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
Mass in special relativity geometry of Minkowski-space, the velocity-space is bounded by "c" and has the geometry of hyperbolic geometry where relativistic-mass plays an analogous role to that of Newtonian-mass in the barycentric-coordinates of Euclidean geometry. The connection of velocity to hyperbolic-geometry enables the 3-velocity-dependent relativistic-mass to be related to the 4-velocity Minkowski-formalism. Mass in special relativity Mass in special relativity incorporates the general understandings from the laws of motion of special relativity along with its concept of mass–energy equivalence. The word "mass" is given two meanings in special relativity: one ("rest" or "invariant mass", and its equivalent "rest energy") is an invariant quantity