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
Angular velocity two-dimensional case above, one may define the angular velocity vector as: where "θ" is the angle between r and v. In terms of the cross product, this is: From this, one can recover the tangential velocity as: If a point rotates with angular velocity formula_28 in a coordinate frame formula_29 which itself rotates with an angular velocity formula_30 with respect to an external frame formula_31, we can define formula_32 to be the composite angular velocity vector of the point with respect to formula_31. This operation coincides with usual addition of vectors, and it gives angular velocity the algebraic structure of