It was shown in [11], that gravitation was proportional to the spatial gradient of the linear expansion velocity of space, produced by the gravitational source. Appendix B to this paper, has now completed the derivation of the corresponding temporal effects. Solutions to the gravitational Maxwell equations presented here, will now derive these mathematical representations, covering both the acceleration and velocity parameters, without recourse to the information in the preceding main text and Appendices other than the use of the proportionality parameters, e_s and e_t.

A total of four solutions are required, one each for each pair of equations (2.32) to (2.35), covering both the spatial and temporal cases both external and internal to the source. Some of the spatial solutions are well known in the literature but are included here both for completeness and because they lead on to the solutions in the temporal domain.

(i) Case 1 - Static, Spatial, External.

In (2.32), because the curl of A_s is zero it can be represented by the gradient of a scalar field thus

where

U_s    is a scalar field.

Substitution of (C.1) into the second part of (2.32) together with (2.8) gives

which is a four dimensional version of Laplace's equation. This, in expanded form is

Here the spatial part has been expressed in spherical co-ordinates while the temporal part is in Cartesian. This is permitted in view of the special co-ordinate relationship between the spatial and temporal domains. In any case the temporal part is zero because there is no direct variation of gravity with time, (¶x₀ = cu¶t).

Eq.(C.3) therefore becomes simply

and (C.4) has the solution

Determination of the constants of integration is as follows. When s ® ¥, the gravitational effect tends to zero, whence U_s ® 0. Therefore k₂ = 0 and (C.5) becomes simply

So that, for a spherically symmetric mass, from (C.1)

To determine k₁, apply Gauss' law to D_s thus, using (2.8)

From which

i.e. as at (2.5). Thus from (C.7) and (C.9)

So that in (C.6)

U_s has the dimensions of a velocity squared and can be related to the spatial expansion velocity of the source as follows

From which

and therefore from (C.11)

as stated in Appendix A et al and derived in [11]. U_s is clearly, from [11], Eq.(3.38) the Newtonian Potential, (spatial), for the Relativistic Domain theory of gravitation external to the source.

(ii) Case 2 - Static, Spatial, Internal.

To solve (2.33) note that, for this case, (C.2) becomes

and thus (C.4) becomes, (the temporal component is again zero)

This has the solution

To determine k₂ note that when s_i = s_g, from (C.11), U_is = U_g = gM/s_g so that in (C17) this gives

which gives in (C.17)

Thus for a spherically symmetric mass, at s_i

Now, applying Gauss' law to D_is

where

s_i     is the surface area of a sphere of radius s_i inside the source.

M_i     is the mass of the source contained within s_i.

From (C.21)

as derived in [11], and equating this to (C.20) shows that k₁ = 0 so that in (C.19)

From [11], Eq.(3.46) U_is is therefore the Newtonian Potential inside the source at a distance s_i from the centre. In an identical process for the external case it is seen that it is related to the internal spatial expansion velocity thus

so that from (C.23) and (C.24)

as derived in [11] and stated in Appendix A.

(iii) Case 3 - Static, Temporal, External.

Solution of (2.34) in the temporal domain is a little more involved. In this case the equivalent of (C.2) is

So that from (2.30)

and thus

Taking the divergence, from (2.31)

Because |D_t| = |D_s|, (C.29) can be written thus

Taking the first integral

A_t is the temporal Acceleration Potential vector and therefore lies along the x₀ axis only. The gradient of U_t can therefore be written

and with ¶x₀ = cu¶t this becomes

Inserting (C.14) for ¶s/¶t this becomes

Integrating

Exactly the same generic result as in the spatial domain.

To determine k₂ note that when s ® ¥ the temporal velocity, u_t ® c so that via similarity with the spatial domain

This gives in (C.35)

Now apply Gauss' law to D_t thus

From this it is seen that

As derived in Appendix A. Comparing (C.39) with (C.31) shows that

so that in (C.37) this gives

and via the same process as at (C.32) shows that

and thus from (C.41) and (C.42)

as derived in [11] and shown in Appendix A.

(iv) Case 4 - Static, Temporal, Internal.

Here (C.29) becomes

Again because |D_it| = |D_is|, (C.44) can be expanded to

The first integral of (C.45) is

A_it is the temporal Acceleration Potential vector internal to the source at s_i and lies only along the x₀ axis. The gradient of U_it can therefore be written

and this becomes

Inserting (C.25) for ¶s_i/¶t this becomes

Integrating (C.49) then gives

When s_i = s_g, from (C.41)

so that in (C.50) this gives k₂ as

which substituted back into (C.50) yields

Now applying Gauss' law to D_it

and from this

as shown in Appendix A. Comparing this to (C.46) shows that k₁ = 0, so that in (C.53) this yields

Finally, via the usual process, with

then

as derived in [11] and stated in Appendix A.

(v) The Steady State Case.

Solutions to the steady state case are not presented here because neither the translational nor the rotational motions of the source contribute directly to the generation of gravity. Secondary effects only exist and are as follows.

(a) Both motions will cause a relativistic increase in the mass of the source which results in a small increase in the magnitude of the gravitational effect.

(b) The relativistic mass increase due to the rotational motion will result in a very small variation of the gravitational field in the azimuthal direction. This will have some effect on the orbital dynamics of orbiting bodies.

(c) Both motions enter into the temporal velocity equation as extra boundary conditions but, as stated above, do not affect the generation of gravity.

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