New Math for Tachyonics                                                             Article by H. Kurt Richter

Representing Superluminal Quantities

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 superluminal analogs of known particles.

The relativistic mass (m) of a real particle in motion can be related to the particle’s rest-mass (mo) 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 (mt) of a moving tachyon, therefore, can be represented by defining it as a negatively-signed imaginary analog of m;
mt = -im ,
where i is the standard 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 (although this does not distinguish between sub-infinite and infinite superluminal velocity). This sort of scenario is sufficient when no complex calculations are involved, but causes confusion if complex quantities are part of the discussion. In such cases, it is necessary to represent the corresponding tachyonic imaginary-unit in a way that distinguishes it from a standard imaginary-unit. This, of course, means that we require an operator that transforms any bradyonic or luxonic quantity, term, symbol, variable, etc., into an exactly analogous tachyonic quantity.

I have devised such an operator, originally called the “imagination unit”, which I symbolize by using “i” in another form, such as a different font,i , and define it as a multiplicative operator that imposes a transformation across the lightspeed barrier, so that it is understood, by convention, to turn a standard quantity into a superluminal analog of itself; meaning, causality is reversed (when compared to the normal causality), speed is faster-than-light (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 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, and v > c. We next use ito generate a complete tachyonic number system, in exact analogy to the standard number system. For example, let {…,-2, -1, 0, +1, +2, …} = i{…, -2, -1, 0, +1, +2, …}. This allows calculations for tachyonic quantities to involve complex quantities without confusion between  i  and  i , where all tachyonic numbers (1, 2, ...) are treated algebraically the same as the standard numbers (1, 2, ...), but only in superluminal reference-frames (the product of i and idoes not equal -1).

Naturally, the Lorentz transformations can be employed here, so they can be used to study the differences between objects in bradyonic (sublight), luxonic (massless or photonic), and/or tachyonic (superluminal) reference-frames.

One way to get a handle on how the imagination-unit works as a transformation operator, is to perform a visual inspection of the Velocity Spectrum, denoted;

iV > { iv > ic > [ c > v > (vo = 0) > (ivo = 0abs = ivo) < (vo = 0) < v < c ] < ic < iv } < iV ,

where iV is infinite speed, iv is velocity between tachyonic lightspeed ic and infinite speed (exclusively), c is lightspeed proper,
v is an ordinary velocity (between vo and c, exclusively), vo is a relative zero velocity, ivo is an absolute zero velocity (a standard pure imaginary), and underlines indicate corresponding quantities for antiparticles.

Here, tachyonic lightspeed, ic, can be defined; ic = 1.00...001c, where the exact number of decimal places to the right is an empirical and statistical unknown.

One-to-one correspondences between bradyonic and tachyonic quantities of any kind, resulting from application of the imagination-unit, can easily be obtained by integration with respect to velocity values on either side of c, exclusive of 0, c, and infinity, so that bradyonic and tachyonic gauges can be related using the Lorentzian or other transformation equations.

Formally, the imagination-unit can be defined using a generalized evaluation equation, or as an evaluation function, where the evaluation is taken exclusively between c and infinite-speed, for any time t. For example, write;

     [v = infinity]
i = |                                    , where  t < 0  for Einsteinian Relativity.
     [v = c]

Thus, for any function f of an arbitrary variable x, we define tachyonic f(x);

                  [v = infinity]
i f(x) = f(x) |                          , where  t < 0  for Einsteinian Relativity.
                  [v = c]

Observe, however, that this definition is a pure evaluation, and does not specify how bradyonic, luxonic, and tachyonic frames of reference are related. And, while I originally called this operator the “imagination-unit”, it is now known more widely as Richter’s Tachyonics Operator. It is a simple evaluation operator, and does not specify a coordinate system, reference frame, dimensional manifold, or any other such consideration. It thus has universal application, regardless of the coordinate systems being used -- classically, quantum mechanically, relativistically, and so on (depending solely on user preference).

In standard physics, the relativistic energy E of a particle, in terms of the particle's mass (m) and momentum (p), can be given by a formula denoted;
E = (pc)^2 + [mo(c^2)]^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.
Alternatively, for the representation of a tachyon's energy Et, we can write;
Et = iE = i { (pc)^2 + [mo(c^2)]^2 } ,
instead of the usual representation (from Einstein’s Special Relativity).
That is, in this scheme, E and Et are dimensionally incompatible values (just as a real number is incompatible with a pure imaginary), because Et must be treated as a tachyonic imaginary. The reason for this representations format is that the standard scheme can cause confusion when complex quantities associated with bradyons and tachyons are considered in the same formula -- merely because a negatively-signed imaginary-unit does not necessarily imply superluminality in all cases. The difference must always be explained explicitly in the accompanying text. So, in a standard representation of a tachyon, it is unclear, from math symbolism alone, when a bradyonic or tachyonic quantity is specified using the negatively-signed standard imaginary-unit.

Employing the new Tachyonics Operator, however, removes this ambiguity, while leaving open the question of reference-system relationships. It does however mandate that natural limits on tachyons (existing at velocities between c and infinite-speed) must correspond to the natural limits of bradyons (between 0 and c). This therefore mitigates possible infinite solutions rendering calculations involving tachyonic quantities meaningless, which can occur in the standard representation of a tachyon.

Also, the standard interpretation is retained when reference-frames are related by the Lorentz Transformations in accord with Einstein’s theory of Special Relativity. But the Tachyonics Operator also works in other situations, which modify or omit the Einsteinian paradigm, such as in non-relativistic Quantum Theory, and in purely non-quantum Classical Physics.

A good example that retains the Lorentz formulas, however, while removing the reversed causality of Einsteinian Relativity, is the Lorentzian Relativity proposed by the late Tom Van Flandern, who conducted astronomical experiments that proved gravity is faster-than-light; thereby requiring that Einstein’s Relativity theories are at least reformulated.

One modification of Einsteinian Relativity that can also remove the causality issue stems from the knowledge that the physical constants are not perfectly constant in time. They are now all known to change slowly with time, due to the accelerating expansion of the visible universe. This knowledge removes the requirement of keeping the vacuum constant of lightspeed an invariant for all observers -- consequently implying that some modification of Einstein’s ideas is needed.

I am of the opinion that every scenario is allowed, in practice. The difference comes from what is investigated, and how one goes about it. In other words, I believe there are many kinds of tachyons; some have negative time (according to standard Einsteinian Relativity), some exhibit a “neutral” time (infinite-speed tachyons act instantaneously, so time does not “flow” for them), and some have positive time (as in Traditional Classical and in Classical Lorentzian schemes).

In any case, the Tachyonics Operator can be used to designate superluminal quantities without introducing confusion or conflict with existing paradigms.

Comments Welcome. E-mail: HKurtRichter@yahoo.com.

 Superluminal Gravitation and the Linearized Wave Equation

Theoretical Science-Fiction by Harrison Kurt Richter

Foreword

Gravity is described as a superluminal force, in a model involving radiating point-like tachyons that give rise to causally-reversed radiation pressure, which we experience as Newton’s gravity.

Set Schrödinger’s wave equation equal to the equation of a line for this type of tachyon, and let the particle be called a
“gravitational exchange tachyon”, or GET particle. Imagine it as a spin-less wave-less and infinitesimal point-particle (epitomizing the notion of the spatial point), and suggest that there exist such tachyons radiating naturally (and constantly) from all real masses; each GET traveling perfect Euclidean-straight lines from their sources to infinity, imparting some fraction of their causally-reversed momentum to all real objects through which they pass.

Because reversed causality orients a tachyon’s momentum-vector opposite to the direction of the particle’s travel, the GETs collectively establish negative radiation pressure in space -- which is what we understand as the gravity described by Newton’s law, in everyday life, or by Einstein’s theory of General Relativity when dealing with satellite orbits, distances in the Solar System, and for larger astronomical reaches, and/or very intense gravitational fields.

The math and physics relied upon to understand this subject has been published online. Readers are invited to check the work, and send comments as desired. But since it is all viewed by many in the scientific community as a trivial exercise in science-fiction (i.e., entirely speculative), the ideas presented can be ignored, as desired. However, modern history has shown that science-fiction ideas sometimes become matters of fact. So, the future is ours for the shaping.

The Concept of Tachyons and How They Cause Gravity

Einstein’s famous equation for the rest-mass of a real particle is denoted; E = m(c^2) , where E is energy, m is mass, and c is the vacuum constant of lightspeed (~ 3x108 m/s). But if E = m(c^2) is positive, then the negative form, -E = -mt(c^2) , can be considered tachyonic (superluminal).

A tachyon is a particle of rest-mass -mt and rest-energy -E , but it always travels faster-than-light (FTL), and has negative time (reversed causality). These are the particles on the other side of the lightspeed barrier, and there are probably more varieties of them than there are of bradyons.

Let v denote any concievable velocity. Then there are three categories of particles, according to Einstein's theory of Special Relativity (involving the above formula):
1. The range 0 < v < c is for ordinary particles, called “bradyons”.
2. The lightspeed constant, v = c , is for massless particles, called “photons” and “luxons“.
3. The range c < v < (infinite-speed) is for tachyons, obtained as analogs of ordinary particles.
Other tachyons can be imagined, of course (including those traveling infinitely fast), but these should be viewed as accounting for phenomena other than gravity.

The critical thing here is that we have to set bounds upon tachyons in analogy to the kinds of bounds we observe are natural to bradyons. For the theory of Superluminal Gravitation, then, we can postulate the existence of a point-tachyon, the GET particle, with no wave characteristics, and which shoots out FTL along a perfectly straight line from the instant of its creation (by any bradyonic mass), although it must interact with bradyonic bodies in its path in such a way as to impart some forward momentum to those bodies; setting up radiation pressure in space.

Because of its reversed causality, this pressure is negative; causing a pull rather than a push. The collective action of all of the radiating GETs consequently establishes the classical "Newtonian" law of gravitational attraction, F = GM1M2/(r^2), and also adheres to Einstein's theory of General Relativity (in particular, the field equations; explained elsewhere).

Attempts to apply Tachyonics to “standard” quantum gravity, involving a spin-2 massless boson, called a "graviton", as the exchange-particle (in analogy to virtual photons of electromagnetism), by setting the graviton's speed FTL, worked to unify gravity, in an empirical fashion, with the other elementary forces (in a gauge-field format). But it soon becomes clear, after applying a bit of reductionism, that a more fundamental scheme can be obtained; which is a simple monopole particle-radiator model, with one-dimensional polarity (just as needed for representing gravity).

After all, gravity does behave as a monopole, quite unlike the quadrupole model usually given, but which has never been satisfactory. Rather, gravity acts, for all intents and purposes, like a wholly classical field of force. The only modification to Newton's law of universal gravitation in the new scheme is not to the well-known formula for the force F, but to assumptions about it; that it implies instantaneous action-at-a-distance. We specify, instead, that gravity appears to be instantaneous, to our senses and instruments, because it propagates FTL -- and our technology remains inextricably tied to the electromagnetic spectrum (with v = c), and is thus limited by it.

Notice that a special case of the wave-equation is available by reducing all wave characteristics of the wave-function to zero, but not necessarily collapsing the function. Rather, for the GET particle, the wave-equation becomes an equation of a line, oriented along a corresponding Ricci Scalar (radius of curvature) in General Relativity. This implies that we can compact all of the quantum-mechanical information about the GET into its path, of which the Ricci Scalar is the line-segment containing the gravitational force vector. This unifies General Relativity with modern Quantum Mechanics, because it means we can derive the one from the other, with this connection between them; quantum physics is compacted into the Ricci Scalar of GR, assuming a linearized wave-equation for the GET particle.

In that case, it can now be shown, through logical inference and circumstantial evidence, that:

Gravity is probably faster than light, and may therefore be a tachyonic force.

Comments welcome: HKurtRichter@yahoo.com