Lorentz group". Foundations of Physics Letters. 1 (1): 57–81. Bibcode:1988FoPhL...1...57U. doi:10.1007/BF00661317. ISSN 0894-9875.
  

Historical

• 
  Bradley, James (1727–1728). "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". Phil. Trans. R. Soc. (pdf)|format= requires |url= (help). 35 (399–406): 637–661. doi:10.1098/rstl.1727.0064.
  


• 
  Doppler, C. (1903) [1842], Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels [About the coloured light of the binary stars and some other stars of the heavens] (in German), 2 (V), Prague: Abhandlungen der Königl. Böhm. Gesellschaft der Wissenschaften, pp. 465–482
  


• 
  Fizeau, H. (1851F). "Sur les hypothèses relatives à l'éther lumineux" [The Hypotheses Relating to the Luminous Aether]. Comptes Rendus (in French). 33: 349–355.
  


• 
  Fizeau, H. (1851E). "The Hypotheses Relating to the Luminous Aether". Philosophical Magazine. 2: 568–573.
  


• 
  Fizeau, H. (1859). "Sur les hypothèses relatives à l'éther lumineux" [The Hypotheses Relating to the Luminous Aether]. Ann. Chim. Phys. (in French). 57: 385–404.
  


• 
  Fizeau, H. (1860). "On the Effect of the Motion of a Body upon the Velocity with which it is traversed by Light". Philosophical Magazine. 19: 245–260.
  


• 
  Galilei, G. (2001) [1632]. Dialogue Concerning the Two Chief World Systems [Dialogo sopra i due massimi sistemi del mondo]. Stillman Drake (Editor, Translator), Stephen Jay Gould (Editor), J. L. Heilbron (Introduction), Albert Einstein (Foreword). Modern Library. ISBN 978-0-375-75766-2.
  


• 
  Galilei, G. (1954) [1638]. Dialogues Concerning Two New Sciences [Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze]. Henry Crew, Alfonso de Salvio (Translators). Digiread.com. ISBN 978-1-4209-3815-9.
  


• 
  Lewis, G. N.; Tolman, R. C. (1909). "The Principle of Relativity, and Non-Newtonian Mechanics". Phil. Mag. 6. 18 (106): 510–523. doi:10.1080/14786441008636725.
  Wikisource version
  

External links

• 
  Sommerfeld, A. (1909). "On the Composition of Velocities in the Theory of Relativity" [Über die Zusammensetzung der Geschwindigkeiten in der Relativtheorie]. Verh. Der DPG. 21: 577–582.