The Tachyonics Operator Provides Grand Unified Field
I attempt here to convey more of my theory on Superluminal Gravitation, mostly in words rather than mathematical symbols; rendering many of the equations I employ into word-form. In some cases, however, it is impossible or impractical to put everything that completes a given description of a complicated equation into words, so I include print and online references as needed, so readers can look up the equations in standard form.
Einstein's theory of General Relativity (GR) is a geometric theory, based primarily on Riemannian Geometry. It describes the curvature of some region of space (the warping of space-time) brought about by the presence of matter in that region. Note, however, that the field equations of GR are entirely geometric in character, and do not specify the quantum nature of gravity. A quantum theory of gravity must be attached, in order to describe gravity at both microcosmic and macrocosmic distances.
In the field equations of GR, the tensor that gives the values corresponding to curvature of space associated with the presence of matter in a particular region of space, is called the "Einstein Tensor" (Kay, 119). It is a 2nd-order mixed tensor defined as the difference between two other tensors. One is a mixed tensor determined as the product of a Ricci Tensor (of second kind) and the contravariant 2nd-order Riemannian Metric Tensor. The other is one-half the result from applying the second-order mixed Kronecker Delta to the invariant radius-of-curvature, also called the "Ricci Scalar". [Note: The divergence of the Einstein Tensor is zero at all points determined using a Riemannian metric. (Kay, 120)] The equation expressing this difference states: the Einstein Tensor equals the product of the Ricci Tensor and the Riemannian Tensor minus one-half the result of applying the Kronecker Delta to the Ricci Scalar. (Kay, 119)
The Ricci Tensor of second kind is the 2nd-order covariant contraction of the 4rd-order mixed form (1st-order contra-variant, 3rd-order covariant) defined using operators called "Christoffel Symbols" and some generalized coordinates (Cartesian, curvilinear, etc.).
A Christoffel symbol is not a tensor (despite looking like one), but is an operator that yields a tensor when applied to a scalar, or to a vector or other actual tensor. In this case, Christoffel symbols are used to define the 4th-order Ricci Tensor as an operator defined as the sum of two differences; one is a function of coordinates, and the other involves two two-symbol products. The two functions of coordinates are operators obtained as partial-derivatives with respect to two generalized coordinates of one Christoffel symbol each. [Handle the subscripts and superscripts of the tensors and coordinates carefully.] The equation states: the 4th-order mixed Ricci Tensor is equal to ((the partial-derivative, with respect to the first generalized coordinate-set, of one Christoffel symbol)) minus ((another such partial-derivative, with respect to the other coordinate-set)) plus ((the product of two Christoffel symbols)) minus ((the product of two more symbols)), with subscripts and superscripts observing a required arrangement. [Reference: Kay, page 101, and Example 9.5, page 119 (on proving Bianci's Second Identity).] [Online, search "Ricci Tensor", "Christoffel Symbols", and "Riemannian Metric".]
The general Christoffel symbol used as the template for all of those used above is like a mixed tensor (1st-order contra-variant and 2nd-order covariant), defined as the product of its 3rd-order covariant form and the 2nd-order contra-variant conjugate of a Riemannian metric-tensor, whose general form is itself defined using a Jacobian matrix of elements obtained as the partial-derivatives of the given generalized coordinates with respect to any other corresponding system of generalized coordinates.
We use a Riemannian metric-tensor to define the applied space-time metric, which is the differential of the arc-length parameter, according to the formula: the signed square of the applied metric is equal to the product of the Riemannian metric-tensor and the two differentials of the two generalized coordinate-sets.
To put this in simpler terms, use Euclidean 3-space, which reduces the Riemannian metric-tensor to a Kronecker delta, so it takes only the two values; 1 when the delta's two indices are equal, or 0 when they are not equal. Then the square of the metric equals the sum of the squares of the differentials of the Cartesian coordinates, x, y and z.
The arc-length parameter is defined as a function of a curvature parameter, such as time. If it is applied to time, it is then defined as a definite integral (with respect to a dummy coordinate variable), evaluated between two time values. This integral is taken of the square-root of a purely multiplicative expression involving the 2nd-order covariant Riemannian metric-tensor, the standard derivatives (with respect to the same dummy variable) of the two generalized coordinate-sets, and a sign operator that can only be positive or negative unity, depending on whether the same multiplicative expression without it is greater-than or equal to zero for the positive case, or less than zero for the negative case. A given curve can be obtained, therefore, by treating the coordinate variables as functions of positive-real time.
Notice, so far, that nothing in these formulas indicates the quantum nature of the field, for which the radius-of-curvature is assumed to be centered on the center-of-mass in the region of space under study. Neither do they concern themselves with the nature of the mass; it could be a particle or a galaxy. They do not even mandate that an actual field of force is present; despite being categorized under the heading of "field equations" of GR.
Of prime importance to my theory on Superluminal Gravitation is the scalar radius-of-curvature, called the "Ricci Scalar". I assign the path of a radiated GET particle to be along this invariant. But the equation that defines this particular scalar is much more complicated than any ordinary radius-of-curvature, and is too complicated to relate here (would take too much explanation to be practical in words alone). Suffice it to say that it is a function of the Riemannian metric-tensor and the generalized coordinates.
Specifically, I use the form derived by D.C. Kay, in his work on Tensor Calculus (Kay, 106), where he examines the symmetry of a Ricci tensor of first kind, after defining it as a contraction of a Riemann tensor of second kind.
Ordinarily, the Ricci Scalar simply works as the radius-of-curvature for the path of an object moving in the region of space being considered. But it can also be specified as orienting the path of a point-like tachyon radiating outward from the source-mass, and passing through the said object. And being point-like by definition, it is easily defined using Quantum Mechanics (QM) -- which amounts to finding a way to make GR compatible with QM, wherein the forms of all existing formulations are preserved.
Note that a GET particle is point-like, waveless, and spin-less, as far as an observer in generalized bradyonic coordinate-systems with metrics larger than the Planck Length would be concerned. This reduces the GET Schroedinger equation to a linear reference (the equation of a line). Indeed, I propose that the GET epitomizes the very concept of a "point" in space, and that its path represents a perfectly-straight Cartesian "line" in space (i.e., a GET particle provides a physics-based definition for classical points and lines).
If, then, these GET particles (radiating spontaneously in all directions from the source-mass) impart some of their forward momentum to the objects through which they pass unswervingly on their way to an infinite distance away, we have a model explaining the Newtonian law of universal gravitation; F = G(Mm)/(r^2) , where G is Newton's constant, and r is the distance between the source-mass M and the object's mass m.
This is accomplished by showing that the GET particles collectively establish negative radiation-pressure in space, due to their reversed causality, where the contributions Fi of the individual GET particles sum-up to the classical force value; F = F1 + F2 + F3 + ... .
It also leaves the field equations of GR intact, and is consistent with the prevailing view that GR is essentially equivalent to Newtonian Gravity at the so-called "weak-field limit" of GR (although there is a debate as to whether this is actually the case).
In any event, the important proposal here is that using GET particles to explain quantum gravity (in the way I described) makes GR compatible with QM, and therefore serves as a basis for expressing gravity in a quantum-mechanical gauge-field format that unifies it with the other forces of nature (already empirically described in such a format).
In a simplistic way, therefore, we can denote a Grand Unified Field (GUF) representation as being the additive combination of the standard model of elementary particle physics (sans gravity), denoted as a basic Unified Field (UF), and the gauge-field model of gravity using GET particles as the mediators, and denoted using the Tachyonics Operator, I , to designate an imaginary gravitational field IG, so the unification amounts to a super-complex number, involving bradyonic quantities in the "real" component (without gravity), and tachyonic quantities (gravity-specific) in the "imaginary" component;
GUF = UF + IG .
Reference: Kay, David C. Tensor Calculus; Schaum's Outlines. McGraw-Hill, 1988.