2 Relativistic Domain Gravitation in the Form of Maxwell's Equations.2.1 Preamble.It has been stated in the literature, i.e. [8] et al, that one of the difficulties in establishing an analogy between electrostatics and gravitation for both Newton's and Einstein's theories, is that electrical charge can take either of two values, +ve or -ve, whereas its proposed gravitation analogy, mass, is always only +ve. As a result, in electrostatics, unlike charges generate an attractive force and like charges generate a repulsive force, whereas, in gravitation, "like masses" result in an attractive force. Thus, for Newton's and Einstein's theories gravitation is, unlike electrostatics, a single valued process and of opposite sense.This difficulty does not exist with the Relativistic Domain theory of gravitation because, as already shown in [12], under the right conditions, its effect can be reversed to exhibit a repulsive force. This is not however, the result of a reversal of the sign of mass, but rather the reversal of the sign of another parameter. This was demonstrated in [12]. One result of the linear nature of Maxwell's equations, and the linear nature of Newton's theory and of the linearization for weak fields of Einstein's equations, is that the process known as superposition can be employed. This was effectively used by Heaviside in his investigations of the subject. It is via this process that it was predicted in Heaviside's paper, and other literature since, that the velocity of propogation of "gravity waves" is the speed of light. While this process produces sets of analogous equations between two theories of a similar genre, and thereby allows theoretical development of the second along the same lines as the first, it does not however, necessarily mean that the parameters so obtained actually exist. Because of this uncertainty in its use, the principle of superposition will, wherever possible, be avoided in this paper and all derivation be pursued from first principles. It should also be noted that Maxwell's equations in electromagnetic theory applies to matter on an atomic scale. Velocities and reaction times are therefore very fast. In gravitational theory, the masses involved are that of ponderable matter and thus velocities and reaction times are accordingly very slow. Finally, it is noted that Maxwell's equations are all spatial in nature and are concerned entirely with effects external to the source. Gravitation on the other hand, is also effective inside the source and has a considerable temporal effect. These additional factors will be considered in the development here.
2.2 A Brief Review of the Maxwell Equations of Electromagnetics.The development herein will, for the most part, be conducted with the differential form of Maxwell's equations, the most general form of which are as follows,
where
E is the Electric field intensity, (gms/coulomb º F/Q). In (2.1), ¶D/¶t is called the displacement current density and the term ¶B/¶t is simply known as the time rate of change of B. In the parametric dimensional definitions, Q is charge, F is force, L is length, I is current, W is magnetic flux and T is time. The objective is now to determine, where they exist, the gravitational analogues of the parameters in (2.1), and from these to construct the equivalent gravitational Maxwell equations.
2.3 Derivation of the Applicable Gravitational Parameter Equivalents.2.3.1 The Electrostatic Parameters.Of the two sets of parameters, electrostatic and magnetic, the former are the easier from which to obtain the gravitational analogues. This can be done by starting from the equivalence relationships for the generation of force, i.e. in generalised vector form,
Where
Q is the electrostatic charge on the source. From (2.2) it is clear that, as stated earlier and in the literature, that the mass of the gravitational source, M, is the analogous parameter to the electrostatic charge, Q. It is important to note that the constant of proportionality in each case in (2.2) is,
electrostatic
Where clearly As is the Acceleration Potential of the gravitational source, see [9], and it is this parameter that may be considered as the gravitational analogy to E, the electric field intensity. Accordingly As could also be known as the gravitational field intensity.
The electrostatic parameter D is obtained from
E by multiplying by e, the permittivity of
the medium, thus
where rs is the apparent surface mass density of the source at a distance s from its centre. Ds is a new gravitational parameter with dimensions of mass/cm2 and may accordingly be designated as the surface mass density vector of the gravitational source. The term -1/4pg would thus be analogous to e and consequently be the ßpatial gravitational permittivity" of the medium, say es. To show that Ds is indeed the gravitational parameter analogous to D, its divergence is determined thus for the situation inside the gravitational source.
and in spherical co-ordinates this reduces to
where
Dis is the value of Ds internal to the source at a distance si from its centre. From [11] Eq.(3.35), substitution for Ais in (2.10), gives
which works out to
where
sg is the physical radius of the source. Thus it is clear from (2.12) that, inside the source, Dis is the analagous gravitational parameter to D. For the situation outside of the gravitational source, it is easily shown by a similar process to that above that
The analagous parameters to Jc and ¶D/¶t are of much less importance than Es and Ds but can exist and are briefly considered below. To determine the analogous gravitational parameter to Jc in (2.1), cognizance is taken of the manner in which Jc is determined in [13], viz.
In a current carrying wire if n is the number of electrons per unit length,
each with a charge qe and an average velocity ve then the charge
rate along the wire is
where
l is a unit vector along the wire length. If the cross sectional area of the wire is a, then
Finally, if this current flow is the result of an electric field, then
where
b is the conductivity of the wire. The gravitational equivalent of Jc would then be determined as follows. If a mass m was falling, with a velocity of vm, under the influence of a gravitational source, and its density was uniform such that its mass per unit length was constant at d, then the conduction mass flow rate would be
where
t is gravitational time. Thus if the average cross sectional area of the mass were am then the conduction mass flow rate density would be
which clearly has dimensions of M/L2T. If (2.16) applies in the gravitational case, then
and bs would be the mass flow conductivity of the medium with dimensions F/TL2, (the same as b). It has been shown in [10] that external to the source the gravitationally induced velocity of a mass is subject to a maximum value. Therefore it is proposed that the parameter bs does exist, and has a value < ¥. In this case, under steady state conditions, vm above would be a maximum value. The analogy to displacement current is determined as follows. If the gravitational Acceleration Potential changes, this would cause a change in the conduction mass flow rate density. The change in As in this case could only be, realistically, a change in the mass of the source, so that from (2.5)
This works out to
So that from (2.8) this becomes
Consequently ¶sg/¶t is the rate at which the radius of the gravitational source changes to cause a variation in M and thereby As. Eq.(2.22) is therefore the mass flow rate density due to a variation in the mass of the gravitational source. The total mass flow rate density then becomes
and which from (2.19) and (2.22) can be written
These effects and their relationships can only occur external to the source. They describe the dynamic results of gravitation on other material bodies and the effects of physical change to the source itself. They are therefore not directly associated with the generation of the gravity fields, and, due to the results of the following Section, do not appear in the Maxwell gravitic equations. This completes the identification of the gravitational analogies to the electrostatic parameters. Note that they could all have been obtained via the superposition principle which provides corroborative support to their derivation.
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P.G.Bass, August 2009
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