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2 Explanation of New Gravitational Theory

The energy-density, rather than just the mass, is a key consideration in this new theory. In this new gravitational theory the attraction between two bodies depends on the energy-densities of each of the two bodies. The energy-density of a body is communicated at the velocity of light via diallel, gravitational-field lines.

Fundamental to this new theory is the discovery of diallel, gravitational-field lines. These connect the two bodies -- providing a path for the gravitational information to flow. To see the evidence of these diallel lines in nature requires a paradigm shift away from the traditional view of gravitational interaction. (see web site www.allanstime.com)

The general model now being used to describe the gravitational field is that the waves are transverse to the direction of propagation of the gravitational energy. No experiments have been able to directly measure gravitational waves, but from the interaction of gravitational energy between a binary pulsar pair, Professor Joseph Taylor (Princeton U.) was able to confirm Einstein's prediction that gravitational energy travels at the velocity of light. [Taylor] These findings are also consistent with the new theory, that the energy travels at the velocity of light.

To better understand how the gravitational energy is transmitted along diallel, gravitational field lines in this new theory, consider the following. As there are seven spherical shells providing the quantum states for any and all of the elements in the periodic table, so there are seven cylindrical shells surrounding a nuclear shell composing each of these diallel lines. All of the force fields can carry photons and/or particles along these diallel lines. Quantum states exist for these diallel lines, also. They similarly depend on the energy conditions.

It may be helpful in envisioning the diallel quantum states to consider other physical systems with cylindrical symmetry and having eigenfunction solutions [Collins]. One such example is the set of modes defining the electromagnetic fields propagating along a circular-cylindrical waveguide. Of special interest are the modal distributions in the case of highly-overmoded waveguides, a situation that may occur when the free-space wavelength of the electromagnetic field is much less than the diameter of the waveguide. Solving the electromagnetic wave equation, subject to the appropriate boundary conditions and assuming a waveguide of infinite length, yields field expressions given in terms of Bessel functions of the first kind. These functions provide an orthogonal set with which to describe the waveguide modes. The eigenvalues are related to the zeroes of the Bessel functions. The particle density distributions in a diallel-quantum state would be described in terms of a similar set of orthogonal functions. Although a great many electrons could propagate along a diallel line by occupying a multiplicity of quantum states, Fermi-Dirac statistics would limit the number of electrons in each state to two electrons. In addition, just as the electromagnetic field can be circularly polarized (the macroscopic manifestation of the spin quantum number of the associated photons) leading to a spiraling of the field vectors in the circular waveguide, the density distribution of a particle in a diallel-quantum state may also be described as spiraling.

2.1 The Fundamental Field Equation

The following equation is fundamental in understanding how the force fields interact and come together for this new theory:

   

D = (E/cG) / Pi(sub2)

(1)

where D is the relativistic density, E is the energy of the unified field, c is the velocity of light, G is the universal gravitational constant, and the last part of the equation, / A2 , the parallel component will be explained below.

Important Note:
The above character, "A2" should read "Pi_sub-2" (Pi sub-2).  If it does not, then you do not have a compatible Greek font and should download the PDF version of this document in which the fonts are preserved.  This will require an Adobe Acrobat Reader, which you may download it free if you don't already have it installed.

 

2.1.1 Density Dependence, D

The density D is the dependent variable; changes in the energy (energy flow in and out of a region) cause changes in the density. For example, the energy flow to and/or away from any space-time continuum along the diallel lines determines the corresponding change in the density in that space-time continuum. Appreciating the importance of the energy density at the particle level, as well as in a region, is integrated in this new theory. The energy can come from any of the force fields. For example, both equations apply: E = mc2 , where "m" is the relativistic mass and E = h<, where "h" is Planck's constant "<" is the electromagnetic frequency of the photon.

2.1.2 Parallel Component (A2)

A dimensional analysis of the above equation reveals that A2 has dimensions of length, time and mass as the force fields interact. The forward slash "/" denotes being parallel in the theory's mass-space-time continuum along the local diallel lines. The "sub-2" on the "A" denotes the energy coming in or going out in the mass-space-time continuum along the diallel lines in the local environment or region.

Combining the energy with this term we have E'A2. Hence, we see that this denotes the energy per mass, per length and per time taken in the parallel direction of the local diallel lines. The quantity in the denominator of equation(1), cG, is the normalizing factor, so that the dependent variable, D, is the density factor taken in the parallel direction of the diallel lines. It is the density that is the principal resultant output after combining the energy from the force fields.

Since the subscript "sub-2" denotes the energy from all sources coming into or going out of the local environment or region, a "sub-1" is implied for the energy, E, and the resulting density, D, as the recipients of the net energy coming in along the diallel lines into the environment or region.

The dependent variable D can also be taken as density of matter in the usual sense: mass per unit volume. But in general, D is a tensor description as the dependant variable resulting from that part of the energy tensor, E, coming from the parallel part A2, as described above. The need for the tensor description comes because the diallel lines have direction and the applicable parallel part comes in through the A2 term f or equation (1) or through the A1 and A2 in the force equation, equation (2).

2.2 Force Equation for the New Theory

From the above field equation, we can derive a force equation:

   

(2)

where the integrals are over the volumes of each of the two bodies being attracted, and where r12 is the distance between the energy-density centers of the two bodies. The integrals across the A terms give the direction for the force vector, which depends on the direction of the diallel lines.

If we let the two bodies be two homogeneous spheres and use equation(1), then equation(2) becomes the classical gravitational force equation:

   

(3)

where r12 is now the distance between the centers of mass of the two homogeneous spheres. Naturally, these spheres can be made arbitrarily small -- point masses -- as in the classical gravitational theory.


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