Mass, Special Relativity and the Equivalence Principle

Abstract

The equivalence principle states that inertial mass of a particle, i.e. the ratio of force and acceleration, is equal to the gravitational mass, i.e. equal to the quantity responsible for the gravitational force. Special relativity dictates that this inertial or gravitational mass of a particle equals γ3 m o where m o is the rest mass of the particle and \(\gamma (= {(1 -{{ v}}^{2}/{c}^{2})}^{-1/2})\) is the Lorentz factor. The latter is unbound when the particle velocity relative to a laboratory observer approaches the speed of light c. This is a well-known result for linear motions and, using instantaneous frames of reference, is shown here to be valid also for arbitrary particle motion. Thus it is the gravitational mass γ3 m o, rather than the relativistic mass, γm o, defined from Einstein’s famous E = γm o c 2 which must be used in Newton’s gravitational law. This difference is totally negligible for non-relativistic velocities, e.g. v < 0. 1c, but can become quite important when v approaches c. Thus special relativity coupled with Newton’s gravitational law dictates that the gravitational attraction becomes much stronger under relativistic conditions than that expected from the simple Newton’s gravitational law using the rest, m o, or relativistic mass, γm o of the particles involved. This is consistent with the expectation, based on general relativity, that Newton’s gravitational law when used with rest or the relativistic masses, may not be valid under highly relativistic conditions. The implications are quite significant for fast subatomic particles.