Proper velocity addition formula: where formula_3 is the beta factor given by formula_4. or in terms of logarithms: Proper velocity is useful for comparing the speed of objects with momentum per unit rest mass ("w") greater than lightspeed "c". The coordinate speed of such objects is generally near lightspeed, whereas proper velocity tells us how rapidly they are covering ground on "traveling-object clocks". This is important for example if, like some cosmic ray particles, the traveling objects have a finite lifetime. Proper velocity also clues us in to the object's momentum, which has no upper bound. For example, a 45 GeV electron
Four-velocity the 3d position vector, written as a column vector The components of the velocity formula_2 (tangent to the curve) at any point on the world line are Each component is simply written In Einstein's theory of relativity, the path of an object moving relative to a particular frame of reference is defined by four coordinate functions "x"("τ"), where μ is a spacetime index which takes the value 0 for the timelike component, and 1, 2, 3 for the spacelike coordinates. The zeroth component is defined as the time coordinate multiplied by "c", Each function depends on one parameter "τ" called