Introducing The Imagination Unit

As demonstrated in Part 2, we can use the relativistic mass M of an
ordinary particle to define a corresponding tachyonic mass tM,
writing;
tM = -iM ,
where
M = m[(1 - [(v/c)^2])^(-1/2)] ,
with m as the ordinary particle's rest-mass.

We have also shown why the negative sign on the imaginary-unit is
necessary for correctness, in such cases. Yet, despite thereby
placing the definition of tachyon mass on a formal footing, this
representation can create some confusion when complex quantities
associated with both M and tM are discussed in the same mathematical
contexts; especially when the two appear in the same formulas.

For example, suppose there is yet another particle with an imaginary
mass, iM, with the same amount of mass as M, but which is tardyonic,
not tachyonic, and we get a negative sign in the equations from
somewhere other than R; say, when employing vectorial velocities, as
with the formula for momentum, and this tardyon goes in the opposite
direction to the original tardyon. This can happen, let's say, if
the oppositely-moving imaginary mass, iM, is for a particle traveling
near-to but slower-than lightspeed, and we want the imaginary unit,
i, to be interpreted according to its common convention of implying
that iM is merely a standard imaginary; such as, in quantum physics,
when we discuss processes involving massive virtual particles (for
instance, the neutral Z-particle and/or the charged W-particles that
mediate the weak-nuclear interactions).

How then do we distinguish between tardyonic and tachyonic -iM?

We could, and should, assign a different symbol to denote the
tachyonic -iM, but that does little to eliminate the potential for
confusion wrought by having two different interpretations of the same
imaginary-unit, i; one applied to tardyons, and another to tachyons.

To solve that problem, we can introduce a new imaginary-unit, an
operator I, called, let us say, the "imagination-unit", which
transforms any ordinary quantity into its tachyonic analog.

That is, multiplying I to any standard quantity and/or symbol is
defined - according to a new convention, to go along with the new
imaginary-unit - as imposing a transformation across the lightspeed
barrier, so that it is understood to project that quantity or symbol
into superluminal spacetime, where causality is reversed, all
velocities are FTL, and all objects therein can be referred to
as "actual imaginaries", to distinguish them from the standard
imaginaries that we deal with on an everyday basis in mathematics,
physics, and engineering.

Naturally, this would not help much if velocity restrictions are not
specified. So, we further define the imagination-unit, I, to imply
evaluation between lightspeed and infinite speed, exclusively.

That is, for example, instead of writing tM = -iM , we define the
tachyonic mass tM as follows;
[v = infinity]
tM = IM = M | ,
[v = c]
where the brackets indicate evaluation between the enclosed values,
but not at those values.

Representing tachyons in this way allows us to discuss a standard
particles with a negative imaginary mass, -iM, in the same context as
tachyonic particles, defined by IM, without fear of the confusion
that would be possible with two interpretations of the standard
imaginary-unit, i.

Case in point, suppose we want the tachyonic analog of a tardyon, but
the tardyon orginally has a negative imaginary mass, -iM, instead of
the standard mass, though it is not originally tachyonic.
We would not want -i to imply that this mass, or anything else, is
tachyonic.
So, we merely write;
[v = infinity]
-i(tM) = I(-iM) = -iM | ,
[v = c]
which reads:
"Negative imaginary tachyonic mass -i(tM) is equal to the tachyonic
analog of the standard negative imaginary mass, -iM, which is equal
to the standard negative imaginary mass, -iM, evaluated between c and
infinite speed, exclusively".

We can, of course, relate motion involving the tardyonic masses M and
iM to their tachyonic analogs using the Lorentz transformations,
because all tachyonic analogs are, by definition, in reference-frames
that are always moving relative to all tardyonic reference-frames, as
long as the tachyons do not move at infinite speed in either frame.

The mass of a tachyon that moves at infinite speed can be defined,
but that must be done quite separately, in a different manner, and
given only as a side-note, because such a tachyon cannot be treated
satisfactorily in any rigorous particle-physics setting, due to the
fact that the presence of an infinite velocity turns all equations
involving it into meaningless exercises.
Infinite-velocity tachyons can, of course, be imagined, and described
using pure mathematics, but they must be considered as having
applications only in metaphysical terms.

These ideas are probably best understood by inspecting the Velocity
Spectrum, denoted;

Vi[Iv[Ic[c[v[v=rel(0)[iv=abs(0)=iv]v=rel(0)]v]c]Ic]Iv]Vi ,

where Vi is infinite speed, Iv is any velocity between tachyonic
lightspeed Ic and infinite speed (exclusively), c is the lightspeed
constant, v is any ordinary velocity between v = rel(0) and c (also
exclusively), v = rel(0) is a relative zero velocity, iv = abs(0) is
an absolute zero velocity (which is a standard pure imaginary), and
the underlines indicate corresponding values for antiparticles.

Note that tachyonic lightspeed, Ic, can further be defined;
Ic = (1.00...001)c ,
with the exact number of zeros to the right of the decimal-point an
empirical unknown - making tachyonic lightspeed both an irrational
and a transcendental number.

Considering first only standard particles [everything to the left of
abs(0)], and no antiparticles, when dealing with complicated systems,
one-to-one correspondences across the lightspeed barrier, associating
standard variables with their tachyonic analogs, can be realized by
integrating with respect to velocities on the other side of c,
exclusive of c and Vi. That is, the evaluations associated with "I"
can be accomplished using integration, whenever a spread of real
quantities in standard spacetime must be related to a corresponding
spread of their tachyonic analogs in superluminal spacetime.

A similar tactic is employed for the antiparticles [everything to the
right of abs(0)].

Finally, it is not always necessary to use the imagination-unit to
describe things made of tachyons, but it is given here as a viable
option when complex tardyonic quantities and complex tachyonic
quantities are treated in the same context, and a method is needed to
eliminate confusion between the two.