Latest revision 06-12-2023

index $ \def \hieruit {\quad \Longrightarrow \quad} \def \slechts {\quad \Longleftrightarrow \quad} \def \SP {\quad ; \quad} \def \MET {\quad \mbox{with} \quad} \def \EN {\quad \mbox{and} \quad} \def \half {\frac{1}{2}} $

Relativizing Newton

We quote from Relativity Reexamined at page 101-103: He wonders whether there might be some deep similarity, something analogous to the electromagnetic field equations, but applicable to inertial matter! He is tempted to believe that some new physical effect may have escaped detection. In electromagnetism, one needs two vectors E and H ; assuming the inertial E to correspond to gravitation, what should be the role of the inertial H? Bridgman does not elaborate [ .. ]
We emphasized the startling similarity between electrostatics and equations of a static gravity field F (gravistatics). In order to discuss nonstatic problems, Carstoiu assumes the existence of a second gravitational field called the gravitational vortex $\bf \Omega$; both fields are supposed to be coupled by equations similar to Maxwell's equations, and obtain a propagation velocity $c$ equal to the velocity of light. [ .. ]
After writing the papers just discussed here, Carstoiu discovered a very extraordinary note of Heaviside (1893, 1950), where he suggests for gravitation a set of equations very similar to Maxwell's electromagnetic equations and Garstoiu's formulas. Heaviside shows that these equations require the introduction of a second field, analogous to the magnetic force; this is Carstoiu's vortex $\bf \Omega$. It is very strange that such an important paper had been practically ignored for so many years, but the reader may remember that Heaviside was the forgotten genius of physics, abandoned by everybody except a few faithful friends.

Indeed there is an obvious similarity between Coulomb's Law of electricity and Newton's Law of gravity: $$ F = \frac{q_1 q_2}{4\pi\epsilon_0 r^2} \quad \sim \quad F = G \frac{m_1 m_2}{r^2} $$ Here $F=$ force, $q=$ electric charge, $m=$ mass, $r$ = distance between point charges/masses, $\epsilon_0=$ dielectric constant of the vacuum, $G=$ gravity constant.
Electricity goes hand in hand with magnetism. And the two are related via the speed of an electromagnetic wave: $$ c = \frac{1}{\sqrt{\epsilon_0\mu_0}} \quad \Longrightarrow \quad \mu_0 = \frac{1}{\epsilon_0 c^2} $$ Here $c=$ speed of light, $\mu_0=$ magnetic permeability of free space. Analogously we could write: $$ G \sim \frac{1}{4\pi\epsilon_0} \quad \Longrightarrow \quad \mu_0 = \frac{1}{\epsilon_0 c^2} \sim \gamma = \frac{4\pi G}{c^2} $$ Where $\,\gamma\,$ is sort of gravitational equivalent of $\,\mu_0\,$.
At first the following question must be raised. Can Maxwell's equations be derived from Coulomb's Law and Special Relativity?. The answer is affirmative. Indeed Special Relativity, together with the assumptions of charge invariance and Coulomb's law, can be employed for a derivation of Maxwell's equations. It may thus be suggested that quite an analogous procedure can be used for gravity. Which is exactly what has been done, by Oliver Heaviside, quite a while ago, resulting in A GRAVITATIONAL AND ELECTROMAGNETIC ANALOGY, the GEM equations of GravitoElectroMagnetism. The existence of Gravitational Waves may be one of the consequences, analogously to the existence of electromagnetic waves.

Can gravitoelectromagnetism stand the (other) tests that are supposed to be empirical evidence for General Relativity?
What are the Tests of general relativity? Quote: The first three tests, proposed by Albert Einstein in 1915, concerned

  1. the "anomalous" precession of the perihelion of Mercury,
  2. the bending of light in gravitational fields,
  3. and the gravitational redshift.
Published articles about these issues do not receive unanimous enthusiasm, especially when it comes to probably the most significant of all, the first one:

Mercury's Perihelion Advance

From Relativity Reexamined at page 98 we quote: The advance of the perihelion of Mercury (43 seconds per century) was hailed as a wonderful check with a theoretical prediction of $42.6$, but here again let us refer to Chazy (1930) who found a number of other examples in the solar system where Einstein's predictions conflict with experiments. It is hard to believe seriously in a coincidence of less than one second for Mercury, while so many other examples give large errors and even opposite signs! Let us here candidly admit that there must be many other unknown factors involved. The computations of Chazy refer to the motions of perihelions of four planets and similar motions for a number of satellites orbiting around planets (e.g., the moon). Errors of at least five seconds per century seem to be the inevitable limit in these very difficult computations. Einstein's theory yields about $1/6$ of the advance of perihelion of Mars and practically nothing for Venus. Let us add that Dicke's discovery of the oblate shape of the sun leads to perturbations that definitely destroy the agreement about Mercury. The question cannot be considered completely settled.
Our second reference is the book Origin of Inertia | Extended Mach's Principle and Cosmological Consequences by Amitabha Ghosh.
A very important remark is already found in section 1.1 Introduction : Though the phenomenon of gravitation is most predominant in the formation of planetary, stellar and galactic systems, the accuracy with which we know the value of the gravitational constant, $G$ , is the least compared to the other universal constants. One may build some heavy mathematical structure for describing gravity on top of it - such as Einstein's Theory of General Relativity - but this fact alone makes that it is actually built on quicksand. The Proposed Modifications of Newton's Laws, as presented in section 2.4, are indeed numerous. Other examples are found on the internet, such as Modified Newton's Gravitational Theory to Explain Mercury Precession and Light Deflection by Keli Kou, which in turn is referring to Advance of Mercury Perihelion Explained by Cogravity by C.J. de Matos and M. Tajmar. Another interesting reference is: A remarkable piece of prose at the end of this article should be quoted: The authors make an appeal to the readers not to consider the MG as an alternative theory of gravity to the GR, because MG has to be made compatible with many other experimental data or observational results for its elevation to that status. So we now prefer MG to be treated as a toy model vector theory of gravity in flat space-time. Toy model? Are you kidding? I didn't thoroughly check content, but if they did their homework well, then this is not just a "toy model". It undermines one of the most important tests of General Relativity!

Apart from any computational details, perhaps it's possible to make an order of magnitude estimate, just to see if Cogravity might have any effect, at all, on Mercury's Perihelion. The dimensionality of $\,\gamma\,$ is $\,[G]/[c^2] = m^3\,kg^{-1}\,s^{-2}\,/\,(m\,s^{-1}\,)^2 = m\,kg^{-1}\,$. Calculate it's magnitude and multiply with the mass of the sun, then we get something with the dimension of length. 
# Gravitational constant
G := 6.67408*10^(-11);
# Speed of light
light := 299792458;
# Mass of our sun
sun := 1.989*10^(30);
# Cogravity factor
Cogravity := evalf(4*Pi*G/light^2);
# That something with
# the dimension of length
Length := Cogravity*sun;

                   Length := 18560.71274 # 18.6 km
Mercury (planet) data are copied from Wikipedia: 
Perihelion := 46001200; # km
Orbital_period := 0.240846; # years
# Length of one arc second in km
second := evalf(2*Pi*Perihelion/(360*60*60));
# Number of orbital periods in a century
number := 100/Orbital_period;
# Estimate of perihelion shift
shift := number*(Length/1000)/second;

                         shift := 34.55502278
The experimental value as well as the value calculated with General Relativity theory is $42.98$ arcseconds. So what we anyway have is a correct order of magnitude, if calculated eventually with Cogravitation / Gravitoelectromagnetism.

Gravitational deflection of light

is the second classical test on General Relativity, covered on a separate web page.

Gravitational redshift

is our third and last classical test on General Relativity. Quote from the Wikipedia reference: Since this prediction arises directly from the equivalence principle, it does not require any of the mathematical apparatus of general relativity, and its verification does not specifically support general relativity over any other theory that incorporates the equivalence principle. The result follows by employing the law increase of (relativistic) kinetic energy = decrease of (common) potential energy, with $m=$ mass photon, $c=$ lightspeed, $g=$ gravitational acceleration, $y=$ height, $h=$ Planck's constant, $\nu$ = frequency and $\,h\nu=mc^2\,$. $$ d(mc^2) = -m.g.dy \hieruit h\,d\nu = -\frac{h\nu}{c^2}g\,dy \hieruit \frac{d\nu}{\nu} = -\frac{g\,dy}{c^2} $$ An alternative formulation works with wavelength $\,\lambda\,$ instead of frequency $\,\nu\,$. $$ \nu = \frac{c}{\lambda} \hieruit d\nu = -\frac{c}{\lambda^2}d\lambda \hieruit \frac{d\nu}{\nu} = -\frac{d\lambda}{\lambda} \hieruit \frac{d\lambda}{\lambda} = \frac{g\,dy}{c^2} $$

So it seems that we have finished already the tests as originally proposed by Einstein.
From Relativity Reexamined at page 54-55 we quote: The experimental " proofs " of general relativity were:
1. Deflection of light rays passing near the sun, observed during eclipses. These were very inaccurate experiments with individual errors of 100% and averaged errors of 30%. The theory is not safe because it assumes an ideal vacuum near the sun's surface, while we can observe very powerful explosions of matter and radiation from the sun.
2. The rotation of Mercury's perihelion. An apparently good check was proven largely accidental by Dicke (1967).
3. The redshift of spectral lines in a gravitational field. The Pound experiments brilliantly prove the result with 1% accuracy, but a very simple reasoning, using the mass $\,h\nu/c^2\,$ of a photon $\,h\nu\,$, is enough to make the prediction.
Tests of general relativity in modern times shall not be considered further. The reason is that they are often done the other way around. Ipse est: since we know that Einstein's theory is infallible, it may be concluded that it explains the phenomena we observe. In this way, General Relativity has become sort of a self-fulfilling prophecy. Furthermore the progress of image technology is such that advanced observation can hardly be distinguished from advanced animation. As a conclusion: There is no experimental check to support the very heavy mathematical structure of Einstein.
And there are many other indications that General Relativity is an over the top theory, built on quicksand, as found for example in Gravity and Antigravity by David Pratt.