Searching for Tachyons

Verification/nullification of my thesis on tachyonic gravity depends on the repeatable experimental detection of tachyons, in a manner that proves beyond doubt that they exist, and thereafter on the development of experiments that reveal tachyon characteristics.

To figure out experimental apparatus that might absolutely prove/disprove the existence of tachyons, therefore, I have been studying (for the past three decades) means by which searches for tachyons have been carried out. One good source for an overview of such efforts is the book Superluminal Phenomena In Modern Perspective, by Suresh C. Tiwari (Institute of Natural Philosophy, Varanasi, India), from Rinton Press (published in 2003), and I would like to post here information from the book, and from the review I published of it (online) a few years ago (but now with updated comments, and some extra research).
The point is to list and elaborate ways tachyons could be proven to exist.

The book itself is a comprehensive review of the literature on superluminal phenomena, and includes a discussion of the math and physics concepts needed to understand any/all superluminal phenomena. But I do not recommend buying it, unless you are as devoted to this subject as I am, because it has a number of printing deficiencies (not Tiwari's fault), and it is not cheap. However, it is easy to obtain on-loan from a library, through the Inter-Library Loan Program. And in any case, I will provide enough information from it that readers here need not bother obtaining it (unless you want to see how Tiwari addresses subjects other than tachyons; e.g., quantum entanglement, photonic-tunneling, astrophysical observations, and so on). [Online, search "Suresh C. Tiwari", from India.]

Let's Consider Tachyons

In the chapter entitled "Fundamental Concepts", Tiwari references Einstein's 1905 paper on Special Relativity (published in Germany), and also Einstein's book, The Meaning of Relativity (first published in 1922), then gives a brief explanation of the Special Theory of Relativity (STR), based on those works, and as part of that explanation asserts that "it is only the constancy of the velocity of light that is crucial in STR", and that "whether it has to be a limiting velocity is unimportant."

I had already reached that conclusion, but it was nice to see Tiwari had done the same.
[Besides researching relativity in all ways, I own an old copy of Einstein's popularization "Relativity: The Special and General Theory", from Crown Publishing (1961).]

He also points to a book by the knighted Cambridge University professor Sir Arthur S. Eddington, entitled "The Mathematical Theory of Special Relativity" (Cambridge U. Press, 2nd ed., 1924), and indicates that it was perhaps the first example of a formal discussion on the STR-related possibility of the existence of faster-than-light particles.

Many other sources, however, state that Arnold Sommerfeld was the first physicist to seriously consider the existence of FTL particles, prior to Einstein's 1905 paper. But Tiwari points out that G.L. LeSage (of Geneva), in 1782, was actually the first in the literature to publish the suggestion of FTL particles, referred to as "ultra-mundane corpuscles", in an attempt to explain, in a mechanical theory, Newton's formula for gravity [although it is said elsewhere that LeSage, in 1758, based his theory on the unpublished work (completed in 1690) of the French mathematician N.F. de Duillier]. Thus, it appears that Sommerfeld was not the first to contemplate FTL particles, though Eddington is indeed the first to recognize that Einstein's STR implied their existence.

As a young man, I actually studied the work of Einstein, Max Born, Feynman, and others while getting into advanced calculus, on my own (then spent 4.5 years in college doing math, physics, and computer science). But I began intensely researching tachyons since first reading an article about them in Scientific American when I was 24 years old, and used much of my free time in college to research the math and physics of tachyons.

In his own book, at the beginning of the chapter entitled "Tachyons", Tiwari declares that Eddington admitted FTL particles are "not forbidden" by STR, but that Eddington then "discards them on physical grounds", maintaining that, because the lightspeed constant acts as a space-time barrier (universal speed-limit), such particles cannot exist.

Of course, no doubt due to Eddington's prominence, that view prevailed among physicists for a very long time, and is still with us -- although many physicists today recognize that this particular assertion must now be viewed as an out-dated assumption. [Otherwise, we may as well call the scientists investigating superluminal phenomena a bunch of dolts!]

As to the first formal physics paper submitting the concept of tachyons to physicists, Tiwari cites the 1962 article in the American Journal of Physics (number 30, page 718), entitled "Meta-Relativity", by O.M.P. Bilaniuk, V.K. Deshpande, and E.C.G. Sudarshan, which proposed the existence of FTL particles (later named "tachyons"), and showed that, despite the fact that c is a limiting velocity, STR does not forbid their existence, and that "no physical principles are violated" by the possibility of their existence.

The name "tachyon" was coined some years later by the physicist Gerald Feinberg, who is thus sometimes believed to have originated the concept. [I may post more on this later.]

Tiwari shows the equations for energy and momentum from STR, involving what I call Einstein's "Relativity Operator", 1/[(1-[(v/c)^2])^(1/2)], and then explains how the theory implies the existence of tachyons. But I assume that readers are sufficiently familiar with STR that the equations need not be reproduced here (they are easy to look up, if needed). My thesis on tachyonic gravity, on which I consulted with Tiwari, was originally based on the same formulas. But I have since determined that Lorentzian Relativity is better than Einstein's STR, and, as Tom Van Flandern has shown, is more consistent with the relevant experimental data.

Clearly, however, there is no requirement from either formulation that the only way to create a tachyon is to accelerate a bradyon past the lightspeed barrier. The equations simply imply the existence of all three types of particles; bradyons, luxons, & tachyons. Furthermore, although Tiwari does not say this (it is my own contention), there is also no prohibition against interactions occurring between all three types of particles. It is quite possible, in fact, that all bradyons and luxons have tachyonic substructure.

Now, a negative energy for tachyons means that they speed-up as they lose energy, as viewed from a standard (bradyonic) reference-frame. Infinite speed is the tachyon's zero-energy level, as we see them. And any causal difficulty we encounter in this can be resolved by applying what is called the "reinterpretation principle", as suggested by Bilaniuk and Sudarshan in 1969, which I paraphrase as follows:
For any observer viewing negative-energy tachyons traveling backward in time, there can be specified a superluminal frame from which an observer views the same tachyons as positive-energy particles traveling forward in time.
In other words, since "it's all relative", as they say, an observer in a standard frame views a tachyon as exhibiting reversed causality, compared to his own, while an observer in a superluminal frame would see the tachyon as having normal causality, and would thus regard the other observer as exhibiting the reversed causality.

Note, however, that the foregoing is rooted in Einstein's STR. And by adopting the Lorentizian approach, the reinterpretation principle can be modified, or dropped. And that seems to me both to simplify things and make it more accurate.

In this context, Tiwari mentions that there are at-least two categorizations of tachyons, based on velocity, corresponding to the two categorizations of bradyons; those moving, and those at rest. However, I suggest there are three cases; those at rest, those moving between lightspeed and infinite-speed, exclusively, and those going infinitely fast, as viewed in a superluminal frame (which does not correspond exactly to the situation for bradyons). But since we can view infinite-speed tachyons only in metaphysical terms, we can treat infinite-speed as the opposite of an absolute zero-speed, and therefore ignore it. Because, it seems logical to me to guess, the only tachyons we might be able to detect directly will have velocities that lie between lightspeed and infinite-speed, just as the only bradyons we detect move between lightspeed and zero-speed; exclusively.

The next two paragraphs represent my personal interpretation on some of the text in Tiwari's chapter on tachyons, and should not be confused with his position, to which I will refer again shortly. But I feel the need to clarify the relationship between absolute and relative zero velocity. Case in point, there is no such thing as an absolute zero speed, since everything we know is moving (somehow) relative to everything else. So, an object at rest can only be specified for a given reference-frame, and therefore has a relative zero speed, and cannot have an absolute zero speed. And this is true even if we reference the frame to the exact center of the universe (if we could determine it), because we have no way of knowing if the center of the universe is not moving with respect to empty space.

A bradyon is considered "at rest" if it has zero velocity measured from a frame that is also at rest with respect to the bradyon. But a nonzero velocity measured from any standard bradyonic frame can imply one of two equivalent conditions; (1) either the bradyon is moving with respect to the frame from which it is being observed, or (2) the bradyon is still at rest, in a given frame, and the observer's frame is moving relative to the resting bradyon. This implies that there is no truly absolute-zero velocity, anywhere.

All velocities are relative, experimentally; including all zero velocities. By the same token, while a tachyon can be considered "at rest" in a given superluminal frame, the corresponding zero-reference for a tachyon involves infinite speed, as viewed from the standard frame. But there is no absolute-infinite velocity for a tachyon either, because infinite-speed is just as relative as the bradyon's zero-speed. We therefore have two kinds of zero velocity; one absolute, and therefore purely non-physical, so that it is useless with respect to experimentation, while the other is relative, and therefore has usefulness in physics, as a starting-point for some nonzero velocity. Correspondingly, we also have two kinds of infinite velocity; one absolute, so that it too has no usefulness in experimental physics, while the other is relative, and is thus a valid reference.

Here, Tiwari reports (and I confirmed, by looking up the paper he cited) that relative infinite-speed tachyons are labeled "transcendental" tachyons by the above noted authors, and holds that such can be used to specify rest-frames for non-transcendental tachyons, but otherwise have limited importance. But I use the term "transcendental" differently in my thesis, because my transformation operator is viewed as imparting mathematically-imaginary status to all tachyonic quantities (mass, momentum, velocity, energy, ...), and thus to all tachyons. This does not mean they do not exist. It means only that they must be handled like imaginary numbers, when discussed in the same contexts as bradyonic quantities. And in pure mathematics, imaginaries can be considered both transcendental and irrational, because the standard imaginary-unit can be defined in terms of Pi and the base e of natural logarithms, which are both irrational and transcendental.

Reminder: The standard imaginary-unit is defined; i = (-1)^)1/2), so that i^2 = -1.
Pi, of course, is the ratio of the circumference over the diameter of any size of perfect circle, and is often rounded to the approximate value 3.14. The base e of natural logs is the limit as n approaches infinity of the n-th power of the sum of 1 and 1/n, for some integer n. But it is also defined using the following expansion;
e = 1 + 1/n! + 1/2! + 1/3! + ... + 1/n! + ... ,
commonly approximated as 2.72.
The relationship between i, Pi, and e is that i equals ln(-1) divided by Pi, denoted;
i = (-1)^(1/2) = [ln(-1)]/(Pi) ,
where ln(-1) is simply the "logarithm, to base e, of negative unity".
Now, Pi is referred to as "irrational" and "transcendental" because its decimal expansion is non-recurring and infinite (apparently). And, the last I checked, although computers have been used to calculate its value to several million decimal places, no final digit or recurring sequence had been determined for it, at that time (a few years back). I have since been told that someone "proved" this is not the case, but have yet to confirm it.
The base e of natural logs is labeled using the same terminology. Thus, because an imaginary number can always be represented as the product of i and any real number, we can state that they can also be defined in terms of these two irrational transcendental numbers -- and no-one would insist that Pi or e do not actually exist. So, I treat all tachyons as transcendental irrationals, when viewed from a bradyonic frame.

[More to come.]