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*Abstract*
We propose here two new transformations between inertial frames that apply
for relative velocities greater than the speed of light, and that are
complementary to the Lorentz transformation, giving rise to the Einstein
special theory of relativity that applies to relative velocities less than
the speed of light. The new transformations arise from the same
mathematical framework as the Lorentz transformation, displaying singular
behaviour when the relative velocity approaches the speed of light and
generating the same addition law for velocities, but, most importantly, do
not involve the need to introduce imaginary masses or complicated physics
to provide well-defined expressions. Making use of the dependence on
relative velocity of the Lorentz transformation, the paper provides an
elementary derivation of the new transformations between inertial frames
for relative velocities v in excess of the speed of light c, and further we
suggest two possible criteria from which one might infer one set of
transformations as physically more likely than the other. If the
energy-momentum equations are to be invariant under the new
transformations, then the mass and energy are given, respectively, by the
formulae m=(pinf/c)[(v/c)2 - 1]-1/2 and e=mc2 where pinf denotes the
limiting momentum for infinite relative velocity. If, however, the
requirement of invariance is removed, then we may propose new mass and
energy equations, and an example having finite non-zero mass in the limit
of infinite relative velocity is given. In this highly controversial topic,
our particular purpose is not to enter into the merits of existing
theories, but rather to present a succinct and carefully reasoned account
of a new aspect of Einstein’s theory of special relativity, which properly
allows for faster than light motion.