In relativistic physics, a velocity-addition formula is an equation that specifies how to combine the 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 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 Galileo 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. 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. Galileo used this insight to show that the path of the weight when seen from the shore would be a parabola. 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 of C relative to A (velocity of falling object as Galileo sees it) is the sum of the velocity 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, moving at speed , would be measured by someone standing on the shore and watching the whole scene through a telescope as Extract of page 367
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 with Lorentz factor in the positive relative to the unprimed frame, then the differentials are
Divide the first three equations by the fourth,
or
which is
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
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
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 standard 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 and ,These formulae follow from inverting for and applying the difference of two squares to obtain
so that
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, so in general
for real number , 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 Equation (12.6) This is derived quite differently by consideration of invariant cross sections.
and
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
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 (using units where c = 1) Left-to-right ordering of operands Right-to-left ordering of operands
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 Wikisource version (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