Generalised Einstein mass-variation formulae: I Subluminal relative frame velocities

Abstract

Much of the formalism in special relativity is intimately bound up with Einstein’s formula for the variation of mass m with its velocity v, namely m(v)=m0∗[1-(v/c)2]-1/2, where m is the mass, v the velocity, c denotes the speed of light and m0∗ denotes the rest mass, noting that in these papers, we employ an asterisk to designate the rest mass. Einstein’s formula together with the Lorentz transformations and their consequences are fundamental to the development of special relativity. Here we introduce the notion of the residual mass m0(v) which for v<c is defined by the equation m(v)=m0(v)[1-(v/c)2]-1/2 for the actual mass m(v); namely the residual mass is the actual mass with the Einstein factor removed. We emphasise that we make no restrictions on m0(v), and that this formal device merely facilitates the analysis. Using this formal device we deduce corresponding new mass variation formulae, assuming only the Lorentz transformations and two invariants known to apply in special relativity. One is force invariance in the direction of relative motion applying to two non-accelerating frames, while the other is not so well known, but applies in special relativity. Together the two assumed invariances imply that the energy–mass transfer rates are frame invariant but not necessarily constant as in special relativity. The new formulae involving two arbitrary constants may be exploited so that the mass remains finite at the speed of light, and an illustrative example is provided for which this is the case, and from which a new comparison formula is derived that is singular at the speed of light. This new expression may be contrasted with the Einstein expression, and roughly speaking, the new formula predicts more mass than that given by the Einstein formula, since the singularity at the speed of light is steeper.