Velocity-addition formula

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 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 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 ⁠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/n⁠ + (1 − ⁠1/n2⁠)V instead of ⁠c/n⁠ + V, where c is the speed of light in the aether, n is the refractive index of the fluid, 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. 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. 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 u {\displaystyle \mathbf {u} } of C relative to A (velocity of falling object as Galileo 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, u ′ {\displaystyle u'} , e.g. a cannonball fired horizontally out to sea, as measured from the ship, moving at speed v {\displaystyle v} , would be measured by someone standing on the shore and watching the whole scene through a telescope as

u = v + u ′ 1 + ( v u ′ / c 2 ) . {\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,

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