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Basic Math for Tachyonics
Representing Tachyons
Calculations involving tachyons are, so far, mere exercises, useful only for stimulating the imagination, since there is no experimental way of detecting tachyons directly, and therefore no actual data on their specific characteristics.* Yet, this frees researchers to imagine tachyons of any desired form; the most ready scenario involving analogs of known particles.
The relativistic mass (m) of a real particle in motion can be related to the particle’s mass (mo) at rest by the equation;
m = mo / [ { 1 – [ (v/c)^2) ] }^(1/2) ] ,
where v is the moving particle’s velocity, and c is lightspeed (~186,000 mi/sec).
The mass (m) of a moving tachyon, therefore, can be represented by defining it as an imaginary analog of m; m = -im , where i is the imaginary unit, defined; i = (-1)^(1/2) , and the context or further definitions make clear that all velocities associated with the tachyon are faster-than-light (but not necessarily infinite in every example).
This sort of representation is sufficient when no complex calculations are investigated, but can cause confusion if complex quantities are present, because of the possible happenstance of having two interpretations of the negatively-signed imaginary-unit in the same context. For instance, we may want to use a tachyonic imaginary-unit, and to represent it in a way that distinguishes it from a standard imaginary-unit (as defined above). This, of course, means that we would need an operator that allows us to transform any sublight quantity or symbol into its tachyonic .
analog.
We can call this operator the “imagination unit”, symbolize it by “i“, and define it as imposing a transformation across the lightspeed barrier, so it is understood to mean causality is reversed, speed is superluminal, and material objects are to be viewed as “actual imaginary” objects, to distinguish them from the “standard imaginary” objects we deal with ordinarily (and for which time is positive, and speeds and/or instantaneous velocities range only from 0 to c).
That is, we say: If m is the mass of a real particle, then its tachyonic analog, mt, is defined; mt = im , where time is negative. Then we can write; imt = i(im) , which allows calculations associated with tachyons to also involve complex quantities, but without confusion between i and i, where the tachyonic imaginary unit, i, is defined in the same manner as the standard imaginary unit; i = (-1)^(1/2), and all tachyonic numbers (1, 2, ...) are treated the same as standard numbers (1, 2, ...), but only in their own frames of reference (so that the product of i and i would not equal -1). This is because the Lorentz transforms must hold in all caes, so that they can be used to study the differences between various objects in sublight, massless, and superluminal reference frames.
One way to get a handle on how the imagination unit works is to inspect the Velocity Spectrum, denoted;
Vi [ iv ( ic | c { v [ vo ( ivo = 0 = ivo ) vo ] v } c | ic ) iv ] Vi ,
where Vi is infinite speed, iv is velocity between tachyonic lightspeed ic and infinite speed (exclusive), c is lightspeed proper, v is an ordinary velocity (between vo and c, exclusive), vo is a relative zero velocity, ivo is an absolute zero velocity (an imaginary quantity), and the underlines indicate corresponding values for antiparticles.
Here, ic is defined; ic = 1.00...001c , with the number of decimal places unknown.
The one-to-one correspondences establishing values associated with application of the imagination unit are obtained by integrating with respect to velocity values on either side of c, exclusive of 0, c, and infinity, so that sublight and tachyonic gauges can be equated, for example, by relating a sublight integral S to its analagous tachyonic integral T;
[v = c] [v = Vi]
iS = iI f(a) dv <=> T = I ft(at) dv ,
[v = 0] [v = c]
where the big I denotes a definite integral, the "<=>" sign indicates a transformation, and v is any velocity, as established for an arbitrarily defined function f of some variable a, whose tachyonic analogs are ft and at, where a itself can be a scalar parameter (such as gauge or length), a component of a vector quantity (for example, the x-component of momentum, force, etc.), a position function (in one dimension), or any other scalar variable; the exclusivity of the integrals being necessary to keep them from becoming undefined entities.
Here, double integrals are used for two variables (a and b), triple integrals for three variables (a, b, and c), and so on; the lone integral applied to scalar quantities, the double to vectors and/or spinors in the plane or on a surface, and the triple to vectors in 3-dimensional space, with four or more integrations being associated only with abstract tensors.
Note that the transform n(ic) => n(1.00...001c) can be used to depict what occurs in superluminal tunneling experiments in which it appears as if photons briefly exhibit tachyon-like behavior, with fewer decimal places on the right corresponding to greater separation between c and ic, and where n denotes any necessary dimensionaless numerical factor (multiplier) indicated experimentally.
In standard considerations, the relativistic energy E of a particle, in terms of the particle's mass (m) and momentum (p), is given by a formula that can be denoted;
E^2 = (pc)^2 + [mo(c^2)]^2 ,
which can be rearranged to read;
E = c { (p^2) + [(moc)^2] }^(1/2) ,
where p = mv , for any velocity v, and in which the right side can be positive or negative; the positive case applying both to particles and antiparticles alike, leaving the negative case for tachyons and their corresponding antitachyons. Thus, for the standard representation of the formula for a tachyon's energy Et, we can write;
Et = - c { [ (pt^2) + [(imoc)^2] }^(1/2) = c { [(moc)^2] - (pt^2) }^(1/2) ,
where the tachyon's momentum, pt, is defined; pt = -imv , because standard tachyonic mass is -im , with v greater than c, and i = (-1)^(1/2); i^2 = -1. Note, however, that Et will not be equal to iE.
Here, again, the standard representation can cause confusion if complex quantities associated when tardyons and tachyons are considered together, so we must have a way to distinguish them in our calculations. The imagination unit, i, can therefore be used in the tachyon's energy formula, to give non-standard tachyonic energy, Et.
If we let vt denote any velocity above c; vt = { ic, iv, Vi } , then we can write;
Et = iE = - c { [(imvt )^2)] + [(imoc)^2] }^(1/2) = - c { [(mtvt)^2] + [(Etomtoc)^2] }^(1/2) ] ,
where pt = imvt = mtvt , and mt = im ; mto = imo .
This formula for Et lays out the relationships between the non-standard tachyonic analog (Et) of the standard energy (E) and the corresponding tachyonic analog (mt) of the original mass (m), revealing the rather straightforward transform;
iE = - c { [(mtvt)^2] + Etomto }^(1/2) ,
where (mtoc)^2 = Etomto , with Eto = mto(c^2) ; given the rest-energy, Eto, and rest-mass, mto, of the tachyon.
Note that, contrary to assumptions that are sometimes made, the rest-energy of the tachyon is zero only if vt is infinite;
Eto = 0 if and only if vt = Vi .
Otherwise, it is no more zero than is a standard rest-energy, unless v is an absolute zero velocity, which is also an imaginary quantity, and therefore unreachable.
Click here for more Math for Tachyonics, or click the link "Application to Gravity" at the lower right of this page.
* It has recently been suggested that the neutrino is actually a tachyon, since it was found that some neutrinos exhibit a negative value for their mass squared, which thus implies an imaginary mass for such neutrinos. Click the link labeled "Related Sites" below, and select pertinent sites, as listed, or click here, to learn more about this development.
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For more on tachyons in general:
do a Google search with keywords tachyon(s), tachyonic(s), superluminal, ...;
engage in library research on tachyons, superluminal tunneling, ...;
or click-on one of the following links.
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