3.3.4  Derivation of the Field Propagation Equations.

The derivation of the previous Sections have provided a mathematical model for the manner in which the Acceleration Potential and the time dilatation effect are generated by a gravitational source. From this it is quite clear how these effects are promulgated to the space surrounding the source. To augment that understanding, this Section will provide the field equations describing that propagation. It is to be noted that because the space-time continuum within which this new gravitational theory has been developed, D₁, is a linear space-time continuum, the field equations derived here will be identical to those of Newtonian theory.

The process of derivation is started from (3.9) as follows, (thus the situation outside the physical boundaries of the source are to be considered first)

Solving for (v_s ²)/2

Call this term U_s, which can be identified as similar to the Newtonian Potential of classical theory, then

From (3.34), this can be expressed as

and thus

The L.H.S. of (3.41) is the Laplace equation for a spherically symmetrical mass. Therefore, the field propagation equation outside of the source may be stated as

Also, because this means that the Acceleration Potential can be stated as

then by vector theory

and the Acceleration Potential vector field outside of the gravitational mass is irrotational as is implicit in the definition of the gravitational space-time continuum.

Internal to the source, the equivalent of (3.37) is

and therefore as in the process above outside the source

calling this U_is, the internal equivalent of U_s then as above

so that from (3.7)

and again following the derivation outside the source

Here the L.H.S. is Poisson's equation for a spherically symmetrical mass, and therefore, the field propagation equation inside of the source may be stated as

In (3.49) and (3.50) r_g is the average density of the gravitational source. Therefore, again by vector theory

and so once again

and the Acceleration Potential vector field inside the gravitational source is also irrotational, again an expected result.

Above it was stated that the parameter U_s was similar to the Newtonian Potential U of classical theory. Their exact relationship is as follows. From (3.40)

and with

then from (3.53) and (3.54)

and therefore the only difference between the Newtonian and Relativistic Domain gravitational potentials is a relativistic one. Again in view of the definition of the gravitational space-time continuum used in this new theory, this is an expected result.

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