For = mvor ·Bs

(2.26)

An astronomical gravitational source can possess two forms of motion, a translational motion due to the rotation of its parent galaxy, and a rotational motion due to its own spin. Consider first the translational motion, this would result in a B_Ts field as shown below.

Fig. 2.1 Generation of the B_Ts Field.

The spin motion would accordingly produce a toroidal field, B_Rs thus,

Fig. 2.2 Generation of the B_Rs Field.

In Figs. 2.1 and 2.2, looking in direction A, the combination of the two fields would then take the form as shown, (approximately), in Fig. 2.3 below. On the LHS the fields would be additive, while on the RHS, subtractive. Thus, according to (2.26), the orbital plane of any planetary mass in such a field would, over a very long period of time, precess in the same direction as the field. When on the RHS its direction of precession would depend upon the relative strengths of B_Ts and B_Rs. If B_Ts > B_Rs, (as shown in Fig.2.3), the planetary orbit would continuously rotate completely around the star. If B_Rs > B_Ts then the planetary orbit plane would eventually stabilise parallel to the spin axis of the star.

Fig. 2.3 Composite B_s Field due to Combined Translational and Spin Motions.

Inside the star the same fields would exist but would now cause the precession of the gaseous material of the star itself. This would have two effects. Firstly, the precessional motion of internal matter would itself set up a B_iRs field opposite to the direction of the spin which would tend to effectively cause a slight reduction in the spin motion. The second effect would, because of the slower rate of material precession on the RHS of the star in Fig.2.3, there would be a build up of density on the RHS which would then cause a slight nutation of the spin axis, and also induce a small translational motion wobble.

Observation of the Sun and the orbital planes of the planets of the Solar System, indicate that none of the above effects have become apparent over the five or so billion years that the system is thought to have been in existence. Furthermore, observation of the proper motion of the nearby stars in the galaxy, similarly do not indicate an erratic motion as they orbit its centre.

Therefore, from the above dissertation, the final conclusion reached here is that analogous gravitational parameters to B and H do not exist. A possible explanation for this is presented in Section 3.

2.4  Derivation of the Applicable Gravitational Equivalents - Temporal.

It is shown in Appendix B that external to the source, the variation of temporal velocity, i.e. the Acceleration Potential in the temporal direction is given by

At = - ( 1-u2 )1/2 u | As | x0

(2.27)

where

|A_s|     is the magnitude of A_s.

x₀     is a unit vector in the temporal direction.

Eq.(2.27) is therefore the gravitational equivalent of E in the temporal direction.

To determine the gravitational equivalent of D in that dimension, note that it would take the form

Dt = et At

(2.28)

and would, similar to (2.8) possess dimensions of mass/cm². Accordingly its divergence would possess dimensions of mass/cm³. Clearly these parameters can only be

the mass surface density and mass volume density and as such must be invariant. Consequently, from (2.27) and (2.28) it must therefore mean that

et = - u ( 1 - u2 ) 1/2 es

(2.29)

so that from (2.28) and (2.29)

Dt = - u ( 1 - u2 ) 1/2 es At = | As | 4pg = rs  x0

(2.30)

Therefore it is clear that

| Dt | = | Ds | = rs     (internal)                      = 0     (external)

(2.31)

and

Ñ4 ·Dt = Ñ4 ·Ds = rv     (internal)                            = 0     (external)

Eq.(2.29) cannot be proved theoretically but, will be shown in Appendix C to be correct via the solutions to the Maxwell equations that result from the above.

The non-existence of the spatial fields H_s etc necessarily determines their non-existence in the temporal direction.

2.5  The Gravitational Maxwell Equations.

In Appendix B the curl of both the spatial and temporal versions of the Acceleration Potential, both internal and external to the source are shown to be zero, i.e. all Acceleration Potentials are irrotational, and consequently, from this and the preceding derivations, the gravitational Maxwell equations can now be stated.

There are four sets of equations required to fully express gravitation in the form of Maxwell's equations. They are as follows.

(i) Spatial, external to the source.

Ñ4 ×As = 0 Ñ4 ·Ds = 0

(2.32)

(ii) Spatial, internal to the source.

Ñ4 ×As = 0 Ñ4 ·Ds = rv

(2.33)

(iii) Temporal, external to the source.

Ñ4 ×At = 0 Ñ4 ·Dt = 0

(2.34)

(iv) Temporal, internal to the source

Ñ4 ×At = 0 Ñ4 ·Dt = rv

(2.35)

Eqs (2.32) to (2.35) represent the simplest mathematical formulation for Relativistic Domain gravitation and it is not considered useful to attempt a generalisation in view of this. Their solutions are presented in Appendix C.

The parameters J_s and ¶D_s/¶t do not appear in these equations due to the non-existence of H_s etc and in any case are more associated with the dynamic effects of gravitation than with the generation of the gravity fields.

G4 Version 1.0.0
Ó P.G.Bass, August 2009