Velocity-addition formula

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A velocity addition formula appears in the special theory of relativity as a consequence of the Lorentz transformations. Different velocity addition formulae also appear in some other theories.

[edit] Velocity addition under special relativity

Due to the time and distance distortions that must be attributed when citing measurements of a frame that is in motion relative to the observer's frame, velocities cannot be added with plain arithmetic, but must employ a more complex formula. Given any two velocity vectors \mathbf{v_1} and \mathbf{v_2}, their relativistic sum is given by

\mathbf{v_1} \oplus \mathbf{v_2}=\frac{\mathbf{v_1}+\mathbf{v_2}}{1+ \frac{\mathbf{v_1}\cdot\mathbf{v_2}}{c^2}} + \frac{1}{c^2} \cdot \frac{1}{1+\sqrt{1-\frac{v_1^2}{c^2}}} \cdot \frac{\mathbf{v_1}\times(\mathbf{v_1}\times\mathbf{v_2})}{1+\frac{\mathbf{v_1}\cdot\mathbf{v_2}}{c^2}}

where \cdot is the vector dot product operation and \times is the vector cross product operation. Note that the correct symbol for a relativistic sum is ⊕, not +. A more common formula is that for the special case of colinear velocities (that is, velocities in the same direction). In that case the relativistic sum is

v=\frac{v_1+v_2}{1+ \frac{v_1 v_2}{c^2}}

in the same direction as the original two vectors.

Graph of the colinear velocity addition function (3D and contours) in units of the speed of light
Graph of the colinear velocity addition function (3D and contours) in units of the speed of light

When the velocities are expressed as a fraction of the speed of light (for example, 0.98 instead of 0.98c or 293,796,609 m/s), the equation can be written as

v=\frac{v_1+v_2}{1+v_1 v_2}

where the resulting velocity is also expressed as a fraction of the speed of light.

One feature of the formula is that no summation of velocities ever exceeds the speed of light. In addition, if either of the velocities is the speed of light, the sum is also the speed of light. If observer B is moving at any speed relative to observer A and observer B observes photons moving in front of him/her at the speed of light, the photons are also moving at c relative to observer A. Photons, unlike most objects, always appear to (and do in truth) move at the speed of light regardless of the reference frame.

[edit] Velocity-addition in other theories

Velocity addition formulae also arise outside special relativity. Generally, if Shift(velocity) is defined as a frequency ratio f'/f, then we expect a theory to generate a special velocity addition formula when its Doppler relationships have the characteristic:

\mbox{Shift}(v_1) \times \mbox{Shift}(v_2) \ne \mbox{Shift}(v_1 + v_2).

For emission theory, the Doppler relationship of f' / f = (1 − v) results in a velocity addition formula of

VTOT = v1 + v2 − v1v2

where VTOT is an equivalent velocity that lets us calculate the same total frequency shift in a single stage.

Although the velocity-addition formula method is general, the appearance of a v.a.f. under special relativity has a very different significance to the use of superficially-similar formulae under other theories.

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