Relativistic Imaginaries

Reconsider the Relativity Operator, R, defined;
R = 1/[(1 - [(v/c)^2])^(1/2)] = (1 - [(v/c)^2])^(-1/2) ,
and, once more, let M denote a moving mass, with m the corresponding
rest-mass, so that
M = Rm = m/[(1 - [(v/c)^2])^(1/2)] .
If v > c, the case for tachyons, then R is an imaginary number;
making M imaginary.

A commonly-used definition of a tachyonic mass tM has it that
tM = -iM .
This would, for instance, be the kind of definition physics
professors first give to undergraduate students.
And that is perfectly understandable, considering the way tachyons
are presented in the literature. See, for example, the entry
Tachyons, by physicist Gerald Feinberg, in the Encyclopedia of
Physics by Lerner and Trigg (I have the 2nd Edition, printed in
1991), from VCH Publishers; page 1246).
But this kind of definition leaves room for confusion when standard
complex quantities and tachyonic complex quantities are discussed in
the same context.

We explain the situation as follows.

Let Q indicate the absolute-value of the difference
"1 - (v/c)^2",
this condition being denoted;
Q = | 1 - (v/c)^2 | ,
and let the following notation convention be observed;
Q+ = (+1)Q whenever v < c ,
Q0 = (0)Q = 0 whenever v = c ,
Q- = (-1)Q whenever v > c .

Then, because
R = (1 - [(v/c)^2])^(-1/2) ,
let us indicate the three cases of R;
Case 1:
R = R+ = (Q+)^(-1/2) if, and only if, v < c .
Case 2:
R = R0 = 0 if, and only if, v = c
(assuming 1/0 is undefined; not infinity).
Case 3:
R = R- = (Q-)^(-1/2) if, and only if, v > c .

In the last case, for tachyons, where v > c, we have;
R- = (Q-)^(-1/2) = [(-1)Q]^(-1/2)] = ...
= 1/(i[Q^(1/2)]) = (1/i)[Q^(-1/2)] .
Note, however, that 1/i = -i ,
according to the following proof.
1/i = 1/[(-1)^(1/2)] = (-1)^(-1/2) = (-1)^[(1/2) - 1]
= [(-1)^(1/2)][(-1)^(-1)] = [(-1)^(1/2)][(-1)^(1 - 2)]
= [(-1)^(1/2)][(-1)^(1)][(-1)^(-2)] = [(-1)^(1/2)](-1)/[(-1)^2]
= (-1)[(-1)^(1/2)]/1 = -[(-1)^(1/2)] = -i .

Consequently, if v > c , then R is;
R = R- = (Q-)^(-1/2) = (1/i)[Q^(-1/2)] = -i[Q^(-1/2)] .

Thus, the relativistic tachyonic mass tM is properly defined;
tM = (R-)m = -i[Q^(-1/2)]m , where v > c ,
while the corresponding tardyonic mass M continues to be defined as
usual, but also;
M = (R+)m = [Q^(-1/2)]m , where v < c .

Hence, we can legitimately write
tM = -iM ,
when deriving tM rigorously using the Relativity Operator, R, but we
cannot write tM = iM , in such cases, because the sign is wrong.

We see that, because of the importance of keeping track of the sign
on the imaginary-unit in the rigorous derivation of tachyonic mass,
tM, we must adopt special rules on the symbols we employ (i.e., we
must use a notation convention), and which allow us to represent
tachyonic mass in terms of tardyonic mass, while maintaining
sufficient rigor to assure accurate conceptualization.

With that necessity established, we address the main source of
confusion in the very next section.