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
Effective mass (solid-state physics) are different. These differences are important, for example in thermoelectric materials, where high conductivity, generally associated with light mass, is desired at the same time as high Seebeck coefficient, generally associated with heavy mass. Methods for assessing the electronic structures of different materials in this context have been developed. Certain III-V compounds such as GaAs and InSb have far smaller effective masses than tetrahedral group IV materials like Si and Ge. In the simplest Drude picture of electronic transport, the maximum obtainable charge carrier velocity is inversely proportional to the effective mass: formula_17 where formula_18 with formula_19 being the electronic