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The Poisson-Laplace equation is a working and acceptable equation of gravitation which is mostly used or applied in its differential form in Magneto-Hydro-Dynamic (MHD) modelling of e.g. molecular clouds. From a general relativistic standpoint, it describes gravitational fields in the region of low spacetime curvature as it emerges in the weak field limit. For non-static gravitational fields, this equation is not generally covariant. On the requirements of general covariance, this equation can be extended to include a time-dependent component, in which case one is led to the Four Poisson-Laplace equation. We solve the Four Poisson-Laplace equation for radial solutions, and apart from the Newtonian gravitational component, we obtain four new solutions leading to four new gravitational components capable (
*in-principle*) of explaining
*e.g.* the Pioneer anomaly, the Titius-Bode Law and the formation of planetary rings. In this letter, we focus only on writing down these solutions. The task showing that these new solutions might explain the aforesaid gravitational anomalies has been left for separate future readings.

“Speculations? I have none. I am resting on certainties.

I know whom I have believed and am persuaded that;

He is able to keep that which I have committed unto him against that day.”

―Michael Faraday (1791-1867)

The Azimuthally Symmetric Theory of Gravitation (herafter ASTG-model) set out in [

when we embarked on the ASTG-model, it was to seek for a plausible solution to the radiation problem thought to bedevil massive stars during their process of formation (cf. [

As argued in [

where:

^{1}This theory can be extended to include the polar solutions

and

About the λ-parameters, it should be mentioned that this property that the λ’s are dynamic parameters assumed to be related to the gravitating body in question is the novelty of the ASTG-model. Putting weight to what we already have said, in a way, the dynamism of the λ-parameters makes the ASTG-model a new classical theory of gravitation where the spin of the gravitating mass enters the gravitational podium. Further, of the λ-parameters, for all conditions of existence, it is assumed that

Furthermore, it should be mentioned that (as was done in the reading [

The novel feature of the ASTG-model is that it brings the spin of a gravitating object into the fold of the classical gravitation (i.e., non-relativistic gravitation). The spin now plays an important and decisive role in generating the gravitational field that has a bearing on test bodies in the vicinity of this gravitating object. But the ASTG-model is not the only theory that does this. For example, we have the gravitomagnetic effects such as the Lense-Thirring Effect [

Of these three important effects, the Pugh-Schiff Gyroscope Effect [

From a purely and strictly general relativistic standpoint, the Poisson-Laplace equation cannot be accepted as a true Law of Nature as it is neither Lorentz nor coordinate invariant for none-static gravitational fields i.e.

Lorentz and coordinate invariance are held as sacrosanct minimum requirements for any law that seek the status of a Law of Nature. In order for the Poisson-Laplace Equation (1) to fulfill the Principle of Relativity, it is necessary to supplement it with a time dependent term, i.e.:

where (here and after)

where

Now, in the weak field approximation, the metric is given by

From a purely general relativistic standpoint, this equation (i.e. (4)) is but an approximation only applicable in the weak field limit. The ASTG-model, together with the Four Poisson-Laplace theory here being advanced, are not a subset of Einstein’s General Theory of Relativity (GTR); see [

As will be demonstrated shortly, an interesting feature of this law (i.e. Equation (4)) is that the time dependent potential

i.e.

As afore-stated, our focus in this letter is on the radial solutions, thus we are going to solve the equation

where ^{−1}. Obviously, this differential Equation will have to be solved for three cases, i.e.

Newtonian Component. In the light of the time dependence just introduced in the Poisson-Laplace equation, we are forced to formally go through the “derivation” of the Newtonian gravitational potential, albeit, with the important difference that the gravitational constant forthwith seizes to be a constant; it is now time dependent. Assuming separability i.e.

From the above, clearly, the time dependent component of the gravitational field

where, as part of the labelling scheme stated earlier, we now have inserted the subscript “1” onto

For the first gravitational component, throughout this letter,

Pioneer Component (I) (Yukawa Potential). Rather in an ad hoc manner, the Yukawa type gravitational potential has long been considered as a possible cause of the Pioneer anomaly [

The Pioneer anomaly and rotation curves of galaxies is then explained by the extra Yukawa term. In the Yukawa term

Now, without wasting much time and space, assuming separability i.e.

where

In a future reading that only awaits the publication of the the present letter, it will be shown that the potential

Planetary Ring Component. We are now going to generate our third gravitational component. This component gives rise to a ring structure around a central massive gravitating body. These rings are such that they are equally spaced. Given this, and as-well that planets not do exhibit such an even spacing, this ring structure “can not explain” the origins of planets as we know them. If one can conceive of the possibility that in those places where a ring is expected and there is none, then, hypothetical, this ring is considered missing, then, the theory has not failed but predicted a missing ring. That said, let us go onto the derivation of this gravitational component.

As one can verify for themselves, the general solution to (7) under the constraint (μ^{2} < 0), is:

where

where

where

thus giving rise to a ring structure, hence, the potential (15) should most certainly explain the existence of rings around planetary bodies such as the planet Saturn. Certainly, it will be interesting to know if this component will be able to explain stellar ring systems. The task for this exploration has been slated for a future reading. In hopeful anticipation, we have called this component the Planetary Ring Component.

With regard to a sinusoidally varying G, is it important to mention here a very interesting development, that is, using about a dozen measurements of Newton’s universal gravitational constant, G; that is, Earth-based laboratory measurements which have been made since 1962, Anderson et al. [

Somewhat in the footsteps of the great German physicist Max Karl Ernst Ludwig Planck (1858-1947) who was reluctant to embrace the revolutionary idea of the quanta that he had discovered in 1900; instead of embracing this potentially landmarking discovery, Anderson et al. [

If one was only seeking separable solutions, then, they would have to end with the solution

As will be shown shortly, there exists two none-separable solutions. For these none-separable solutions we assume

where

The none separable space and time function

where ^{2} > 0 and ν^{2} < 0).

Pioneer/Darkmatter Component (II). Let us set

The two solutions of α are represented by

Our step in solving (17) and (18) is the following: (1) We solve for

where

where

As will be seen a future reading that tackles the Pioneer anomaly, the potential

Titius-Bode Component. As before, let us set

where

In this solution

Using the relation

Since

where

The gravitational constants

As already said in Section 2 is that it has been shown in the reading [

As in electromagnetism, it is demonstrated in the readings [

If is in (7) we have

From (28), it follows that the modified Lorenz gauge condition

From this equation, the most natural solution to this equation for

What we are really concerned about here is (28). Other than the Lorenz gauge, this Equation (28) gives us an extra constant constraint on μ. For the cases (_{2}, G_{3}, G_{4}, and G_{5}; they remain unaltered. However, for the case (_{1} becomes a pure constant without any time variation because the Newtonian gravitational potential Φ_{1} is static since (28) tells us that if (

With regard to a time variable-G (i.e., the Newtonian gravitational constant), recent research has shown that, at least for the last 9 billion of the Universe’s assumed 13.8-billion-year history, the Newtonian gravitational constant G has not varied more than (at most) one part in a billion. This result is obtained after an exhaustive study of about 580 observed supernovae events by Professor Jeremy Mould and his Ph.D. student Syed Uddin at the Swinburne Centre for Astrophysics and Supercomputing and the ARC Centre of Excellence for All-Sky Astrophysics. Their research findings show that the Newtonian constant G has not changed appreciably over cosmic time. This research which focused on Type 1a supernovae, demonstrated a constant G within an upper bound of

The five solutions i.e. Equations (9), (13), (15), (22) & (27) all emerge from the Equation (4), they are thus legitimate gravitational potentials. The questions is, “How does Nature select one solution over the other?” We hypothesize that Nature employs all the five solutions concurrently on every gravitating body at al-times. If this is the case, the total gravitational potential is then given by a superposition of all the five potentials, i.e.:

Written in full, the total radial gravitational potential is as given by:

In this expression (32),

This letter has shown that the Four Poisson-Laplace Equation (4) has five radial solutions

An interesting outcome of the Four Poisson-Laplace Equation (4) is that the gravitational constant G emerges as a time dependent constant. Since it was first proposed that the gravitational constant G might vary with time [

Other than the Newtonian component, the gravitational force may contain four more components and these components may―as hinted herein―explain a number of mysteries such as Darkmatter, the Pioneer Anomaly, origins of the Titius-Bode Law and the emerges of structures such as planetary ring systems.

Golden GadzirayiNyambuya, (2015) Four Poission-Laplace Theory of Gravitation (I). Journal of Modern Physics,06,1195-1206. doi: 10.4236/jmp.2015.69124