Forum:Tachyonics Talk

Applying Tachyonics To Gravity

Forword:

Astronomical observations by the late Tom Van Flandern have indicated that gravity is faster-than-light. For details, see the site MetaResearch.org, and look under the heading “Gravity & Relativity”. This confirmed experimental result, though debated, implies Tachyonics may possibly have applications to gravity.

Indeed, I have long held that the virtual-exchange particle responsible for gravity (its quantum) is a special type of tachyon. But to understand this possibility, readers here may find it helpful to consider what is already known about gravity. To that end, standard methods of representing gravity are given below, followed by theoretical means of applying Tachyonics to the riddle of gravity.

Review of Standard Gravity Concepts

In classical terms, the gravitational field can be described as a "central force field," in which all force-vectors must always point to the origin of the field, which field is given by a vector-field function, F, of the form; F(x,y,z) = f(x,y,z)(xi + yj + zk ) , where f is a scalar function of the coordinates x, y, and z, and i, j, and k denote respective unit-vectors. (Berkey, 1037)

On the other hand, the more modern description of gravity is obtained from the field equations of Einstein's General Theory of Relativity, in which gravity is displayed according to its observed effect of imparting curvature to the space surrounding the source-mass; explained in a bare-bones fashion as follows. Special Relativity supplies us with a 4-dimensional spacetime metric (measuring stick), defined in differential form; d(s^2) =  d(x^2) + d(y^2) + d(z^2) - (c^2)d(t^2) , where s is a position function for a point (x,y,z) considered at any given time t, so that ds is an absolute interval in our 4-dimensional reference-frame, referred to as a "pseudo-Euclidean spacetime," specified as a localized inertial frame.

By joining a number of these local reference-frames together, and constructing a correspondence system relating them to each other, a general theory of their collective state is obtained. (Besancon, 621) The result of Einstein's efforts at this sort of activity is his well-known theory of General Relativity, which basically describes how a gravitational field curves the space around a source-mass, according to the behavior of a certain tensor, Gab, defined by the equation; Gab =  - G[Tab(8π)]/(c^4), Insert non-formatted text here where Tab is the energy-density tensor for the source-mass, G is Newton’s gravitational constant, c is lightspeed, and the subscripts, a and b, each range from 0 to 3, with the 0-valued subscript associated with the time coordinate, t, and the integer values associated with the spatial coordinates, x, y, and z.

Notice that the equation for Einstein’s Tensor represents ten second-order partial-differential equations [obtained by applying the Laplacian operator (i.e., the del operator squared)], and establishes the basic metric used in the sixteen components of Gab. Of those, ten are independent, where Gab = Gba, while the components of Tab are constructed from classical density. (Besancon, 622)

For a Euclidean, or "flat", spacetime, in which the source is massless (mass = 0), the metric serving as the solution to these equations takes the form d(s^2), given by Einstein’s theory of Special Relativity (SR), and which interval, the metric, is an invariant under the Lorentz transformations.

For a spherical, nonzero source-mass, Schwarzschild provided the solution in polar coordinates, represented here using the angular coordinates  and . In particular, letting a = 2Gm/(c^2) and b = a/r, we write; d(s^2) = [1/(1- b)][d(r^2)] + (r^2)[d(t^2)] + [(sin^2)][d(^2)] - (1- b)(c^2)[d(t^2)] , where r is distance, t is time, and geodesics in the plane are obtained when  is half the value of π;   =  (0.5)π  ~  1.57. As given by Gab, then, the geodesics satisfy the following equation involving a length constant k and a generalized function u of the distance parameter r; {[(d^2)u / d(p^2)] + u} =  [Gm/(k^2)] + [3Gm(u^2) / (c^2)]. If we thus consider only planar geodesics, and set  = π/2, then u(r) = 1/r. Otherwise, let k = (Gm {1 + e[cos(pp)]} / u)^(1/2) , where e ~ 2.718, and let p denote a mass-density constant, defined; p = 1 + [3(Gm)^2]/[(kc)^2] , so the solution takes the following the form; u(r) = (Gm{1 + e[cos(p)]}) / (k^2). But for a classical rather than relativistic version, let p = 1. (Besancon, 622)

Now, Einstein used Riemannian geometry to obtain the said “field equations”, denoted by Gab, describing the curvature of space around a massive body; this curvature understood as being due to the presence of a gravitational field from that body --  although, obviously, such an understanding must be stated in the accompanying text, because the equations describe only the geometry of the space around the body, and involve no inherent reference to a field of force. Note also that “Riemannian geometry” is the term for generalization of all known geometries, including all Euclidean and non-Euclidean geometries. In this case, however, understanding Riemannian geometry requires knowledge of Gauss' differential geometry of a curved surface, embedded in a three-dimensional space; defined parametrically by the three space functions (x, y, and z) of the surface coordinates (u and v), so that the infinitessimal distance ds between the points (u,v) and (u + du, v + dv) can be given by Gauss' quadratic form; d(s^2) = [X(u,v)][d(u^2)] + [Y(u,v)][2(du)(dv)] + [Z(u,v)][d(v^2)] , where X, Y, and Z are arbitrary functions of the surface coordinates, u and v. This is, of course, the form that allows calculation of the length of curves on the curved surface, including determination of the geodesic between two points on the surface. And it also establishes the curvature at any point on the surface.

Riemann generalized this form by introducing a continuous n-dimensional manifold of points x1, x2, x3, ..., xn, so that ds became a metric between nearby points, (x1, ..., xn) and (x1+dx1, ... , xn+dxn), and it is therefore more properly defined using a function g of those coordinates, according to the formula;

n d(s^2) = (summation gij) d(xi) d(xj) , i, j = 1

where the gij are suitable functions of the selected coordinates (x1, ..., xn), used to make the gij define different geometries on the manifold, with i, j = 1,..., n. (Simmons, 243)

Einstein is said to have recognized that Riemann's geometry in n-dimensional space is to n-dimensional Euclidean space what the general geometry of curved surfaces is to the local geometry of the plane. He then showed exactly how the geometry of some real finite space is a Riemann space in which curvature is determined by the distribution of matter (Simmons, 243); accomplishing this by writing d(s^2) in terms of the Riemann metric-tensor gmn as a function of x;                     3                 3 d(s^2) = summation summation gmn(x) d(x`m) d(x`n) , m = 0         n = 0

where m,n = 1, 2, 4, …, and where the " ` " symbol indicates that the letter to its right is a superscript (to put the formula into ordinary keyboarding symbols). [Note: Riemannian space can be rendered flat, as a special case.]

Now, Einstein characterized the gravitational field by the symmetric-tensor field function gmn, called the "metric tensor", understanding that gravity is equivalent to acceleration. He then constructed (with help) the set of generally covariant equations of motion denoted by Gab, among which the second differential orders form a unique subset, involving two variables: (1) the total energy Eo of perfect vacuum, defined in terms of potential energy U; Eo =  8(3.14)GU/(c^4) , and (2) the coupling constant Cc, used to determine the strength of the interactions between all matter in the universe, and defined so that Newtonian gravitational theory is obtained by a reduction resulting from letting Cc be a certain value; Cc = (8π)G/c , corresponding to what is called the “weak field limit”. In that limit, goo = 1 + 2P/(c^2) , for some gravitational potential P satisfying a Poisson equation of the form; LP = (4π)Gp , where p is mass density, and L is the Laplacian operator. (Lerner, 1048-1052)

This, then, briefly summarizes the fundamentals of gravitation as expounded in classical and in relativistic terms. Models of Quantum Gravity include variations on these themes, while the “standard model of quantum gravity” assumes that the virtual-exchange particle, called the “graviton,” is a spin-2 massless boson that propogates at the speed of light, where the field these particles make up can be depicted as a quadrupole --  obtained by merely super-imposing one electromagnetic-type field onto another, differing orthogonally in orientation.

But these gravity schemes have been around too long, in my view, and there are plenty of ideas around, nowadays, for consultation in updating these notions.

The energy of such particles would probably be so great that it is unlikely they could have escaped detection for so long, unless it is assumed they are weakly interacting. This is now understood using M-theory, where the larger part of the energy of gravity exists in an alternate-dimensional manifold, or “brane”, while the portion we experience as gravity is relatively weak. This also implies that gravity is largely imaginary in nature, or “virtual”, for all instrumentation used to detect its quantum nature in the physical/visible universe.

Researchers should be forgiven for concluding that the old graviton concept is not perfectly accurate, because the quanta of gravity are probably not like the conventional graviton at all. And they might also be pardoned for thinking that the virtual-exchange particle for gravity is not necessarily a boson, either, but is a type of tachyon; implying that the standard model, while logical, does not hold in light of modern empirical considerations --  while certain tachyons fit the bill for gravity quanta more correctly, if used to explain today’s experimental data.

That is, experimentally, and according to ordinary observation, the implication is that gravitation is actually superluminal in nature -- gravity's quanta are probably tachyons, which explains why the quanta of gravity have been so hard to detect.

References Berkey, D.D. & Blanchard, P. Calculus, 3rd Ed. Sunders College Publishing, 1992. Besancon, R.M. The Encyclopedia of Physics. Reinhold Publishing, 1966. Fogiel, M., Director: REA. REA's Problem Solvers: Advanced Calculus. Research and Education Association (REA), 1981. Lerner, R.G. & Trigg, G.L. Encyclopedia of Physics, 2nd Ed. VCH Publishers, 1991. Nolan, P.J. Fundamentals of College Physics. William C. Brown Publishers, 1993. Simmons, G.F. Differential Equations w. Apps. and Hist. Notes, 2nd Ed. The Intl. Series in Pure and Applied Math., McGraw-Hill Books, 1991.

The Case for Tachyonic Gravity

Borrowing from classical photometry, let identical tachyons be emitted randomly as free particles radiating from a source-mass, where tachyon flux TF through a small surface area, a distance R from the source-mass, is defined; TF = IA/(R^2) , where I is the radiation intensity, and A is the area of the surface. (Edminister, 225) And, for the moment, let us ignore all wave characteristics of the tachyons, and call these particles "Gravitational Exchange Tachyons", or GET particles.

The area A can be defined; A = S(R^2), where S is the solid angle subtended at the center of the source-mass. Thus, the energy En of the nth GET contributes to the total energy Et of the N incident GETs; N Et =  summation En  =  TF/A  =  I/(R^2) , n = 1 where n = 1, 2, 3, ..., N; ordinarily considering a small finite range for N.

This equation relates the intensity I to the energy Et, which itself decreases inversely as the square of the distance R from the source-mass, in accord with the inverse-square law. (Nolan, 155) Similarly, the force F1 from the source-mass M1, tugging on some mass M2 at distance R, is given by the formula; F1 = G(M1)(M2)/(R^2). It is easy, then, to imagine that the surface is a cross-section of M2, and that the value of En for a given GET is related to the GET's momentum pt by the formula; pt^2 =  (En /c)^2 - (mtc)^2 , where mt is the imaginary mass of the GET, and mt = im, for a standard mass m whose superluminal analog is mt, with i denoting the Tachyonics Operator.

However, converting this mass into its relative energy equivalent, we eliminate the mass term, writing; pt = {En[En/(c^2) + (vt/c)^2 - 1]}^(1/2) , where vt is the GET's velocity. Thus, for some number N of GETs incident on the surface, the sum Pt of the momenta of all the GETs is; Pt = Npt. Letting boldface indicate vectors, we thus have pt = | pt | ; Pt = | Pt | = | Npt |. Of course, if the GETs are not identical, we simply write; Pt = pt1+pt2+... Npt.

Assuming that some fraction j of the GETs passing through the surface impart a fraction f of their causally-reversed momentum to the surface (whether or not any are actually captured or repulsed by the surface), and that there are a finite number l of such surfaces (the cross-sections of M2 are summed to obtain the volume of M2), then we have a way to explain the force F1 in terms of momenta of GET quanta; F1 = ljfPt.

For a vectorial picture, of the total effect of the tachyonic irradiation of M2 by M1, it suffices to reverse the direction of the position vector R in the classical interpretation of F1, while specifying F1 as a function of location; F1(x, y, z) = G(M1)(M2)/(R^2) = G(M1)(M2)/((x^2)+(y^2)+(z^2)) , where R = |R|, and the magnitude R of the position vector R (originally pointing from M1 to M2), is defined according to the formula denoted; (R^2) = (x^2)+(y^2)+(z^2). That is, if R is defined; R = xi+yj+zk, where x, y and z are the space dimensions, and i, j, and k are their respective unit-vectors [standard classical representation from vector analysis (Berkey, 1037)], and R is also viewed as corresponding to a tachyonic vector-field function FT of R, then the flux of tachyons imparting the force F1 onto M2 can be represented by writing an equation of the form; FT(R) =  G[(M1)(M2)/(R^3)]R , in analogy to Coulomb's law for static electric fields (Griffiths, 62), where R is nonzero, and is cubed because we wish to obtain R from R^2, as follows; R^2 = |R|^2 = R/(|R|^3) = R/(R^3) = [1/(R^3)]R. (Berkey, 1038)

In practical applications, however, give the definition of the gravity function F as; F(R) = -G[(M1)(M2)/(R^3)]R , so field-vectors point towards the center of the source-mass, as viewed from a standard reference-frame (where time is positive). (Berkey, 1038; figure 1.5)

In the case of depicting the tachyonic radiation mathematically, assign a positive value to the right side of the above equation for F(R), and let each radiated GET correspond to a field-vector RL, where L = 1, 2, 3, ...; retaining the same forms as in Coulomb's law for static electric-field vectors.

To represent superluminal gravitation empirically, then, let FT(R) be an “actual” imaginary, making it an undetectable quantity relative to all presently detectable quantities. Tachyonic gravity can thus be represented by applying the operator i (the imagination-unit) to F, writing; FT(R) = iF(R) = G((M1)(M2)/(R^3))R , where i reverses the sign on the right-hand side of the equation for F(R), and also imposes the other required specifications for tachyonic quantities, as viewed from a standard reference-frame.

The Newtonian force F1, therefore, can be given in terms of the overall effect of the GETs acting collectively, represented by FT(R), or in terms of the sum of their individual force contributions, obtained from Pt. That is, F1 =  ljfPt  =  |FT(R)|.

Naturally, the overall gravitational field must be irrotational about a non-rotating source; the field does not rotate if the source is held still, but remains a constant radiant field, with zero curl. (Fogiel, 997) What is more, the Newtonian field is said to be “equivalent” to the weak-field limit of the corresponding field obtained from Einstein's theory of General Relativity (Lerner, 1048) --  although, there is debate as to whether this is an accurate assessment. At any rate, an extension to Relativity is built into my thesis on tachyonic gravity, as follows. This type of tachyonic radiation, as given by FT(R), explaining quantum-gravity using GETs as the actual quanta, implies unification with the other fundamental forces if they are represented using Gauge-Field Theory, where a given GET is understood to travel along a Ricci Scalar (a radius-of-curvature within the region involved) in Einstein’s field equations, and where the quantum-mechanical characteristics of the GET are compacted within it --  which can be interpreted as constituting the compaction of quantum-mechanical information in the path itself. This serves as a method for obtaining compatibility between modern Quantum Mechanics and the field equations of Einstein’s theory of General Relativity.

Because today’s Relativistic Quantum Field Theories require that Einstein's theory of Special Relativity is involved, containing, as an inherent aspect, the notions of tachyons and superluminal spacetime (because the theory must be included when investigating elementary particles experimentally), it is clear that the endeavor of Tachyonics provides a means by which Quantum Mechanics can be derived from General Relativity, assuming the compaction of quantum-mechanical information into the Ricci Scalar of General Relativity.

Professor of Mathematics G.F. Simmons once wrote: "Einstein conceived the geometry of space as a Riemannian geometry in which the curvature and geodesics are determined by the distribution of matter; in this curved space, planets move in their orbits around the sun by simply coasting along geodesics instead of being pulled into curved paths by a mysterious force of gravity whose nature no one has ever really understood." (Simmons, 244)

Applying Tachyonics to the problem, however, we find that this "mysterious force" can now be understood quite succinctly, and in a manner completely consistent both with classical and relativistic descriptions; resulting in a valid gauge-field unification scheme (meaning empirical, and therefore testable). Here, then, is the link between tachyons and visible reality; the superluminal gravitational field. It’s an all-pervasive attractive force between massive bodies, described as a central force-field, given by the classical vector-function, FT, viewed as evidence of the effect of the GETs emitted by the source-mass.

The GET particles impart a fraction of their causally-reversed momentum to masses through which they pass on their absolute straight-line paths from the source to an infinite distance, thereby (due to that reversed causality) imposing a pull rather than a push to all of those masses (the radiation pressure being negative) while imposing curvature to the space surrounding the source-mass, according to the General Theory of Relativity, in which the gravitational potential GP (not the potential energy) is related to the Riemannian metric-tensor, g ab, according to the formula; goo = 1 + 2(GP)/(c^2) , where a = 0,1,2,3, and b = 0,1,2,3, and all physical points/events occur in the four-dimensional sub-manifold corresponding to a standard four-vector in the ten-dimensional manifold of relativistic Gauge-Field Theory --  in it's turn, being part of an overall super-dimensional manifold, called the “Multiverse”.

Now, to represent a Grand Unified Field, GUF, inclusive of tachyonic gravity, GT, and involving the most updated version of the basic Unified Field, FU (unifying the weak and the strong nuclear fields with electromagnetism), we simply write; GUF =  FU + GT , where FU = [(FE + FM) + FWN] + FSN, and GT = FT(R) , with FE the electric field representation, FM the magnetic field representation, FWN the weak-nuclear field representation, FSN the strong-nuclear field representation, and FT(R) the tachyonic vector-field representation of gravity; FU being the real part, and GT the actual-imaginary part, of the complex GUF.

Be aware that much work is being done on this topic, and that a growing number of reputable scientists are recognizing that gravity is probably superluminal.

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

References Berkey, D.D. & Blanchard, P. Calculus, 3rd Ed. Sunders College Publishing, 1992. Besancon, R.M. The Encyclopedia of Physics. Reinhold Publishing, 1966. Edminister, J.A. Electromagnetics, Theory and Problems of. Schaum's Outline Series; McGraw-Hill, 1979. Fogiel, M.; Director: REA. REA's Problem Solvers: Adv. Calculus. Research and Education Association (REA), 1981. Griffiths, D.J. Intro. to Electrodynamics, 2nd Ed. Pentice Hall, 1989. Lerner, R.G. & Trigg, G.L. Encyclopedia of Physics, 2nd Ed. VCH Publishers, 1991. Nolan, P.J. Fundamentals of College Physics. William C. Brown Publishers, 1993. Simmons, G.F. Differential Equations w. Apps. and Hist. Notes, 2nd Ed. Intl. Series in Pure and Applied Math., McGraw-Hill Books, 1991.