^{*}

Gravitomagnetism is universally and formally recognised in contemporary physics as being the linear first-order approximation of Einstein’s field equations emerging from the General Theory of Relativity (GTR). Herein, we argue that, as has been done by others in the past, gravitomagnetism can be viewed as a fully-fledged independent theory of gravitomagnetism that can be divorced from Professor Einstein’s GTR. The gravitomagnetic theory whose exposition we give herein is exactly envisioned by Professor Maxwell and Dr. Heaviside. The once speculative Maxwell-Heaviside Gravitomagnetic theory now finds full justification as a fully fledged theory from Professor José Hera’s Existence Theorem which states that all that is needed for there to exist the four Max-well-type field equations is that a mass-current conservation law be obeyed. Our contribution in the present work, if any, is that we demonstrate conclusively that like electromagnetism, the gravitomagnetic phenomenon leads to the prediction of gravitomagnetic waves that travel at the speed of light. Further, we argue that for the gravitational phenomenon, apart from the Newtonian gravitational potential, there are four more potentials and these operate concurrently with the Newtonian potential. At the end of it, it is seen that the present work sets the stage for a very interesting investigation of several gravitational anomalies such as the ponderous Pioneer Anomaly, the vexing Flyby Anomalies, the mysterious Anomalous Rotation Curves of Spiral Galaxies and as well, the possibility of the generation of stellar magnetic fields by rotating gravitational masses.

“The world which we have to build from the crude material is the world of perception,

and the process of building must depend on the nature of the percipient.”

―Sir Arthur S. Eddington (1882-1944)

The first to consider the possibility of a formal analogy between gravitation and electromagnetism is the great and pre-eminent Scottish physicist―Professor James Clerk Maxwell (1831-1879), in his landmarking treatise on “A Dynamical Theory of the Electromagnetic Field” [

Writing in his book “Causality, Electromagnetic Induction and Gravitation”, Professor Oleg D. Jefimenko (1922-2009) revived [

then, there must be two corresponding fields,

where α, β, γ are arbitrary positive constants and are related to the speed of light c by the equation

Unfortunately, Professor José Hera [

In closing the present section, we give the synopsis of the paper. In Section 2, we give the fundamental basis for why gravitomagnetism must exist. In Section 3, we suggest a source for gravitomagnetic waves. In Section 5, we demonstrate that gravitomagentic waves must travel at the speed of light c. In Section 6, we fit in the Lorenz gauge into the present gravitomagnetic framework. Lastly, in Section 7, we give a general discussion and conclusion drawn. Here at the penultimate, we must state clearly that in this reading, we do not explore the resulting theory, i.e., test is predictions based on already well-known gravitational anomalies―this task we have left for future readings.

Without any doubt, the Hera Existence Theorem [

where;

where,

is the gravitational permittivity of free space. It is universally accepted that the conservation of mass is always at play everywhere and every-time in the Universe, so, it (the mass continuity equation) holds simultaneously with the Poission-Laplace equation thus, one can―as shall be done shortly, solve these two equations simultaneously. A part of the notation to be used hereafter, it is important that in line with the notation used in the reading [

Now, to demonstrate Professor Behera [

where,

hence:

Now, it is a bona fide mathematical fact that for any general, smooth, differentiable and uniform continuous vector field, say

where

where, analogously, as the speed of light:

is the speed of the resulting gravitomagnetic waves. This speed is not necessarily equal to the speed of light. However, most approaches in gravitomagnetism ad hocly set this speed to equal the speed of light without any real fundamental physical justification but as part and parcel of the formal analogue with Maxwellian electrodynamics. In §(5), we will show that a proper consideration of facts at-hand within the gravitomagnetic framework, invariably set this speed to equal the speed of light, c. It is our view that this achievement in Section 5 of demonstrating that

Notice that in (13), we have two options “±”. Which of the two corresponds to reality? Certainly, since there can only be one Universe, only one of the two options “±”, and not both, will correspond to reality. To answer this question, Professor Behera [

Now, at this point we have to ask what does (13) really mean? This equation is a direct logical consequence of the Poisson-Laplace Equation (7) and the continuity Equation (9). Without any doubt, (13) points directly to the undeniable fact that the gravitational field g has an associated magnetic-like field

Now, Equations (7) and (13) have a seductive and irresistible resemblance with Maxwell’s source coupled equations so much that for the brave that will set their mind on this, they most probably would proceed without detouring [

In Professor Behera [

We will demonstrate that Professor Hera [

where,

Equation (19) is not a new invention but a mathematically valid way if not an “economic”, efficient and elegant manner of writing Equations (15) and (18); we have not done anything new or strange here.

Now, the tensor field,

As is well known―or as can be shown or easily demonstrated is that, any smooth, differentiable and uniform continuous antisymmetric two-rank tensor field,

where,

Now, if,

where in (23),

Another way to generate the field Equations (16) and (17) is to note that given the definitions (23) and (24), and the field tensor,

which is the Bianachi identity. From this equation, the field Equations (16) and (17) follow smoothly. So, effectively, the mass continuity equation together with the Poisson-Laplace equation for the gravitational field, are sufficient to demonstrate that gravitamagnetism is a just physical theory with a firm fundamental physical basis.

In his search for a causal theory of gravitation, Professor Einstein [

“After the untenability of the theory of action at distance had thus been proved in the domain of Electrodynamics, confidence in the correctness of Newton’s action-at-a-distance theory of gravitation was shaken. One had to believe that Newton’s Law of Gravity could not embrace the phenomena of gravity in their entirety, any more than Coulomb’s Law of Electrostatics embraced the theory of electromagnetic processes.”

Certainly―as Professor Behera [

Just as is the case with Electromagnetism, gravitomagnetic waves (hereafter, GM-Waves) are a clear prediction of not only the present gravitomagnetic theory―but all gravitomagnetic theories. In vacuum, we have:

As can clearly be read-off from (26) and (27), these GM-Waves travel in empty space at the speed

An important question is “How are these GM-Waves generated and how are they to be detected?” The gravitational phenomenon, in both Newtonian and Einsteinian theories, is often assumed to be a central field phenomenon, i.e., the resulting field depends only on the radial distance from the source, the field is spherically symmetric. If we did assume this here, then, the gravitomagnetic vector field

So, asking “How are these GM-Waves generated?” amounts to asking how does one obtain the conditions,

The proposed Unified Field Theory (UFT) [

1) Lead to a violation of gauge invariance in Quantum Field Theories (QFT) [

2) Lead to short range and short lived photons [

3) Lead to three extra degrees of freedom, which amongst many other things, this would lead to a modified Plank Radiation Law [

4) Lead to an energy dependant speed of light [

In the readings [

Of massive photons, the only problem that remains unsolved by (28) is that the speed of light becomes energy dependent. But this is not necessarily a problem as has been argued in [

where

Equation (19) is the equation for a massless gravitas. For a massive gravitas, as derived in [

If the work presented in [

^{2}We must note that the Lorenz gauge,

In (30), the field tensor

According to the foregoing, the speed of gravity or the speed of the resulting gravitational waves is

The mass term in (30) vanishes because of the gauge conditions (28) and (29). The 0-component of (31) is:

and the space component of this same equation is:

Equation (32) has been considered in [

Now, if we assume a non-static gravitational field, then, the gauge condition (29), becomes:

then, from the above equation, it follows that:

The above equation is easily deducible from Equation (34). Now, applying the operator,

From (32), it follows that the above reduces to:

Since

Comparing the left hand-side of (38) and (33), i.e.,

Therefore, the speed of GM-Waves is the same as that for light in a vacuum. In our feeble view, this is an important result as it emerges from within the internal structure and logic of the theory, it is not imposed from outside as is the case in most gravitomagnetic theories, were it is ad hocly or impromptu assumed that the speed of gravitomagnetic waves or gravitational waves must be the same as that for light. From the result (14) and (39), it follows that the fundamental constant,

In the next section, by deriving the corresponding Lorentz term, we shall now apply the present gravitomagnetic theory to see what it “has to say” insofar as known gravitational anomalies is concerned.

What is the equivalent Gravitomagenetic Lorentz (hereafter GM-Lorentz) equation of motion? We know that the Electromagnetic Lorentz (hereafter EM-Lorentz) equation of motion is given by

field, whereafter they have proceeded to investigate the Lorentz term,

anomalies. In the equation,

While there are a number of ways to arrive at the Electromagnetic Lorentz Force Law (cf. [

As part of our contribution in the present work, we depart from this tradition of assuming that the GM- Lorentz field is given by,

This Coulomb gauge,

Now, just as the EM-Lorentz force field is given by,

where the term,

If this dark-force really exists, it may explain some of the observed gravitational anomalies that have puzzled a great number of scientists today. It is a term which if real, then, it will―depending on its strength; certainly lead to significant anomalous gravitational effects that are not modelled in the term,

Since we have shown in [

Since

where

1) The Problem of the Flat Rotation Curves of Spiral Galaxies [

2) The Flyby Anomalies Problem [

3) The Pioneer Anomaly Problem (cf. [

4) The Origins of the Titius-Bode Law (cf. [

5) The Origins of the Tilt About the Ecliptic Plane of Planetary Orbits. Not much thought has been given into the enquiry of why planetary orbits are tilted or whether one can find a systematic pattern in this. If our work is correct, it somehow suggests a possible pattern that can clearly be deciphered.

6) The Secular Increase in the Mean Sun-(Earth-Moon) Distance [

Clearly, there is a plethora of work ahead. All we can say about this work, is that―most of it has been conducted, all it awaits is finalization, where upon it will be submitted for publication. This work, together with [

From the Hera Existence Theorem [

With regard to a time variable Newtonian gravitational constant G, recent research [

On one the achievements of the present theory, we can safely say that, an important outcome of the present theory is that we have shown that gravitomagnetic waves must―as electromagnetic waves do; travel at the speed of light and these most certainly are generated by spinning concentrations of gravitating mass. In the theory, we make no prior assumption to this fact but set the speed of these waves to

Another aspect that sets the present gravitomagnetic theory apart from most if not all gravitomagnetic theories in existence today (cf. [

Despite the fact that we have tried to distance the present gravitomagnetic theory from our proposed UFT [

In-closing, allow us to say that the main purpose of this work has been to justify the fundamental physical basis of gravitomagnetism well outside the framework of our proposed UFT [

Assuming the acceptability of the ideas here presented, we hereby make the following conclusion:

1) Gravitomagnetism is a valid physical phenomenon since it can be justified from well accepted fundamental physics principles.

2) Gravitomagnetic waves travel at the speed of light c.

3) As pointed herein, gravitational anomalies such as darkmatter, the secular recession of planets from the Sun, the Pioneer anomaly etc., may prove to be effects caused by the gravitomagnetic field.

Golden GadzirayiNyambuya, (2015) Fundamental Physical Basis for Maxwell-Heaviside Gravitomagnetism. Journal of Modern Physics,06,1207-1219. doi: 10.4236/jmp.2015.69125