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
Mass flow rate product is as follows. The only mass flowing "through" the cross-section is the amount normal to the area, i.e. parallel to the unit normal. This amount is: where "θ" is the angle between the unit normal formula_4 and the velocity of mass elements. The amount passing through the cross-section is reduced by the factor formula_5, as "θ" increases less mass passes through. All mass which passes in tangential directions to the area, that is perpendicular to the unit normal, "doesn't" actually pass "through" the area, so the mass passing through the area is zero. This occurs when "θ" = "π"/2: