Proper velocity by Hermann Minkowski's flat-space metric equation . Here a reference map frame of yardsticks and synchronized clocks define map position x and map time "t" respectively, and the "d" preceding a coordinate means infinitesimal change. A bit of manipulation allows one to show that proper velocity where as usual coordinate velocity . Thus finite "w" ensures that "v" is less than lightspeed "c". By grouping "γ" with v in the expression for relativistic momentum p, proper velocity also extends the Newtonian form of momentum as mass times velocity to high speeds without a need for relativistic mass. The proper velocity
Velocity-addition formula appear there must be inverted here: This matrix rotates the a pure time-axis vector to , and all its columns are relativistically orthogonal to one another, so it defines a Lorentz transformation. If a fly is moving with four-velocity in the ship frame, and it is boosted by multiplying by the matrix above, the new four-velocity in the shore frame is , Dividing by the time component and substituting for the components of the four-vectors and in terms of the components of the three-vectors and gives the relativistic composition law as The form of the relativistic composition law can be