Transformations Across the Lightspeed Barrier

The traditional method of describing a tachyon is to input a superluminal velocity into some formula involving the
Relativity Operator, 1/[{1 – [(v/c)^2]}^(1/2)], and use the negatively-signed but otherwise standard imaginary-unit, i ,
to express the result.  For instance, if m is the mass of a bradyon, then the imaginary mass u for a perfectly analgous tachyon
can be obtained as a direct analog, by writing;  u = -im .
[Reference: Encyclopedia of Physics, 2nd Ed., Lerner & Trigg. VCH Publishing, p 1246.]

However, due to the implication that there exists a superluminal universe (from the Light Cone of Einstein’s theory of Special Relativity), we can guess that it exhibits its own number system, incompatible with the standard number system (although, there is a one-to-one correspondence), just as the set of all real numbers is not additively compatible with the set of all pure imaginaries (i.e., a sum from both sets forms an irreducible complex term; example, z = x + iy). 

In that case, the standard imaginary-unit, i = (-1)^(1/2); i^2 = -1, becomes inadequate to define complex tachyonic quantities
in the same formulas as complex bradyonic quantities, without resulting in confusion and too much redundancy (although the standard method is acceptable in most other situations); explained as follows.  





At issue is that a negatively-signed standard imaginary-unit does not always imply superluminality.  There are many instances of negative imaginaries in the equations physicists use to study natural processes.  Thus, it becomes confusing if we also use the same symbolism to define tachyons -- and it is bothersome to repeatedly have to explain representations in an accompanying text. 

We can solve this problem by defining tachyons using an operator, i, inspired by the standard imaginary-unit, i, but which is used specifically to define tachyonic quantities.  This frees-up the standard imaginary-unit, so that negative imaginaries can be treated in the usual fashion, without confusing them for implying superluminality.

This new operator, originally called the "imagination-unit", can be referred to as the "Tachyonics Operator", and its application is very simple.  Multiplying the operator,i,  to any real quantity/variable transforms the quantity/variable into a superluminal analog of itself.  And it is usually sufficient to provide a transformation equation, and a statement of the mathematical conventions involved, in context, to illustrate how the new operator is to be interpreted and applied. 

In other words, if m is a bradyon's mass, then mt = im can denote the corresponding mass of a tachyon taken in perfect analogy to m, but which does not explicitly depend on the negatively-signed standard imaginary-unit. 

However, notice that Einstein’s Relativity Operator does not include a distinction between sub-infinite and infinite superluminal velocities, so transformations defined by must be limited, to keep infinities from inadvertently nullifying formulations involving tachyons (which happens in the standard scheme), if we wish to adhere to Einsteinian Relativity (e.g., as in the Relativistic Quantum Field Theories used in elementary particle physics).

Here, let i = i^i , where the superscript is not an exponent (but is just symbolism), and let m = mr , to indicate "real" mass.

Then, the Tachyonics Operator can be defined using a general evaluation equation; 

                                                      ,

where f is some function of an arbitrary bradyonic variable x .

If we want f(x) to represent a constant, such as a real bradyonic mass mr, then write;

                                                              ,

where t < 0 for Einsteinian Relativity.
This formula simply states that a tachyonic mass results from multiplying the tachyonics operator to a bradyonic mass, which involves evaluation in superluminal spacetime (between lightspeed and infinite speed, exclusively), where reversed causality holds for the resulting tachyonic analog if we are committed to Einsteinian Relativity. 

However, there is no a-priori requirement to use Einstein’s scheme, involving the Lorentz Transformations for relating quantities in different inertial frames.  The Tachyonics Operator will work with any desired set of coordinate transformations (non-Einsteinian Lorentzian, classical Quantum Mechanical, Brane or String Theoretical, etc.). This makes it suitable for applications in which reversed causality for superluminal particles is not actually required or is not desired.

For constants and variables that can take ranges of values, we obtain data-spreads on either side of c, with one-to-one correspondences between bradyonic and tachyonic values, using basic integration; to establish a scaling format for the superluminal analogs of bradyonic quantities.




Notice, then, that the Tachyonics Operator can be used to define an entire number system for superluminal spacetime -- including the specification of a superluminal analog of the standard imaginary-unit; it = (i^i)i .  And while this may seem trivial (more like science-fiction or pseudo-science, than usable physics), it does provide a means of representing metaphysical numbers in a quantitative manner (the formula can output a numeric result, so, in theory, given the technology, it can be tested experimentally).

This could have important applications in artificial intelligence (synthetic brain) research, if consciousness and the sub-conscious can be modeled as the interaction of the gray-matter of the human brain with ubiquitous superluminal energy fields that invisibly permeate the visible universe.  [It may lead to a breakthrough, by suggesting how to impart self-awareness to machines.]

A convenient way to understand how transformations across the lightspeed barrier actually work is to perform a visual inspection of the velocity spectrum, involving all possible theoretical velocities, denoted;
            iVabs| iV > {iv > ic > [c > v > (vo= 0 > | ivo = 0abs = ivo | < vo=0)< v < c ] ic < iv}< iV | iVabs
where iV is infinite-speed, iv is velocity between tachyonic-lightspeed (ic) and infinite-speed (taken exclusively), c is lightspeed proper, v is an ordinary velocity between vo and c (also taken exclusively), vo is a relative zero velocity (specified in a context), ivo is an absolute zero velocity (a standard pure-imaginary), and the underlines to the right indicate corresponding quantities for antiparticles.  All bradyons exist between the square brackets, and are mathematically real, with the exception that an absolute zero velocity is a pure imaginary, since there is no way of setting an absolutely fixed starting point -- even with respect to the very center of the universe (because we have no way of determining if that point is or is not moving with respect to empty space). Correspondingly, there is an absolute infinite-speed, which must be distinguished from the relative infinite-speed.

The Tachyonics Operator imposes a transformation across the lightspeed barrier, which can be specified for any bradyon (or its antiparticle), but is restricted to evaluation within the exclusive limits.  The simple transform is;  [v, v ] --> {iv, iv}.

The most experimentally testable section of the velocity spectrum therefore falls between relative-zero and a relative-infinite velocity, exclusively, with lightspeed as a central space-time node (the reference “barrier” across which the transform occurs).

As we can see, specifying the velocity spectrum assists in understanding how some transformation-operator can relate bradyonic quantities to their superluminal analogs, where scaling, gauge, or data correspondences are implemented.  To explain further, note that, if, for some function F we can obtain derivatives f’(x) with respect to a continuous range of values for a variable x , denoted;


                                                                                                                                           ,
then we can proceed, correspondingly, to give a definite integral between positive and negative infinity, equal to the sum of the integrals that take negative and positive infinity separately;
 

                                                                                                                                                       .

Thus, if we let x be a function of time t , as an example, then the positive integral can be used for studying bradyons, and the negative for tachyons with reversed causality; assuming Einsteinian Relativity.  Imposing exclusivity on the two integrals renders them empirical (meaning; applicable to something measureable in experimental settings);



The situation for velocity, however, is rather different. Suppose x is a function of velocity v (although, it therefore remains a function of time, since velocity is the derivative of distance with respect to time). Then negative infinity is replaced by antiparticle infinity, and both 0 and c become spacetime nodes, establishing four integrals on the right, making up the following set;




if we maintain exclusivity of limits throughout. [In practice, we can sometimes remove the exclusivity for 0 and/or c , but cannot do so for the infinities, and maintain empiricism.]  Here, overbars indicate values for antiparticles, instead of underlines.  

It is simple, then, to imagine a tachyonic analog for any known bradyon, and obtain hypothetical sets of characteristics for each of these analogs.  The tactic is to give an integration example to establish specific one-to-one correspondences across the lightspeed barrier, for the given variable x as a parameter or other specification of particle properties.  The Tachyonics Operator then serves as a convenient sort of notation convention, when complex quantities for bradyons and tachyons appear in the same formula, and where the tachyons are taken in exact analogy to bradyons.  The basic integration formulas are;  




and




for particles and antiparticles, respectively.

Of course, integration is not always needed.  The Tachyonics Operator also works algebraically, by treating it like a new kind of imaginary-unit.  And, in that case, it is an intuitive matter to postulate tachyons which are not analogs of known bradyons, merely by indicating quantities describing their properties/characteristics and using the Operator to define them as superluminal in nature.

For example, we could postulate a purely classical, point-like and spinless tachyon that travels a perfectly straight Euclidean trajectory, from its point of origin to a point at infinity, and which interacts with real matter in its path in various ways as it speeds across the universe.  But tachyons with other properties can also be imagined, although they are all only science-fiction concepts, at this time.  However, in order to maintain empiricism, it will be necessary to continue to regard the Operator as a metaphysical concept, even after experimental confirmation of its usefulness has been verified.

Conclusion:  The Tachyonics Operator, denoting an empirical evaluation, indicates a transformation across the lightspeed barrier of spacetime.  It is understood, by definition, to transform any bradyonic quantity/variable into a tachyonic analog.  A necessary convention, in that respect, is that it must be treated algebraically as a new kind of imaginary-unit, and a correlating mandatory convention is that it must be understood as a metaphysical concept, including when it is employed in experimental settings.

Applications of the Tachyonics Operator reach into many areas  --  most importantly;  modern cosmology, particle physics, artificial intelligence research, and biophysics, with serious ramifications for the epistimology of science as a whole.


Comments are welcome. E-mail:  HKurtRichter@yahoo.com

 
 Tachyonic Metaphysics

The Tachyonic Metaphysics Group is dedicated to understanding both the seen and unseen aspects of our existence according to the tenets of the new natural philosophy called Interdiscipline Synthesis Cosmology (ISC) -- in which Tachyonics is employed to explain the life-force of all living things, the true nature of the mind, consciousness and the subconscious, a number of previously unexplained natural phenomena (such as quantum gravity), and all paranormal phenomena (including psychic ability, actual magic, astrological influences, and the substance of spirit entities).

Categories
Tachyonics, Superluminal Gravitation, Magic and the Unseen Forces of Existence, the Physics of the Gods.

Founder: H. Kurt Richter  www.groups.yahoo.com/group/tachyonicmetaphysics/

Topics
 
Tachyonics
Tachyonics is the study of hypothetical particles called “tachyons”, which travel faster-than-light, and may or may not have negative time (depending on the kind). This new endeavor offers incredible insights into reality, and serves as the basis for formulating an Interdiscipline Synthesis Cosmology as a natural philosophy whose database may eventually give rise to a valid Theory of Existence, capable of describing all natural processes in the most accurate manner possible. [The physicist’s Grand Unified Field Theory would necessarily be an important and essential component of the ISC ToE.]

Superluminal Gravitation
Gravity can be unified with the other forces of nature by depicting it as an "imaginary" tachyonic field. Modern astronomical observations have confirmed that gravity acts faster-than-light (go to MetaResearch.com for details). Representing gravity as an imaginary superluminal field of force is therefore mandatory, for obtaining empirical validity and conceptual accuracy, in understanding how gravity works subatomically.

Magic and the Unseen Forces of Existence
Magic, mysticism, and psychic ability can be explained as the manipulation of “unseen” superluminal forces by the human brain. In fact, ISC not only implies the existence of such phenomena, but asserts that they can now be investigated experimentally, since Tachyonics provides relevant mathematical representations for these phenomena.

The Physics of the Gods
Incorporating Tachyonics into modern theoretical physics results in the notion of superluminal gravitation, which constitutes a means of blending Einstein’s theory of General Relativity with modern Quantum Mechanics. This, then, allows gravity to be incorporated with the other fundamental forces in a gauge-field format, which, in turn, enables construction of a Grand Unified Field Theory (GUFT) in the same format.


Tachyons

Tachyons are hypothetical particles which always travel faster than light. They can exhibit reversed causality (negative time) compared to ordinary particles, and some of them are capable of moving at infinite speed. It is by virtue of their superluminal speeds that tachyons (predicted by certain implications of Einstein's theory of Relativity) are the most important of theoretical particles being sought by physicists, since they potentially offer means by which we may explain the nature of quantum gravity in an empirical way. 

Additionally, tachyonics (the study of tachyons) could lead to technological advances so profound as to be unimaginable today. Tachyonics, in fact, hands us the possibility of obtaining instantaneous communications capabilities, spanning unlimited reaches of space; the certainty of finding a practical means of breaking the lightspeed barrier to spaceflight; and the prospect of tapping into unlimited sources of energy.

What is more, the tachyon can be used to explain a plethora of previously unexplained natural and supernatural phenomena, including the way the mind works, the life-force of all living things, the nature of spirit beings, actual magic, and psychic abilities, such as clairvoyance, distant viewing, telepathy, and psycho-kinesis.

To be sure, actual detection of tachyons would revolutionize the way we view the universe and our place in it, since proof that tachyons exist would encourage seekers of truth to adopt the new natural philosopy Interdiscipline Synthesis Cosmology, which blends science and spirituality in an empirical way (i.e., experimentally testable).

Traditionally, a tachyonic mass Tm can be viewed as the superluminal analog of a sub-light mass Sm (called a “bradyonic” mass), where an operator similar to the imaginary unit [ i = (-1)^(1/2) ; i^2 = -1 ] transforms Sm into Tm -- just like the imaginary unit (i) transforms any real number (y) into an imaginary number. [iy is found in a complex number of the form z = x + iy, where z is the complex number, x is the real-number part of z, and y is the imaginary-number part of z.] In this case, multiplication by the new unit, called the “imagination unit”, which can be denoted as "i^i", does not imply a standard transformation (such as a rotation), but implies transposition of specifications relating to the speed at which Sm travels through space. Hence, if Sm is the mass of a real particle in ordinary space-time, then the equation Tm = (i^i)Sm defines Tm as a tachyonic analog of Sm, where Tm is treated algebraically like an imaginary quantity.

Now, for decades it was believed that all tachyons have negative time. But recent work has shown that this need not be the case for all tachyons. Some may obey classical and/or quantum-mechanical physics, which does not involve reversed causality. Also, Einstein’s theory of Special Relativity (which was at one time interpreted as “predicting” the existence of causally-reversed tachyons) depends on the assumption that the speed of light is invariant (constant) for all observers. However, modern experimental tests of the physical constants has revealed that they are not fixed, but slowly change their values with passing of time, due to the accelerating expansion of the visible universe. This means Einstein’s conclusion that lightspeed is an invariant constant is not valid, and a modification of Special Relativity is therefore warranted.

Several methods for modifying Special Relativity have been put forward, with the most promising being Lorentzian Relativity (instead of Einsteinian Relativity), since Einstein derived his operator (1 – [(v/c)^2])^(-1/2) from the Lorentz Transformations, and a logical course would be to go back to where Einstein obtained this operator.

With Lorentzian Relativity replacing Einsteinian Relativity, the concept of the tachyon is not removed, but the necessity for viewing it as existing in negative time is not needed. On the other hand, it is most probable that many different kinds of tachyons exist; some with negative time, some instantaneous (with zero time), and some with positive time.  So, we are going to need all the mathematical and theoretical tools at our disposal.  


Comments Welcome: HKurtRichter@yahoo.com
http://groups.yahoo.com/group/tachyonicmetaphysics/