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
Chirp mass terms of other common mass parameters: where formula_7 is the total mass of the system, formula_8 is the reduced mass of the system, formula_9 is the mass ratio, and formula_10 is the symmetric mass ratio. In general relativity, the phase evolution of a binary orbit can be computed using a post-Newtonian expansion, a perturbative expansion in powers of the orbital velocity formula_11. The first order gravitational wave frequency, formula_12, evolution is described by the differential equation where formula_14 and formula_15 are the speed of light and Newton's gravitational constant, respectively. Integrating equation () with respect to time gives: where C