The GET-Particle's Path

An empirical path for a GET particle is given by D.C. Kay's equation for the Ricci Scalar.

My thesis on Superluminal Gravitation hinges on the acquisition of repeatable detection and manipulation apparatus designed specifically first, to prove that tachyons exist, and second, to test the thesis. But I had not indicated where to look, although I stated that the GET particle's path and action must be along an arbitrary radius-of-curvature (Ricci Scalar) for the path of an object moving while under the influence of the gravitational field of the source-mass. But I had not published a proper equation for it (one that can be used in experimental settings), although I have mentioned it.

So, here it is in detail, in ordinary keyboard symbols. I use the formula David C. Kay provided for the Ricci Scalar in his contribution to the Schaum's Outline Series, on Tensor Calculus (McGraw-Hill, '88); theorem 8.6, page 106. He displayed the equation in stating that the Ricci Tensor is symmetric, in the chapter on Riemannian Curvature; specifically, while defining Ricci Tensors of second kind.

Here is a rendering, into standard keyboarding symbols, of D.C. Kay's formula for the Ricci Scalar, or "Ricci curvature", Rs, in Einstein's theory of General Relativity;

Rs = [g^(ij)] f(x) ,

where

f(x) = f1(x) - f2(x) + f3(x) ,

with

f1(x) = [ {&/[&(x^i)]}{&/[&(x^)]} ][ ln(sqrt|g|) ] ,

f2(x) = [ 1/(sqrt|g|) ]{ &/[&(x^r)] }[ (sqrt|g|)(F1) ] ,

f3(x) = (F2)(F3) ,

for

F1 = ( Christoffel symbol of second kind; contravariant in r, covariant in i and j. ) ,

F2 = ( Christoffel symbol of second kind; contravariant in r, covariant in i and s. ) ,

F3 = ( Christoffel symbol of second kind; contravariant in s, covariant in r and j. ) .

|g| = ( The determinant of the metric tensor's conjugate matrix; i.e., contravariant in ij. )

i, j, r, s = ( Arbitrary index symbols, with mandatory specific ordering; given. )

& = ( Symbol for partial differentiation; used instead of standard symbol. )

^ = ( Symbol for index; implies indices. Does not indicate an exponent. ).

In words, take the square-root of the determinant of the conjugate matrix of the standard metric-tensor, then determine its natural logarithm and its inverse, then use these in the terms defining the three functions, f, of the generalized-coordinate sets, x^i and x^j.

The first function is written as the partial-derivatives successively taken with respect to x^i and x^j of the logarithm of the square-root of this determinant of the metric-tensor; that is, the determinant of the conjugate matrix of the metric-tensor.

The next function is written as the product of the inverse of the square-root of this same determinant with the partial-derivative with respect to x^r of the product of the square-root of this determinant and a Christoffel symbol of second-kind which is contravariant in r and covariant in i and j.

And the last function is written as the product of two such Christoffel symbols; the first contravatiant in r and covariant in i and s, and the second contravariant in s and covariant in r and j.

The difference between the first two functions is added to the last function, and the result is operated on directly by the second-order contravariant form of the metric-tensor, to get the scalar quantity, Rs, the radius-of-curvature; called the "Ricci curvature invariant".

That's it. Meaning, looking experimentally for my get particle, it must be found moving along a Ricci Scalar (radius-of-curvature), and cannot be found going anywhere else (i.e., it cannot take curved paths; the GET particle always observes a Cartesian-straight path).

This does not follow from the foregoing. I'm just saying, this is where a GET resides, and it is not scattered or absorbed by real matter in its path.

It's the objects in the paths of the collection of GET particles radiating from a source that take curved trajectories -- because of the negative radiation-pressure established by the overall flux of radiating GETs, in a time-dependent setting. This is the action that makes gravity work quantum-mechanically; the GET particles' force contributions not only sum to the Newtonian force value, they also physically establish the radius-of-curvature of the space-time described by the field-equations of Einstein's theory of General Relativity.

Having the quantum-mechanical information about a GET particle compacted into some Ricci Scalar, employing a special case of a linearized wave-equation (specifically and only used for GET particles), we can actually derive Quantum Mechanics systematically from the field-equations of General Relativity. And that seems like something seekers of a Grand Unified-Field Theory would be interested in doing, if for nothing else but to use the math as an exercise (for stimulating the imagination).

Note, however, that if my GET particles are proven to exist, that will not rule out the existence of superluminal analogs of the standard spin-2 graviton, nor even that such can also constitute gravitational attraction. It just seems that standard gravitons can now be used additionally to explain the life-force of all living things. That is, superluminal gravitons probably account for the imaginary-energy field that all living organisms interact with to maintain animation.

Other tachyonic energy fields can be postulated, as well, to account for sentient thought, human emotions, and so on. But that deviates into metaphysics. An effort to incorporate Tachyonics into scientific thought can be called "Interdiscipline Synthesis Cosmology", or ISC, which enables experimental investigations into paranormal phenomena.

Allow me to end mathematically.

Let " ^ " denote upper indices (not exponents), and let " _ " indicate subscripts.

Example:

Let x^i_j denote a tensor, contravariant in i and covariant in j.

Then the Kay equation can be written;

Rs = [g^(ij)]{f1(x) - f2(x) + f3(x)} ,

where

f1(x) = [{&/[&(x^i)]}{&/[&(x^j)}][ln(sqrt|g|)] ,

f2(x) = [1/(sqrt|g|)]{&/[&(x^r)]}(sqrt|g|)(F^r_ij) , and

f3(x) = (F^r_is)(F^s_rj) ,

with the "F" indicating Christoffel symbols (rather than forces or functions),

and the "&" indicating partial differentiation (instead of using the standard symbol).

The equation for R above must be viewed as an empirical formula (i.e., testable).

Comments welcome. E-mail: HKurtRichter@yahoo.com