The Imagination-Unit (continued)

[Continued from post "The Imagination-Unit", Part 1 (include definite article and hyphenation in title).]

Part 2: Tachyons and Special Relativity

At this point, in the interests of clarity, I should discuss SR in a little more detail. Readers familiar with SR, and the notion of tachyons, can skip this part.

Consider an ordinary object at rest; for example, a basketball at rest on a basketball court. It has a rest-energy E and a rest-mass m, related by the well-known equation

E = mc^{2},

where c is the lightspeed constant (approximately 3 x 10^{8}meters/second).

Suppose we next roll the basketball across the court; setting it in motion with respect to the stationary surface of the court. It can then be viewed as existing in a different frame, local to itself; moving relative to the stationary frame of the court. And to relate these frames, we can apply transformation equations to the variables associated with various quantities (mass, velocity, ...) specified initially in either frame.

Orient a set of Cartesian coordinate axes so that the ball's center-of-gravity starts at the origin O, fixed relative to the floor, where we begin counting time t at t = 0, and the ball's center-of-gravity, with a mere push, can be made to move in the positive x-direction at a constant velocity v, without obstruction, so that the values of y and z are always zero.

Next, let x, y, and z denote the spatial parameters, and t the time parameter, for the stationary reference-frame, but let x', y', z', and t' denote the corresponding respective parameters for the moving reference-frame (the one moving with the ball), and where the x-axis and the x'-axis lie on the same infinitely-long line in space. Then the reference-frames will be related according to the Lorentz transformations;

x' = R(x - vt) ,

x = R(x' + vt') ,

y' = y ,

z' = z ,

t' = R[t - (vx/c^{2})] ,

t = R[t' + (vx'/c^{2}] ,

where the Relativity Operator, R = 1/{[1 - (v/c)^{2}]^{1/2}}, allows us to calculate the relative value of a quantity for a moving object from the corresponding value at rest.

If M denotes the basketball's moving mass, and m is its rest-mass, then we have;

M = mR = m[(1 - [(v/c)^2])^(-1/2)] = m/{[1 - (v/c)^{2}]^{1/2}}.

Notice therefore that, because the ratio v/c is part of the expression in R of which we take a square-root, then there is only one relationship between v and c that makes sense for a real basketball with positive time; v < c.

Suppose now, however, that we let M denote the mass of a real or a virtual subatomic particle, instead of a basketball. Then there are the three fundamental cases for M;

v < c, for positive real bradyons,

v = c, for massless photons, and

v > c, for negative imaginary tachyons.

Most of the subatomic particles cataloged by physicists as having mass, as far as we can tell, have positive rest-mass, including both real and virtual particles with mass. [Note: The neutrino may be the first exception to this rule to be recognized.] The scalar energy E and vector momentum P are defined using the real rest-mass m;

E = R(mc^{2})

and

P = R(mV) ,

where V is vector velocity; | V | = v .

Of note is the fact that the second case, for massless photons, actually works-out to make R an infinity if we embrace the mathematical convention that the inverse of 0 is infinity;

1/0 = (infinity) .

This occurs because, if v = c, then

R = 1/[(1 - 1^{2})^{1/2}] = 1/(0^{1/2}) = 1/0 .

Alternatively, yet remaining mathematically rigorous, we can say instead that the inverse of 0, in such cases, is "undefined", and maintain that the rest-mass of a photon is 0; which means all photons are massless particles, made entirely of energy.

Contrastingly, tachyons are particles with negative rest-mass that always travel faster-than-light, and have reversed causality (negative time), compared to bradyons. And their rest-mass is both imaginary and negatively signed.

I must now go into greater detail on this than has been provided for the other two cases.

Notice that the relativity operator, R, dictates what happens when you try to accelerate a real mass up to lightspeed. It works-out that M approaches infinity as v approaches c. In other words, it would take an infinite amount of energy to accelerate a bradyonic mass up to lightspeed. And because we do not have access to infinite energy, and do not observe infinite energy expended anywhere in the universe at large, then the lightspeed constant represents a kind of universal speed-limit. It is, by all accounts, a space-time barrier.

Thus, many physicists assumed (logically) that nothing "real" exists on the other side of lightspeed. Unfortunately, this has also caused some to conclude that tachyons cannot be created, even by a Big Bang like the one that initiated our universe. Hence, some people continue to insist that tachyons do not and cannot exist.

To be clear, the relativity operator, R, does not mandate that nothing faster-than-light (FTL) can exist, somewhere. It does indicate that it would require infinite energy to accelerate a real mass up to c, but it does not forbid objects that already travel at FTL speeds from existing on the other side of the lightspeed barrier. Nor is it necessary to get tachyons by accelerating real masses to and beyond c. In the cosmological Big Bang idea called "Inflation Theory", it is said that there was a period of superluminal expansion for all the energy associated with the first moments of the Big Bang. It is therefore entirely possible that many particles of various kind were created that retained the superluminal velocities of the energies out of which they were formed, at that time. Furthermore, because of its reversed causality, a tachyon's energy decreases as its velocity increases, with its zero-energy state at infinite speed. So, it is reasonable to think that higher-speed tachyons were easily created, because the required energy would be extremely low.

Also, while we depict tachyons as having imaginary mass, mathematically, we must remember that words like "imaginary", "abstract", and other terms employed in math contexts are labels for different types of numbers and numerical quantities, chosen to distinguish between them. But such a label does not necessarily imply that imaginary quantities do not exist. Thus, to label a tachyon's mass as "imaginary" does not imply non-existence for tachyons, because we are using the strict mathematical meaning of the word "imaginary", not its common literary meaning.

Interestingly, the standard imaginary-unit, i, can be defined in terms of two well-known irrational transcendental numbers. One of these is the value of Pi (the ratio of the circumference over the diameter of any size of perfect circle), and is often given the approximate value of 3.14. The other is the base e of natural logarithms, defined as the limit as n approaches infinity of the n-th power of the sum of 1 and 1/n, for any integer n. It is also defined using the following expansion;

e = 1 + 1/n! + 1/2! + 1/3! + ... + 1/n! + ... ,

which is 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 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). In fact, to date, though computers have been used to calculate its value to several million decimal places, we have yet to find its final digit, or to identify a recurring pattern. And the base e of natural logarithms is labeled using the same terminology, for similar reasons. 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 -- although no-one would insist that Pi or e do not actually exist.

Consequently, just because we think of tachyons as imaginary, theoretically speaking, this does not mean that they cannot or do not exist.

To understand how tachyons work, be aware that it would take an infinite amount of energy to slow a tachyon down to c, just as it would take an infinite amount of energy to speed a bradyon up to c. And if we could see the emission of a tachyon from a composite body, as viewed from a bradyonic frame, it would appear as if the tachyon came from an infinite or very far-off distance and was completely absorbed by that body. That is, if we have a video of the ordinary emission of a bradyon from the body, the analogous ejection of a tachyonic analog of the bradyon would look much like we had merely run the video of the ordinary process in reverse.

[Continued in my next post.]