Velocity-addition formula

Such formulas apply to successive Lorentz transformations, so they also relate different frames.

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.

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.

The cosmos of Galileo consists of absolute space and time and the addition of velocities corresponds to composition of Galilean transformations.

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.

, e.g. a cannonball fired horizontally out to sea, as measured from the ship, moving at speed

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]

The cosmos of special relativity consists of Minkowski spacetime and the addition of velocities corresponds to composition of Lorentz transformations.

In a system where lengths and times are measured in the same units, the speed of light is dimensionless and equal to 1.

The four-velocity is defined to be a four-vector with relativistic length equal to 1, future-directed and tangent to the world line of the object in spacetime.

This matrix rotates the pure time-axis vector (1, 0, 0, 0) to (V0, V1, 0, 0), and all its columns are relativistically orthogonal to one another, so it defines a Lorentz transformation.

The form of the relativistic composition law can be understood as an effect of the failure of simultaneity at a distance.

For the parallel component, the time dilation decreases the speed, the length contraction increases it, and the two effects cancel out.

The failure of simultaneity means that the fly is changing slices of simultaneity as the projection of u′ onto v. Since this effect is entirely due to the time slicing, the same factor multiplies the perpendicular component, but for the perpendicular component there is no length contraction, so the time dilation multiplies by a factor of ⁠1/V0⁠ = √(1 − v12).

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.

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

The ordering of operands in the definition is chosen to coincide with that of the standard configuration from which the formula is derived.

The parallel component of u′ can be found by projecting the full vector into the direction of the relative motion

and the perpendicular component of u′ can be found by the geometric properties of the cross product (see figure above right),

Examples of alternative notation include: 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 is not detailed here, but see for reference Lewis & Tolman (1909) Wikisource version (primary source) and Sard (1970, Section 3.2).

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.

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.

At LHC the crossing angle is small, around 300 μrad, but at the old Intersecting Storage Ring at CERN, it was about 18°.

According to Minkowski, the time-like vector (zeitartiger Vektor) for a given duration lies on a hyperbola.

In fact, the hyperbolic tangent of rapidity is the ratio of velocity to the speed of light in vacuum.

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 .
Decomposition of 3-velocity u into parallel and perpendicular components, and calculation of the components. The procedure for u ′ is identical.
Hippolyte Fizeau (1819–1896), a French physicist, was in 1851 the first to measure the speed of light in flowing water.
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 .
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.
The functions sinh , cosh and tanh . The function tanh relates the rapidity −∞ < ς < +∞ to relativistic velocity −1 < β < +1 .
Light cone and conjugate hyperbolas in Minkowski (1908)