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
Law of mass action formula. The law of mass action also has implications in semiconductor physics. Regardless of doping, the product of electron and hole densities is a constant at equilibrium. This constant depends on the thermal energy of the system (i.e. the product of the Boltzmann constant, formula_20, and temperature, formula_21), as well as the band gap (the energy separation between conduction and valence bands, formula_22) and effective density of states in the valence formula_23 and conduction formula_24 bands. When the equilibrium electron formula_25 and hole formula_26 densities are equal, their density is called the "intrinsic" carrier density formula_27 as this would be