Velocity-addition formula

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.
The notation employs as velocity of a body within a Lorentz frame , and as velocity of a second frame , as measured in , and 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 an 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 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 instead of , where is the speed of light in the aether, is the refractive index of the fluid, and 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 ( in the ) for the addition of relativistic velocities. 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. 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. 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 formula_1 of C relative to A (velocity of falling object as Galilei sees it) is the sum of the velocity formula_2 of C relative to B (velocity of falling object relative to ship) plus the velocity 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
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
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,
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. If the primed frame is travelling with speed formula_6 with Lorentz factor formula_7 in the positive relative to the unprimed frame, then the differentials are
Divide the first three equations by the fourth,
or
which is
u_y'}{1 + \frac{v}{c^2}u_x'}, \quad u_y' = \frac{\sqrt{1-\frac{v^2}{c^2}}u_y}{1 - \frac{v}{c^2}u_x},</math>
in which expressions for the primed velocities were obtained using the standard recipe by replacing by and swapping primed and unprimed coordinates. If coordinates are chosen so that all velocities lie in a (common) plane, then velocities may be expressed as
(see polar coordinates) and one finds
\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}.\end{align}</math>
The proof as given is highly formal. There are other more involved proofs that may be more enlightening, such as the one below.
Since a relativistic transformation rotates space and time into each other much as geometric rotations in the plane rotate the - and -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 . 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 , which is the motion of the ship away from the shore, as measured from the shore, and which is the motion of the fly away from the ship, as measured from the ship. The four-velocity is defined to be a four-vector with relativistic length equal to , future-directed and tangent to the world line of the object in spacetime. Here, corresponds to the time component and to the component of the ship's velocity as seen from the shore. It is convenient to take the -axis to be the direction of motion of the ship away from the shore, and the -axis so that the plane is the plane spanned by the motion of the ship and the fly. This results in several components of the velocities being zero; .
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,
Since the relativistic length of is ,
so
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:
This matrix rotates the a pure time-axis vector to , and all its columns are relativistically orthogonal to one another, so it defines a Lorentz transformation.
If a fly is moving with four-velocity in the ship frame, and it is boosted by multiplying by the matrix above, the new four-velocity in the shore frame is ,
Dividing by the time component and substituting for the components of the four-vectors and in terms of the components of the three-vectors and gives the relativistic composition law as
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 simultaneity slices as the projection of onto . 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 .
General configuration.
Starting from the expression in coordinates for parallel to the , 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 in the unprimed frame and in the primed frame, and split them into components parallel (∥) and perpendicular (⊥) to the relative velocity vector (see hide box below) thus
then with the usual Cartesian unit basis vectors , set the velocity in the unprimed frame to be
which gives, using the results for the standard configuration,
where · is the dot product. Since these are vector equations, they still have the same form for in "any" direction. The only difference from the coordinate expressions is that the above expressions refers to "vectors", not components.
One obtains
where 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.
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 can be found by projecting the full vector into the direction of the relative motion
and the perpendicular component of can be found by the geometric properties of the cross product (see figure above right),
In each case, is a unit vector in the direction of relative motion.
The expressions for and can be found in the same way. Substituting the parallel component into
results in the above equation.
Using an identity in formula_28 and formula_29,
\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>
where the last expression is by the standard vector analysis formula . The first expression extends to any number of spatial dimensions, but the cross product is defined in three dimensions only. The objects with having velocity relative to and having velocity relative to 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
for any real numbers and , although it is true that
Also, due to the last terms, is in general neither commutative
nor associative
It deserves special mention that if and 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 relative to the primed frame, and the primed frame moves with velocity relative to the doubly primed frame hence is the velocity of the unprimed frame relative to the doubly primed frame, and one might expect to have 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
and
Reverse formula found by using standard procedure of swapping for and for .
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
Reverse formula found by using standard procedure of swapping for and for .
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:
 (using units where c = 1)
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 -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 (primary source) and .
Fizeau experiment.
When light propagates in a medium, its speed is reduced, in the rest frame of the medium, to , where is the index of refraction of the medium . The speed of light in a medium uniformly moving with speed in the positive -direction as measured in the lab frame is given directly by the velocity addition formulas. For the forward direction (standard configuration, drop index on ) one gets,
Collecting the largest contributions explicitly,
Fizeau found the first three terms. 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, , and insertion in the formula for yields
For this case one may also compute and from the standard formulae,
the trigonometric manipulations essentially being identical in the case to the manipulations in the case. Consider the difference,
correct to order . Employ in order to make small angle approximations a trigonometric formula,
where were used.
Thus the quantity
the classical aberration angle, is obtained in the limit .
Relativistic Doppler shift.
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 , so subscripts on velocity components are dropped. In the observers frame, introduce the geometrical observation
as the spatial distance, or wavelength, between two pulses (wave crests), where 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" , and its inverse is the observed (temporal) frequency. The corresponding quantities in the emitters frame are endowed with primes.
For light waves
and the observed frequency is
where is standard time dilation formula.
Suppose instead that the wave is not composed of light waves with speed , but instead, for easy visualization, bullets fired from a relativistic machine gun, with velocity in the frame of the emitter. Then, in general, the geometrical observation is "precisely the same". But now, , and is given by velocity addition,
The calculation is then essentially the same, except that here it is easier carried out upside down with instead of . One finds
\right), \quad \nu = \gamma_{_V}\nu'\left(1+{V\over s'}\right)</math>
Observe that in the typical case, the that enters is "negative". The formula has general validity though. When , the formula reduces to the formula calculated directly for light waves above,
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 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,
where is the angle between the light emitter and the observer. This reduces to the previous result for collinear motion when , but for transverse motion corresponding to , the frequency is shifted by the Lorentz factor. This does not happen in the classical optical Doppler effect.
Hyperbolic geometry.
Associated to the relativistic velocity formula_60 of an object is a quantity formula_61 whose norm is called rapidity. These are related through
where the vector formula_63 is thought of as being Cartesian coordinates on a 3-dimensional subspace of the Lie algebra formula_64 of the Lorentz group spanned by the boost generators formula_65. This space, call it "rapidity space", is isomorphic to as a vector space, and is mapped to the open unit ball,
formula_66, "velocity space", via the above relation. The addition law on collinear form coincides with the law of addition of hyperbolic tangents
with
where the speed of light is set to unity so that formula_69 and formula_70 agree. It this expression, formula_71 and formula_72 are velocities of two objects in any one given frame. The quantity formula_73 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 formula_74.
The line element is found by putting formula_75 or equivalently formula_76,
with and the usual spherical angle coordinates for formula_60 taken in the -direction. Now introduce through
and the line element on rapidity space formula_80 becomes
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 formula_82 assumed to have two or more particles,
where
The objective is to find a correct expression for "relativistic relative speed" formula_95 and an invariant expression for the incident flux.
Non-relativistically, one has for relative speed formula_96. If the system in which velocities are measured is the rest frame of particle type formula_97, it is required that formula_98 Setting the speed of light formula_99, the expression for formula_95 follows immediately from the formula for the norm (second formula) in the "general configuration" as
The formula reduces in the classical limit to formula_102 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. 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
The correct expression for the flux, published by Christian Møller in 1945, is given by
One notes that for collinear velocities, formula_105. In order to get a "manifestly" Lorentz invariant expression one writes formula_106 with formula_107, where formula_108is the density in the rest frame, for the individual particle fluxes and arrives at
In the literature the quantity formula_110 as well as formula_73 are both referred to as the relative velocity. In some cases (statistical physics and dark matter literature), formula_110 is referred to as the "Møller velocity", in which case formula_73 means relative velocity. The true relative velocity is at any rate formula_95. The discrepancy between formula_95 and formula_73 is relevant though in most cases velocities are collinear. At LHC the crossing angle is small, around 300 rad, but at
the old Intersecting Storage Ring at CERN, it was about 18◦.