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
Moment (physics) of "n" corresponds to a different moment: the 1st moment corresponds to "n" = 1; the 2nd moment to "n" = 2, etc. The 0th moment ("n" = 0) is sometimes called the "monopole moment"; the 1st moment ("n" = 1) is sometimes called the "dipole moment", and the 2nd moment ("n" = 2) is sometimes called the "quadrupole moment", especially in the context of electric charge distributions. Moments of mass: Assuming a density function that is finite and localized to a particular region, outside that region a 1/"r" potential may be expressed as a series of spherical harmonics: The