Superluminal Gravitation and the Linearized Wave Equation

Foreword

I attempt to describe gravity as a superluminal force by suggesting that small point-like tachyons give rise to a causally-reversed radiation pressure which we experience as Newton’s law of universal gravitation.

The idea involves setting Schrödinger’s wave equation equivalent to the equation of a line in space, for the tachyon held as causing gravitation. In particular, this model of gravity envisions an infinitesimally small point-particle, epitomizing the mathematical notion of a point in space, and suggests that there may exist such point-like tachyons radiating naturally (and constantly) from all real masses in all directions; each such tachyon traveling perfect Euclidean-straight lines from their sources to infinity, imparting some of their causally-reversed momentum to all of the “real” objects through which they pass on their journey. And since reversed causality orients an imparted momentum-vector in the opposite direction to the direction of the tachyon's travel, then such tachyons would collectively establish negative radiation pressure in ordinary space. This is what we understand as the gravity described by Newton’s law of gravitation, locally, and by Einstein’s General Theory of Relativity, when dealing with large astronomical reaches.

Also, if gravity is tachyonic, and there is a superluminal universe in connection with the detectable universe (but unseen due to existing in alternate-dimensional spacetime), then the gravity from tachyonic matter would cause repulsion for all ordinary matter, and this could explain the "dark energy" and "dark matter" that astronomers are interested in understanding more completely.

I provide here the mathematical justification for my theory. Readers are invited to check my work. But since it is all speculative, it can be ignored, if desired.

The Concept of Tachyons and Tachyonic Spacetime

Einstein’s famous equation for the rest-mass of a real particle is; E = mc2 , where E is energy, m is mass, and c is the lightspeed constant. But this formula is derived from the energy equation for a moving mass, which involves momentum; E2 = p2c2 + (mc2)2 , where p is the particle’s momentum. The difference is that the momentum of a particle at rest is zero. The problem with that, however, is that the reference-frame for a resting particle is relative to the observer, and must therefore be specified case-by-case. There is no absolute rest-frame; everything in existence moves somehow, relative to everything else.

Nevertheless, let us use the simplified formula, because it is easier to work with.
If E = mc2 is positive, for a real mass, m , then -E = -mc2 can designate the analogous form for a tachyonic mass, -m ; i.e., particle of mass -m and energy -E is a tachyon (faster-than-light, or FTL), where, if v is any velocity, then;
0 < v < c for ordinary particles, called “bradyons”,
v = c for massless particles, also called “photons” and “luxons“, and
c < v <  for tachyons obtained as analogs of bradyons.
[Other tachyons can be imagined, of course (including those traveling infinitely fast), but I view such tachyons as accounting for phenomena other than gravity.]
I next define convenient variables, to make for a more concise text. Thus, let
vr = v where 0 < v < c , for bradyons,
vc = v where v = c , for photons, and
vt = v where c < v <  , for tachyons.
I also assign special symbols for other important variables, as well.
For a bradyon, let mr be the mass, Er the energy, and pr the momentum.
For a photon or luxon, let mc = 0 , Ec be energy, and pc be the momentum.
For a tachyon, let mt be the mass, Et the energy, and pt the momentum.
The sets are;
{ bradyons: vr , mr , Er , pr | Er2 = pr2c2 + (mrc2)2 }
{ photons: vc = c , mc = 0 , Ec , pc | Ec2 = pc2c2 }
{ tachyons: vt , mt , Et , pt | Et2 = pt2c2 + (mtc2)2 }
Notice that there is no such thing as an absolute zero velocity, regardless of the reference-frame; even for the center of the detectable universe, because we do not know if the center of the universe is itself moving with respect to the voids of outer-space. Consequently, a zero velocity specified for a given real reference-frame is purely relative, while an absolute zero velocity is an imaginary quantity. And there exists no absolute infinite velocity for a tachyon, although a relative infinite velocity can be specified, theoretically, in a tachyonic reference-frame.

We should also address the issue of the lightspeed constant, to remove possible ambiguity. For example, we now know that all the physical constants (including lightspeed) change slowly with time, as the universe expands, although we can continue to use the latest confirmed values of the “constants” for most purposes. But the knowledge that the lightspeed constant changes along with the evolution of the universe begets important ramifications for cosmologists. It does not, as a rule, however, alter the present discussion very much. Yet, it warrants mention.
And regardless of it, I make use of the lightspeed constant, just as Einstein did.
Hence, to stress my reliance on the lightspeed constant, c , recall the relativity operator, a , which Einstein derived from the Lorentz Transformations, and consequently defined it using the formula,
 ,
or, in standard keyboard symbols, (alpha) = 1 / ( sqrt { 1 - [(v/c)^2 ] } ) .
Applying this operator multiplicatively to some quantity associated with a rest-state for a particle results in a corresponding quantity for the moving particle; assuming standard reference-frames (i.e., coordinate systems). For instance,
if mr is a resting particle‘s mass, we can let mr = amr be the moving mass.
Similarly, a can be applied to other measurable or calculable quantities, such as velocity, momentum, and time (which renders them “relativistic”).

It is this operator, then, that implies the existence of tachyons (particles that always travel FTL, and which are said to be “superluminal” in character), due to the presence of the velocity ratio, v/c , under a radical-sign in the denominator. To the point, the three overall categories of particles (bradyons, photons, and tachyons) arise directly from applying the operator, as follows.

Take a generic rest-mass, m , as an example, where m = am defines the corresponding moving mass. So, whenever v < c , then a is a real number, and therefore m is real. But if v = c , then, by convention, a = 0 (or else it is said to be “undefined”), making m = 0 , which is the specific case for photons. Thus, if v > c , then a is an imaginary number, making m an imaginary mass.
And we have justified our nomenclature; mr is bradyonic rest-mass, mc = 0 is photonic rest-mass, and mt is tachyonic rest-mass -- so corresponding moving masses become obvious; mr = amr , mc = amc = 0 , and mt = amt .

With velocities, we write; vr = avr , vc = avc = c , and vt = avt . And with time, we can write; tr = atr , tc = atc , and tt = att . As an important result, we use the defining formula for a as the basis for a graph plotting velocity against time, in two dimensions of space, to construct what is called the “light cone” of Special Relativity. [Online reference, go to: http://en.wikipedia.org/wiki/Light_cone. For a print reference, see textbook Modern Physics for Scientists and Engineers, by Thornton & Rex, from Saunders College Publishing, '93. Pg. 50, in 1st edition.]
Mere inspection of the light-cone (and test-runs of the formula for a ) suggests that the universe has three time-dependent "regions"; the past, present, and future for an event, or for a moving object. It can also suggest the existence of objects (such as subatomic particles) that are not "real", mathematically, but which are not strictly forbidden, theoretically. A tachyon is one such object, having negative time (thus, reversed causality), and an imaginary mass (as compared to a bradyon, with real mass, and positive time).

The basic implication is that tachyons travel backwards in time, as viewed from a standard bradyonic system in which the normal flow of time is counted as going "forward" (or designated positive). However, a tachyon would also have positive time if viewed from a superluminal system, and therefore normal bradyons would move backwards in time as viewed from that kind of system.

And there are other implications, when comparing tachyons to bradyons; all of them depending on the nature of the coordinate-system being used. The most important consequence in the present context is that a tachyon's momentum will appear to be opposite to that of an analogous bradyon, as seen from a bradyonic system. One can assume, then, that a collision between a tachyon and bradyon would result in post-collision trajectories, momenta, and energies that take into account the reversed causality of the tachyon, when the event is observed from a standard bradyonic frame.

In other words, if bradyons radiating from a sample of radioactive substance give rise to radiation pressure pushing things away from the sample, within the local space surrounding the sample, then perfectly analogous tachyons radiating from the same sample would produce radiation pressure that pushes in towards the sample, because of the tachyon's reversed causality. That is, radiating tachyons would impart negative radiation pressure, in the space around the sample. We might conclude, then, that a scenario involving this negative-radiation-pressure could account for the pull of gravity between all real masses; assuming tachyons actually radiate spontaneously in all directions from all real massive objects.

Another implication is that the universe we observe from bradyonic spacetime has a counterpart, or more correctly an extension, in tachyonic spacetime. That is, for any given bradyon, we can speculate that there is an analogous tachyon in some superluminal spacetime manifold, and this implies that a superluminal universe co-exists with (and may be interactive with) our detectable bradyonic universe.

The Tachyonics Operator and the Velocity Spectrum

The traditional method of describing a tachyon is to take the result of inputting a superluminal velocity in the formula for the Relativity Operator (alpha), and using a negatively-signed but otherwise standard imaginary-unit, i , to express the result. That is, if mr is a real mass, then an imaginary mass m , for a tachyon, can be obtained as a direct analog of the real mass, by writing; m = -imr . [Reference:
Encyclopedia of Physics, 2nd Ed., by Lerner & Trigg, VCH Publishing, pg. 1246.]
However, due to the implication that there exists a superluminal universe, we can imagine that it exhibits its own number system, incompatible with the standard number system (although, there is a one-to-one correspondence). In that case, the standard imaginary-unit ( i = sqrt (-1) ; i^2 = -1 ) becomes inadequate to describe the kind of tachyon I wish to employ to explain quantum gravity (though the traditional method works fine in most contexts involving tachyons).

One problem is that a negatively-signed imaginary-unit does not automatically imply superluminality. There are many instances of negative imaginaries in the equations physicists use to study natural phenomena. So, it becomes confusing if we also use the same symbolism to define tachyons -- and it is bothersome to repeatedly have to explain representations in a text accompanying the equations. I solve this problem by defining tachyons using an operator, (i^i) , inspired by the standard imaginary-unit, but which is used only to designate tachyonic quantities. This frees-up the standard imaginary-unit, so that negative imaginaries can be treated in the usual fashion, without confusing them with implied superluminality.

My new operator, originally called the "imagination-unit", can be referred to as the "tachyonics operator", and its application is very simple. Multiplying the new operator to any real quantity, variable, or symbol transforms it into a superluminal analog of itself. All that's required is that a transformation equation is given in the context of a discussion involving tachyons (so, if no tachyons, ignore it).

In other words, if mr is a bradyon's mass, then mt = (i^i)mr is the imaginary mass of a tachyon taken in perfect analogy to mr . However, the transformation must be limited, so that unwanted infinities do not nullify the result. Thus, we write;
[ v > c ]
mt = iimr = mr | , tt < 0 .
[ v <  ]

This formula simply states (if not transcribed correctly) that a tachyonic mass results from multiplying the tachyonics operator to a bradyonic mass, which involves evaluating the given bradyonic mass in tachyonic spacetime (between lightspeed and infinite speed, exclusively), where reversed causality holds for the tachyonic analog.

An example exercise is to use integration to establish a scaling format for the superluminal analogs of bradyonic quantities, such as velocities in general;


[An equation was given but not transcribed my the board client. It is an example integration across the lightspeed barrier.]

Note too that the tachyonics operator can be used to define an entire number system for tachyonic spacetime -- including the specification of a superluminal analog of the standard imaginary-unit; (tachyonic i) = (i^i) i .

A convenient way to understand how the transformation works is to inspect the velocity spectrum, involving all possible theoretical velocities;

or, in standard keyboard symbols,
(i^i)(infinite v) > (i^i)v > {c > v > (relative zero-v)} > i(absolute zero-v) ,
where bars can indicate antiparticle velocities (not shown), bradyonic and lightspeed velocities are within the braces, relative zero velocity is testable, while absolute-zero velocity is imaginary (denoted by the standard imaginary-unit, i ), the superluminal velocities (in one-to-one correspondence with bradyon velocities) are designated by the tachyonics operator, (i^i) , and infinite velocities are outside the purview of experimental testing (at this time).

One other point could be raised here, regarding the lightspeed constant, which acts as a barrier between bradyonic and tachyonic spacetimes. It clearly stands to reason that a superluminal universe would have an analog of the lightspeed constant (if it has analogs of everything else), and thus superluminal lightspeed could be defined as, let's say iic = 1.000...0001c , where the exact number of zero-valued decimal-places is undetermined. And in that case, an extra place must be made in the velocity spectrum for iic , since it could be an issue if that has a bearing on an experiment. We should therefore consider the superluminal photonic tunneling experiments, in which velocities for the tunneling photons are calculated at significant multiples of c . Such experiments could in-fact provide the theoretical lower-limit on the possible superluminal analog of c . Günter Nimtz, of the University of Koln, Germany, for instance, has been doing photonic tunneling for about two decades, and is an often-cited authority on the subject; noting his own and other physicist's determinations of speeds at which different categories of photons (radio, microwave, etc.) engage in superluminal tunneling (which seems to depend largely on two factors -- the frequency of the photons, and the specific nature of the apparatus used by various investigators). Superluminal photon velocities cited in the most easily obtained online sources I consulted for this article (but dated only through '03) ranged from 1.4c to 8c . [The actual speeds noted in cited literature were: 1.4c, 1.7c, 2c, 4.7c, and 8c. Reference: psiquadrat.de/downloads/nimtz03.pdf . Consider also hal.archives-ouvertes.fr/docs/00/24/69/30/PDF/ajp-jp1v4p565.pdf . ]

The purpose of getting into this is (1) to establish the concept of tachyons for the present discussion, (2) to specify the velocity spectrum, with the goal of using it to understand a logical transformation scheme that relates bradyonic quantities to their exact superluminal analogs (e.g., setting-up a scaling, gauge-based, or data-spread format for one-to-one correspondences), and (3) to have a springboard for postulating tachyons with properties not in analogy to known particles.

From elsewhere. Not edited (copied and pasted as-is).

The Gravitational Exchange Tachyon (GET)

If, for any function F , we can define derivatives with respect to x , so that
F(x) = f'(x) = df(x) / d(x) , then, for functions f and F , we can give a definite integral between positive and negative infinity, for the variable x , equal to the sum of the integrals that take negative and positive infinity separately.
If we let x be the time t , then the positive integral can be used for studying bradyons, with positive time, and the negative for tachyons, with negative time.
And imposing exclusivity on the two time integrals, to remove absolute-zero and infinite-velocity solutions, pertinent parameters corresponding to each type of particle (bradyons, photons, and tachyons) are rendered empirical.

That said, the functional operator a , used to impose Einsteinian relativity on an object of rest-mass m , is given as a function of the ratio between the velocity (v) of the object and the lightspeed constant (c) ; a = 1 / { sqrt[ 1 - (v/c)2 ] } , where the moving mass m , is defined; m = am . This formula is an example of how we get the three categories of particles; standard, massless, and superluminal.
And its use to plot time against velocity in three-dimensional space results in the Light Cone of Special Relativity, which itself implies a tachyonic universe that coexists with the detectable universe. A tachyon for which we could attempt to devise detection and manipulation apparatus may reside in a superluminal analog of the standard space-time manifold, so the limits on it are easily specified;
c < vt < ii(+ infinity) , 0 > tt > ii(- infinity) , 0 < xt < ii(infinity) , ... .

It is simple, then, to imagine a tachyonic analog of all known particles, and obtain hypothetical sets of characteristics for each of the analogs. The tack is to give an integration example to establish specific one-to-one correspondences across the lightspeed barrier required for a given parameter or parameter set. This is the purpose of my Tachyonics Operator, ii ; to imply superluminality, while removing possible interference with the standard imaginary-unit, i .

Here, I must restate my reasons for introducing the Tachyonics Operator, so the justification for using it is made clear. First, there is no need for the Operator if we specify bradyons and tachyons in their respective reference-frames (i.e., coordinate systems), use the Lorentz Transformations to relate quantities in either frame to their counterparts in the other, and there is no confusion as to the application of the standard imaginary-unit, i = sqrt(-1). Specifically, the Relativity Operator, R = {1 - [(v/c)^2]}^(-1/2), invokes the unit for tachyons, where v > c. For example, suppose v = 1.1c. Then v/c = , so (v/c)^2 = (1.1)^2 = 1.21, which means 1 - [(v/c)^2] = 1 - 1.21 = -0.21. This puts a negative number under the radical in the denominator of R, turning into an imaginary number;
R = {1 - [(v/c)^2]}^(-1/2) = 1 / sqrt(-0.21) = 1/ i[sqrt(0.21)] = -i / sqrt(0.21) .
In that case, the negatively-signed imaginary-unit implies superluminality.

However, if we try to do quantum physics for tachyons, we run into the difficulty of having two different interpretations of the negatively-signed imaginary-unit; one that implies superluminality, and the other that does not. And there is no obvious indicator, other than context, to tell us which interpretation to apply. So, when we discuss both bradyons and tachyons in the same context, we are stuck explaining how we apply the imaginary-unit every time it appears with a negative sign.

Case in point, let x' denote the operator that is the partial-derivative with respect to some position x in a standard Cartesian coordinate system, in one dimension. In Quantum Mechanics, such an operator is multiplied by the negatively-signed imaginary-unit and the value 6.5621 x 10^(-16) electron-volt seconds (from dividing Planck's Constant, h, by 2pi = 6.2832) to obtain a momentum operator P in one dimension; P = -i{h/[2(6.3282)]}x' , where h = 6.5821 x 10^(-16) eV s. Thus, the negative imaginary-unit is retained when we write the formula for the expectation value <P> for this momentum, in terms of a particle's wave-function, which is a function of position, x, and time, t. [Expectation momentum, <P> , is the definite integral between + and - infinity of the product of the wave-function's conjugate and the wave-function's partial-derivative with respect to x.]

Thus, if we want the relativistic momentum operator, rP, we write;
rP = RP = {1 / sqrt[1 - (v/c)^2]}[-i{h/[2(6.3282)]}x'] .
For a tachyon, then, where v > c, R becomes a negatively-signed imaginary number. Let's denote it as -iN, and let Q = {h/[2/(6.3282)]}x', so that
rP = (-iN)(-iQ) = [(-1)^2](NQ) = -NQ.
Consequently, since N and Q would both be real numbers, in all cases, and there is no indication from this formula that we are dealing with a tachyon, there seems to be some confusion as to how to interpret the formula. Does it have meaning, in the sense of experimental testability, or is it a useless exercise?

And rP is a momentum operator, so its negative, by itself, does not necessarily imply superluminality. That must be explained in the text. Yet, I find this a rather bothersome state of affairs, even if we use different symbols for bradyonic and tachyonic momentum. It's like engineers who replace the "i" with a "j", to help avoid confusion between their use of i and the standard definition of it. Using a different symbol for the same operator does not actually resolve the confusion; it just sets it aside. And when engineers need both applications of the unit in the same work, loads of explanations must be included throughout the text, to keep the reader from getting things mixed-up.

Another issue is that there are no limitations on the velocity that a tachyon can obtain, when defined using the negatively-signed but standard imaginary-unit. The velocity could be infinite; rendering the mathematics involved quite useless.
So, I solve these problems by using a limited transformation across the lightspeed barrier to define a tachyon (in point-particle form); removing the lightspeed and infinite-speed from the range of velocities the tachyon can exist in. This has the advantage of eliminating possible confusion wrought by having two different applications of the imaginary-unit in the same treatment, while putting variables for tachyons on an empirical (experimental) footing.

Using the Tachyonics Operator, let us describe a new particle, not in analogy to any known particle, except perhaps that we allow it to resemble some particles with respect to their macrocosmic behavior. I'm looking for a point-like, spin-less, completely classical tachyon that shoots off to an infinite distance, faster-than-light, seeming instantly, upon being spontaneously generated in some real mass.
It is not affected much, or at all, by real objects it encounters (and probably not by other tachyons, as well), and barely registers a whisper of affect on those objects, except that it transfers (by a theoretical twisting-string substructure) some of its forward momentum to the bodies it passes through on a perfect or near-perfectly straight Cartesian path. [The twisting substructure is not necessarily actual. It is an hypothetical means of obtaining the action integral in a string-theory model.]

Being point-like means this tachyon epitomizes the concept of a point in space, and its path therefore serves as a justification for the concept of a line in space.
And being spin-less means its Schrödinger Equation is reduced to the equation of a line in space (i.e., no wave characteristics whatsoever). We have thus been given a particle that is wholly compatible with all Modern Quantum-Field Theories, albeit, by reducing the wave-equation for this tachyon to a mere linear reference.

Collectively, such tachyons radiating spontaneously from a real object would give rise to radiation pressure on other bodies in the Universe, but due to the reversed causality of tachyons, of any kind, this is imparted as negative radiation pressure; causing a pull toward the source, instead of pushing things away from the source. So, we have obtained a particle, consistent with Quantum-Field Theory, that can account for Newton's law of Universal Gravitation; F = G[Mm/(r^2)] , where F is the attractive force an object of mass M exerts on some other mass m located the center-to-center distance r away from M (and G is Newton's constant).

In particular, if we let tFi denote the amount of force each such special tachyon contributes to F , then we can define F as their sum; F = tF1 + tF2 + tF3 + ... .

Let me call this attraction-imposing tachyon the Gravitational Exchange Tachyon, or GET particle, and note that, using its reversed causality to establish a negative radiation pressure, and equating this with Newtonian Gravity (NG), I am greatly encouraged to think this model (superluminal quantum gravity) is compatible with the field equations of Einstein’s theory of General Relativity (GR) -- by means of the equivalence of GR to NG at the weak-field limit of GR.

Conclusion: Gravity is faster than light, and is therefore a tachyonic force.

An intriguing result of such speculation is to realize that, should there truly exist a superluminal universe coexisting with the visible universe, complete with analogs of all known particles [as well as many tachyons for which exact bradyon analogs do not exist (thus explaining why so many particles predicted by string theorists are not currently detectable)], then the gravity of a superluminal body composed of tachyonic mass would correspondingly produce its own version of gravity (also faster-than-light) that imparts positive radiation pressure on all bradyonic matter it passes through; thus setting-up a universal repulsive force (true antigravity) in the visible universe, from seemingly invisible matter. Consequently, I believe this is an explanation for the so-called “dark matter” and “dark energy” astronomers are attempting to investigate more thoroughly, these days.

In short, if the gravity of a bradyon is a universally attractive superluminal force in the visible universe, then the gravity of a tachyon is a universally repulsive force in the visible universe, where tachyons collectively account for dark matter, and their superluminal gravity accounts for dark energy. However, there could also be other superluminal forces at work, and some of these may have influences on the visible universe, as well. For example, tachyonic energy fields could be used to explain the life force of animate things, the seat of consciousness, and how it is that humans have minds, exactly what emotions really are, and a whole array of other previously unexplained natural and supernatural phenomena.






















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