Generalised Einstein mass-variation formulae: II Superluminal relative frame velocities

In part I of this paper we have deduced generalised Einstein mass variation formulae assuming relative frame velocities vcv>c. We again use the notion of the residual mass m0(v)m0(v) which for v>cv>c is defined by the equation m(v)=m0(v)[(v/c)2-1]-1/2m(v)=m0(v)[(v/c)2-1]-1/2 for the actual mass m(v)m(v). The residual mass is essentially the actual mass with the Einstein factor removed, and we emphasise that we make no restrictions on m0(v)m0(v). Using this formal device we deduce corresponding new mass variation formulae applicable to superluminal relative frame velocities, assuming only the extended Lorentz transformations and their consequences, and two invariants that are known to apply in special relativity. The present authors have previously speculated a dual framework such that both the rest mass m0∗ and the residual mass at infinite velocity m∞∗ (by which we mean p∞∗/c, assuming finite momentum at infinity) are equally important parameters in the specification of mass as a function of its velocity, and the two arbitrary constants can be so determined. The new formulae involving two arbitrary constants may also be exploited so that the mass remains finite at the speed of light, and two distinct mass profiles are determined as functions of their velocity with the rest mass assumed to be alternatively prescribed at the origin of either frame. The two profiles so obtained (M(U),m(u))(M(U),m(u)) and (M∗(U),m∗(u))(M∗(U),m∗(u)) although distinct have a common ratio M(U)/M∗(U)=m(u)/m∗(u)M(U)/M∗(U)=m(u)/m∗(u) that is a function of v>cv>c, indicating that observable mass depends upon the frame in which the rest mass is prescribed.