http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.261.5640&rep=rep1&type=pdf

this is the reference

Dear Stefano,

As long as a theory is not confronted with the binary pulsar data I am not interested.

Have you seen the ingredible achievement: "Detection of the gravitational redshift in the orbit of the star S2 near the galactic centre massive black hole", arXiv: 1807.09409 ? The perihelion motion of S2 will also be obtained within about two years.

Dear Norbert,

thanks a lot for your link.

That paper confirms at a galactic scales around the MBH what is already predicted by GR in our solar system. in term of frequency shifts and precessions...

I'm not very comfortable with this sentence:

"consists in equal terms of the special relativistic transverse Doppler effect (B1,tD) and the general relativistic gravitational redshift "

both are GR effects predicted by the Schwarzschild solution and already tested in GPS systems, to break it down sounds like GR does not have the capability to model the effects of motion on clocks (it is very likely the opposite).

Do you think that the values of the Perihelion will be accurate enough to be able to test higher orders than the ones we currently get from Mercury?

Best regards

Dear Stefano,

The problem is that you play with different perturbation orders. It is easy to see that Gerber's retarded potentials can be included into a relativistic approach which must coincide with another relativistic one as General Relativity. The problem for me is to justify retarded potentials as the Gerber's ones because they implies a non conservation of energy-momentum associated to the gravitational mass.

Dear Thierry,

Perhaps I don`t understand you. Do you mean that the gravitational field is not connected with mass?

The Lorentz gauge is one of the possible ones when you want to obtain a wave equation easily, but the gauge choice conditions cannot change the physics of the phenomena. What are you speaking about?

Dear Daniel,

this is remarkable

Dear Stefano,

What I want to say is that the general formalism is the General Relativity and the Gerber's potentials are approaches. They are quite close if you assume that the gravitation interaction is propagating at the velocity of light c and the potentials needs therefore a certain time to act on a given point of the space. The difference is the the curvature which in the case of Mercury with respect to Sun is very small. This is for me the reason that this approach can work. For instance, in a black hole the Gerber's potentials are not enough and the above approach cannot coincide with the one of General Relativity in such singularity.

Dear Daniel,

you are right in that regard, it is always the case to check till what term of the series expansion they match, and it is quite unlikely that an agreement for strong fields.

The Gerber retarded potential has been thought for a century as an unjustified guess of the author to arrive to calculate the 43" arc.

It is instead demonstrated to be a first approximation of an actual well formed exchanged wave of potential.

The Gerber retarded potential N(t)/[(1-beta)*(1-beta)] can be seen as a first order Doppler effect of a moving source in acoustic applied twice.

Dear Norbert Straumann

I think you know this paper...

General relativity verified by a triple-star system

Einstein’s theory of gravity — the general theory of relativity — is based on the principle that all objects accelerate identically in an external gravitational field. A triple-star system provides a stringent test of this principle.

https://www.nature.com/articles/d41586-018-05549-4

it actually tests the WEP, not the EEP or SEP , it is a strong confirmation of the WEP..

You should explain me why EEP or SEP since it does not make sense to test LLI while free falling with the star...

Dear Stefano,

You are asking an important and really interesting question. In most gravity theories outside general relativity, extended bodies with self-gravitational energy do not move on geodesics. This was first noticed by Dicke in 1964, and independently by Nordtveth in 1968, and is called the "Northveth effect". It was also discussed in detail by Cliff Will in (1971).

I suggest that you first read in Misner-Thorne- Wheeler the short Section 40.9. I learned the details about the effect (the violation of the strong equivalence principle) when Weinbergs GR book appeared in 1972. There the detailed derivation is given for the Brans-Dicke theory on p. 244--248.

Dear Norbert,

" outside general relativity, extended bodies with self-gravitational energy do not move on geodesics. "

I thought that a geodesic was the ideal trajectory suitable only for a "point mass", so that the tidal forces due to the extension of the body would be non-detectable with no influence on the testbody itself.

Since tidal forces are present in GR for a pure gravitational interaction, I wonder how is it possible that they do not play any role in the geodesic of a massive body..

Should then be the self gravitational energy of the free falling body to cancel out the tidal effects of gravitational field?

I will have a look at the literature you suggested about the Nordtvedt effect... Thanks a lot

Yes basically the test of the WEP for large bodies confirms such effect on large scales... so it reinforces the validity of the UFF.

Dear Thierry,

As you know very well electrodynamics, you know that the Lienard-Wiechert potentials are equivalent to radiation, i.e. there is a contribution of the energy independent of the system which doesn't conserve locally for the masses in motion under such potentials.

Dear Thierry,

"Since Einstein INVENTED self-gravity "

Einstein did not invent self gravity, since self gravity is the gravitation which the free falling mass has, being itself a mass.

That is not a detail and makes the other theories crash...

We are in front of the effect such that a tiny free falling object falls in the same way as a big mass which has its own non negligible gravitation...

A small pebble and a neutron star fall in the same way in a black hole...

Dear Thierry,

self-gravitational energy means the energy which is needed to take the components of the massive body to infinite distance against the gravitational pull.

So the "falling in itself" is not so far from reality.

Basically a star is bound gravitationally in a non-negligible way.

While the electrical fields/energies bound in the same way in different materials, as shown in the Eotvos experiment, it was not clear if gravitation itself bounded in the same way as EM energy does in all bodies.

In other words it was not clear if gravitational energy followed the WEP.

As Prof. Norbert Strauman pointed out, only Nordtvedt showed in the 80's that GR has this property, unlike other gravitational theories.

Dear Stefano,

The two equations in MTW on p.1128 give the acceleration of a massive body at rest in a uniform external (weak) field. As you can see, the complicated expression (40.55) for a large class of alternative metric theories reduces to the Kronecker symbol for GR. For instance, the Brans-Dicky theory (which, by the way, was published years before the two authors by Markus Fierz, the successor of Pauli at ETH) contains an additional long range scalar field that couples indirectly to massive gravitating bodies. In Weinberg's book you ind the detailed derivation.

Stefano> http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.261.5640&rep=rep1&type=pdf

Among more sensible statements, the paper also says

Ginè> "Thus we ought to regard an elementary interaction not as a one-way exchange, but as a two-way round-trip exchange. Hence, we must apply the corrective factor twice in the initial potential."

So, gravity in this model is caused by Small Green Men flying around transmitting "potential" to each other???

This looks like an absurd modification of the gravity theory by Duillier and Le Sage, https://en.wikipedia.org/wiki/Le_Sage%27s_theory_of_gravitation, independently rediscovered by my friend Jørgen Karlsen some years ago, http://www.principlephysics.com/The-Birth-of-a-Theory.html, and apparently many others. The latter mechanical theory actually makes some sense to me, since it (more-than-less) is the mechanism behind a real physical phenomenon, the Pointing-Robertson effect: https://en.wikipedia.org/wiki/Poynting–Robertson_effect.

However, the most important feature of the Pointing-Robertson effect also point to the real problem with all time-delay models: that they fail to reproduce the radial directed force required for conservation of angular momentum. It makes little sense to study pure radial motion in these models, with ridiculous hypotheses (probably) to get factors of two in the perihel motion of Mercury right, if the important prediction is that Mercury will spin into the Sun long before any perihel motion becomes noticeable, due to a friction force against its direction of motion.

It is interesting how Maxwell electromagnetism and Einstein GR gravity theories manage to maintain a radial force to sufficient accuracy (only modified by electromagnetic or gravitational radiation as long as quantum mechanics can be ignored), despite the fact that these forces are "transmitted by photons and gravitons" respectively; the latter is a somewhat poetic interpretation of the actual mathematical mechanisms.

Dear Kare,

Physics deals with interactions implemented with exchange of entities (bosons and gravitons), so I would not be so surprised of the two-way mode.

The two-way is not where GINE starts from but it is the point of arrival by applying the retarded potential as a functional and the slight path difference. He does not implement a double retarded potential although what comes out is a sort of double Doppler effect of the potential.

This is also the point raside by Prof. Baldomir about the momenta conservation.

Are you sure that such application of the retarded potentials violates the conservation laws?

Because that is not implemented in the same way as the EM field retarded potential.

Dear Norbert,

"Crucial to solar-system experiments is the question of whether the sun and the planets move on geodesics of spacetime. This question is complicated by the contributions to the spacetime curvature made by the moving body itself."

So basically the Brans-Dicke theory is falisfied by the Nordvedt effect...

The Grav. energy couple with itself in the same way as it couples with any other form of mass-energy.

I don't understand why the Grav energy should be negative and the EM energy of an atom with +and- charge, defined as positive...

Dear Thierry,

All the potentials must follow a gauge condition because they are not physical observables without it.

Dear Norbert Straumann ,

in the Will's equation above we have the equation of motion of the COM of an extended free falling object A, in the field of a celstial body G.

Forgive my ignorance but the above big formula (in the previous post) considers a geodesic in the GR space-time, whose curvature gets created only by G (correct me if I'm wrong), not the equation of a geodesic in the total field generated by A and G together.

Since it can be arbitrary to establish between massive bodies who is the actual object free falling, for example for a binary pulsar both are "free falling", what should be considered is the global field = actual structure of the space time, not each self field.

In other words, the self energy of A is involved but we have also the external gravitational field of A, which can free fall but at the same time alters G in a non negligible way.

Stefano> "Physics deals with interactions implemented with exchange of entities (bosons and gravitons)"

No, this is not how the fundamental physical models are implemented; this a poetic interpretation of the implementation, complementary to the precise set of mathematical rules used to derive real physical predictions. The equations are not derived from the exchange picture; it is the other way around.

Stefano> "Are you sure that such application of the retarded potentials violates the conservation laws?"

It depends on how it is implemented. If one starts from a Lagrangian (with the right symmetries) and an action principle to derive equations of motion, the wanted conservation laws will automatically follow from the Noether theorem. But contribution from radiation (often negligible) must be included in the accounting. However, if one just begin to directly modify the equations of motion in more-or-less ad hoc manners, one will almost certainly end up with violations.

Dear Kare,

I agree that the relation does have to respect the conservation laws and somebody has to demonstrate that it does.

At least the legend according to which Gerber's formula is not derivable from logical steps, is not true. It is instead derivable, as a first approximation, from an actual functional taking care of the fact that any "perturbation" is supposed to travel at the speed of light in vacuo so there is a propagation delay.

I guess that now we have more possibility to check the higher terms of the "rosetta law" of motion and check if the "exchanged potentials" is singled out.

yes..since the adherence to reality requested by the conservation laws represents quite a tight boundary which usually singles out many theories..

Dear Stefano,

If you want to understand more about the theoretical aspects, you have to look at more recent papers on the equations of motion for systems of bodies with self- gravity, like binary pulsars consisting of neutron stars. In the Natur- paper you find the relevant references.

As am example, I recommend the article by T. Damour et al. on the second post-Newtonian order of tensor-(multi-)scalar theories, Physics. Rev. D53, 5541(1996).

(At the time when Thibault D. was working on this I was an invited guest in Bures.)

Dear all,

does somebody know about this seems promising theory of gravitation?

https://phys.org/news/2018-12-gravity-mathematically-dynamics-subatomic-particles.html

Dear all,

Perihelion precession in binary systems: higher order corrections

https://arxiv.org/pdf/1802.03333.pdf

Newton adds a cubic force to his inverse square law of get precession, and Newton's apsidal angle theorem was used in 1895 to obtain a slight modification of the exponent in his inverse square law. Rather, Einstein's calculation amounts to the introduction the speed of light into Kepler's third law so that the rest mass energy is balanced by the gravitational energy, as I show in my book "Seeing Gravity". The perihelion of Mercury can hardly be considered in the same way that an event horizon is to a black hole. The numerical coincidence must be attributed solely to luck.

Up until Einstein, the exercise amounted to explaining the perihelion solely in terms of parameters that described the elliptic orbit. What does the speed of light have to do with the orbit? All books on general relativity chalk up the missing 43 seconds to "relativity." Relativistic phenomena don't enter at all. Why is Mercury singled out when his formula should apply to all the planets that both prograde and retrograde?

Dear Bernard,

the reference to the retarded potential of the Spanish scientist is the following.

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.261.5640&rep=rep1&type=pdf

Dear Stefano,

Glancing through the article, it just seems like a rehash of the derivation of Einstein's equation for a modified ellipse. Although he discusses Gerber, he leaves out all of the work done on Newton's apsidal theorem, and the modification of his inverse square law. I don't see any explanation, other than a finite propagation speed. But what is propagating at a speed less than light?

Have I missed something?

Dear Bernard,

in that paper the retarded potential are applied correctly and give the Gerber's formula as a first approx, so the Gerber's formula is something actually correctly derived from the RP...

RP are applied in the same way the electromagnetic potentials are applied...I agree with you that it is difficult to understand what is actually propagating...

Dear Stefano,

GR does not deal with potentials, much less with time retardation. That would be something for PPN.

I don't dispute the correctness of Gerber's result since his equation corresponds to the Weber force, which was applied by Tisserand and others to gravitation beginning in the 1870's. I only dispute Kevin Bacon's claim that since Gerber's deflection of light does not give the presently accepted value that would also invalidate Gerber's calculation of the perihelion shift whose first order approximation does. According to several authors, our knowledge of the epheremides hasn't progressed much beyond Le Verrier's time.

Dear Bernard,

the Schwarzschild solution contains the potentials and has been tested several times...

Dear Stefano,

That's the point: The Schwarzschild solution is obtained by solving Einstein's condition of an "empty" universe (all the components of the Ricci tensor must vanish). Empty meaning that there is nothing except gravity which does not disturb the emptiness (quote from Dirac's book). If there is a central mass (a mere constant of integration based on some asymptotic condition that Newtonian mechanics must apply) than how can it be empty?

Also the energy-stress tensor vanishes, which contains all energy and momenta except gravitational energy. Yet, the 00 component of Einstein's equation is used to derive the critical mass density of the universe which must (and does) take into account gravitational energy. There is no gravitational energy, and, yet, there is! This is a completely illogical concoction of contradicting statements.

Dear Bernard,

Your phrase:

That's the point: The Schwarzschild solution is obtained by solving Einstein's condition of an "empty" universe (all the components of the Ricci tensor must vanish).

Is not obviously true. The elements of the main diagonal are not zero for the Ricci tensor components.

Dear Stefano,

For the Schwarzschild metric, the Ricci tensor is diagonal, and it is only necessary to compute diagonal terms, which are sums of sectional curvatures. According to Einstein's condition of emptiness, each of these terms must vanish. See: Besse, Einstein Manifolds, p. 102ff

Dear all,

it seems that the speed of light cannot be constant at least in gravitation.

Preprint SHAPIRO TIME-DELAY, Curved 4D space-time or Variable speed of light

Stefano,

You do not understand the most elementary things about GR. Light propagates along null geodesics. With this the Shapiro delay is computed, if done correctly. You should not confuse this with coordinate velocities which enter in the computation as intermediate tools.

This propagation law is also the basis of lensing theory. You should not think that generations of gifted scientists around the globe do such quite simple things wrong.

Dear Norbert,

In GR in fact, which is just one model of gravitation although undeniably powerful. GR works fine to a certain extent and that is exactly the extent to which it works in providing credible explanations.

The diffeormorfism of the space-time is instead the "intermediate tool", which works fine as a calculations tool but is devoid of any ontology, it provides numerically correct results of certain quantities, that's all.

The experiment is quite straightforward, GR manages to give account of the delay very well but at the end of day, realistically, it is a bouncing of a signal which did not go at the speed of light in vacuo. The only reasonable explanation is that its speed was slowed down by gravitation along the path, no room for a time-dilation the clock is just one...

It is enough to apply the formula provided by Dicke about VSL to explain the phenomenon...I really don't think Dicke did not understand well gravitation.

Stefano,

Most of what you write is simply wrong. I could present to you a derivation of the Shapiro time delay which does not use coordinates, only the null geodesics of the gravitational field (a geometric concept). The final formula involves only invariant terms determined by the metric and its Riemann tensor. This is conceptually closely related to the derivation of the equation for geodesic deviation (the Jacobi equation) in my book, Sect.3.1. The problem is purely geometric, a mathematicians could solve it without knowing anything about GR. The field equations enter only in a concrete situation when the metric has to be known with sufficient accuracy (for instance for the planetary system).

You would not understand my derivation, because I use advanced differential geometry, like in the cited section (you may have a look at). But with some effort you may be able to learn it (but that takes time).

Dear Norbert,

that should be the original derivation from General Relativity which gave the results of Shapiro, the reason why the experiment was considered the fourth test of GR.

As you said, it is based on the reasoning of the light deflection, measuring also the time taken by light across the gravitational field (of the SUN).

it is indisputable though, that from a purely geometrical view in 3D, the actual length of the trajectory of light is totally insufficient to explain the delay.

For a geometrical viewpoint in 4D (space-time) you are right but that is only a useful tool of calculation that bends a hypersurface to mimic what actually occurs in 3D. It is also indisputable that GR computes correctly the quantities necessary to calculate the delay, but at the end of the day, the ontology is lost. Considering the space-time and its diffeomorphism to account for gravitational phenomena in time and space, is not very different than taking a Laplace to transform to make calculations for linear systems in complex variables and then get the right results in the time variable... it works but there is no ontology behind it...

Stefano,

My derivation is entirely in four dimensions; I never use a spacetime splitting. You cannot criticise something you do not know. Why do you not look at my book? Is it not available in your library or too expensive for you. Stefan could send you the pages of Sect, 3.1; he also knows the content.

Dear Norbert,

thanks I will have a look.

Dear Norbert,

in section 3.1 of your book about GR and astrophysics, I read the derivation of the Schwartzschild solution and the Birkoff theorem. Although these are certainly related to the derivation of the Shapiro Delay, I cannot see what you mentioned.

It is possible to get almost the same results as GRT with a VSL

Article A spatially-VSL gravity model with 1-PN limit of GRT

the possibility of such a model is allowed by

Preprint SHAPIRO TIME-DELAY, Curved 4D space-time or Variable speed of light

Dear Stefano,

I suppose that 'more scientific' derivation of the gravitational potential is given by Jefimenko in his book 'Gravitation and cogravitaton'.

Jefimenko uses the only condition, the finiteness of gravity, to derive the gravitational potential from Newtonian one, i.e. in (v/c)^0 order in post-Newtonian approximation.

Jefimenko gives the estimate of the Mercury perihelion shift as 1/3 of the observed data - due to the effect of retardation.

Dear Vladimir,

to my understanding what is written here is that he did not match the observed data with his model.

Vladimir Onoochin : I was notified by RG, of your comment in this long-forgotten forum. Whatever is its worth or relevance to your comment, I posted bellow some passages of a recent publication on this issue:

[Mercury’s perihelion advance:

Long after Newton’s law of universal gravitation was generally accepted as the basis of the planetary motion in the solar system, the French astronomer Le Verrier in 1859announced (based on many years of careful observations and calculations) that the perihelion of the planet Mercury evidently undergoes precession, at a slightly faster rate than can presumably be accounted for by Newtonian mechanics, given the known distribution of the planets and the other objects in the solar system. Since Newton’s laws allows only a unitary and centrally directed force, an explanation for Le Verrier’s finding was arbitrarily attributed to a perturbation caused by the secondary effect of the gravitational forces from other planets; but did not explain why the precession rate varied with different planets, especially the high precession rate of Mercury. The perihelion precession of Mercury is 5,600 arcseconds (1.5556°) per century relative to the Earth. Newtonian mechanics, presumably taking into account all the effects from the other planets, predicts (without strong basis, since it is a very complex many-body problem) a precession of 5,557 arcseconds (1.5436°) per century, but the rest 43 arcseconds (a mere 0.076%!) remained un-explained.

In 1898 a German school teacher named Paul Gerber [15]wrote a paper in which he proposed a velocity-dependent propagation of gravity that predicted non-Newtonian 43 arcseconds advance of orbital perihelia per revolution given by the expression k πM/(Lc2); where c is the posited speed of propagation of gravity, M is the sun's mass, L is the semi-latus-rectum of the orbit, and k is a constant depending on the precise form of the assumed potential. Although there are controversies about this formulation, Gerber showed successfully that a value of the constant k = 6 gave the correct additional 43 arcseconds of Mercury’s perihelion advance.

Albert Einstein in his 1915 - 1916 publications [16,17] claimed to have given a precise account of the discrepancy of precession of Mercury, and deduced Garber’s above relation based on his newly proposed theory of general relativity (GR). The understanding that Einstein’s GR based on impeccable mathematics and a totally different and novel perspective of space and time gave an exact account of the expression given by Garber for the non-Newtonian precession of Mercury; became an instant cause of celebration by Einstein himself. This purported achievement by Einstein is considered [18] a poetic marvel in modern physics and the strongest “proof” of GR. Einstein retroactively admitted (after questions were raised) that Gerber obtained the correct expression for Mercury’s perihelion advance before him. But Einstein dismissed any possibility that someone could explain the additional 43arcseconds precession of Mercury or that Gerber’s expression for it could be obtained, without using his theory of general relativity. According to Einstein [19], “Mr. Gehrcke wants to make us believe that the Perihelion shift of mercury can be explained without the theory of relativity. So there are two possibilities. Either you invent special interplanetary masses. [...] Or you rely on a work by Gerber, who already gave the right formula for the Perihelion shift of mercury before me. The experts are not only in agreement that Gerber’s derivation is wrong through and through, but the formula cannot be obtained as a consequence of the main assumption made by Gerber. Mr. Gerber’s work is therefore completely useless, an unsuccessful and erroneous theoretical attempt. I maintain that the theory of general relativity has provided the first real explanation of the perihelion motion of Mercury. I have not mentioned the work by Gerber originally, because I did not know it when I wrote my work on the perihelion motion of Mercury; even if I had been aware of it, I would not have had any reason to mention it”.

Einstein’s claim of Mercury’s perihelion advance was contested by so-called “100 authors against Einstein” led by Philipp Lenard, Gehrcke, Silverstein, Reuterdahl et all, who opposed Einstein’s theories accusing him of many things, such as incorrect formulation, plagiarism etc., and questioned his priority. Ironically, W.W. Engelhardt in his recent publication [20] uses similar reasoning like Einstein and raised the controversial issue that Gerber’s expression cannot be derived from Einstein’s GR. Engelhardt claimed that instead of GR, Einstein used a modified Newtonian theory to derive Gerber’s expression and not from his theory of general relativity; hence contest the justification of the claim by Einstein of the proof of his theory of relativity. Engelhardt using purported Einstein’s modified Newtonian mechanics derived an expression for the Gerber’s equation, which is one-third of Gerber’s formula.]

(16) (PDF) Quō Vādis Theoretical Physics and Cosmology? From Newton's Metaphysics to Einstein's Theology!. Available from: https://www.researchgate.net/publication/361442433_Quo_Vadis_Theoretical_Physics_and_Cosmology_From_Newton's_Metaphysics_to_Einstein's_Theology[accessed Jul 18 2023].

Dear Stefano,

Jefimenko did not have the aim to obtain the corrspondence of his result with the observed data. He wrote inn his book that the effect of retardation gives 1/3 of the observed perielion shift.

The observed value is too small to measure one-year perihelion shift. If it would be possible one can separate the effect of curvature of the space, the retardation effect etc. because the conditions for the Mercury are different for every Mercury's year (positions of other planets).

So Jefimenko only made a suggestion that not all factors are considered in computation of this shift. For example, in Einstein's model the effect of finiteness of speed of gravity is omitted.

The other factor as dependence of the Mercury's mass on its velocity (considered by some authors, including Wolfgang Engelhardt) is omitted too.

I wish to say that no one model of the Mercury motion takes into account all factors.

Dear Vladimir Onoochin ,

The actual measurements made by the observations of Le Verrier (ROSETTA TRAJECTORY) were divided into higher-order effects of Mercury / Sun behaviour (as predicted by GR) and the effects of the other planets on Mercury.

To my undestanding Jefimenko claimed that only 1/3 of the effect predicted by GR can be attributed to higher-order effects of the two-body SUN-mercury, while the rest should be attributed to the other far away planets.

Jefimenko, as I mentioned, might have calculated in the wrong way the effects of retardation of gravitation violating angular momenta conservation attributing to the sun/mercury interaction a lower value of the angular momentum included in the rosetta's trajectory.

Dear Stefano Quattrini ,

As I can find, the most precise results of calculations of a value of the perihelion advance are given in Park, Ryan S. et al. , "Precession of Mercury's Perihelion from Ranging to the MESSENGER Spacecraft." The Astronomical Journal 153.3 (2017)p. 121.

According to the presented data, the most account to the value is given from the planets (Jupiter, Venus). But the accuracy (possible errors) of these data is so high that it can create some doubts. As I know, calculation of three-body system (the Sun, the Mercury, the planet)can be made only numerically. The accuracy of these calculations are mostly determined by the correctness of the model, or of the system of equations of motion. Therefore, the scientists who have doubts in GR state that in such calculations not all factors are taken into account (paper of Narlikar and Rana in Monthly Notes of Royal Astr.Soc. v. 213, 657 (1985)), where the authors estimate the resonances of Mercury and Venus rotations.

I wish to say that since we cannot verify calculations of both sides, it would be good to consider only theories but not the correspondences of theoretical predictions to the calculated data.

Jefimenko compared his calculations and Einstein's calculations, or the final formulas of both theories. He obtained that his results - only due to the retardation and without curvature of the space is 1/3 of Einstein's result.

It is a great tragedy for physics and indeed for humanity in general since the discovery of the King of uncertainty i.e., “The Evil Quanta”, that spurious and highly uncertain “experimental measurements” of very small, (0.076% in the case of Mercury’s perihelion advance) or very large (3x10^8 m/s velocity of light), values of a parameter were used with impunity to formulate esoteric theories (GR, SR respectively) as “scientific tools”; to combat quantum uncertainty! These esoteric theories in turn are then used as the most certain truths of the universe, as well as the ruling ideas of centuries-long of class rule!

In the context of this discussion, the following article may be of interest:

C.J. de Matos, M. Tajmar: Advance of Mercury Perihelion Explained by Cogravity - Version II (arXiv:gr-qc/0304104v1)

Dear Vladimir Onoochin ,

nevertheless the rosetta trajectory observed for the stars orbiting around Sagittarius A (BH of Milky-way), is fully accounted by the formula provided by GR.

Article Observation of S4716—a Star with a 4 yr Orbit around Sgr A*

there all the stars orbiting which have a remarkable rosetta trajectory due to the high eccentrically orbits, have been carefully found to follow the full predicted shift by GR.

Dear Stefano Quattrini ,

First, let us separate computation of the Mercury perihelion advance from the relativistic predictions. Why?

Since in the GR even two-body problem (when two bodies of masses M_a and M_b, and M_a is of the same order as M_b) cannot be solved. If you know who and where solved this task, please, inform me.

In the solar system the mass of the Sun is much greater the total mass of all planets. It is like a motion of the Mercury around the fixed center. It is not the two-body system.

Second. Most of the perihelion shift is due to interaction of the Mercury with other planets. The values of these shifts, caused by each separate planet, had been calculated still in XIX century. The question is how it was done?

Actually, mutual motion of the Mercury and the Venus is like a problem of three-body system with the fixed center. But even this problem has not been solved. As I remember (I can mistake) Laplace developed the perturbation method to calculate influence of for example of Jupiter into the motion of the other planets.

Yes, this method is good. But it is very sensitive to resonances (when a ratio of the periods of rotations of two planets is rational number). Then the method gives big errors - still Laplace noted it. I am not sure that Park and co-authors estimated possible errors from such resonances.

Third. When we read that the values of the perihelion shift due to every planet has been calculated with the highest accuracy and when the authors do not describe what method of calculation of these values they used it is reasonable to have some doubts in their data.

Yes, they give a reference . But it is like a software (some code). This software has been written using some equations. What equations? We cannot know it.

I suppose it would be more useful to compare the GR approach to the problem of rotation of some body around a fixed attractive center with the approach of Gerber and the approach of Jefimenko (gravitation-cogravitation).

“…the rosetta trajectory (?) observed for the stars orbiting around Sagittarius A (BH (?)of Milky-way), is fully accounted by the formula provided by GR….have been carefully found to follow the full predicted shift by GR(?)".

Only two things are missing from this narrative: 1) Einstein, in a publication in Annals of Mathematics, dismissed the possibility of “black hole” formation even in theory! The following is a copy of his conclusion at the end of this mathematically rich and lengthy publication: "The essential result of this investigation is a clear understanding as to why the "Schwarzschild singularities" do not exist in physical reality. Although the theory given here treats only clusters whose particles move along circular paths it does not seem to be subject to reasonable doubt that more general cases will have analogous results. The "Schwarzschild singularity" does not appear for the reason that matter cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the velocity of light.

This investigation arose out of discussions the author conducted with Professor H. P. Robertson and with Drs. V. Bargmann and P. Bergmann on the mathematical and physical significance of the Schwarzschild singularity. The problem quite naturally leads to the question, answered by this paper in the negative, as to whether physical models are capable of exhibiting such a singularity." Albert Einstein. A. Einstein, The Annals of Mathematics, Second Series, Vol. 40, No. 4 (Oct., 1939), pp. 922-936

https://pdfs.semanticscholar.org/8dd0/dfafef0c53c428fdc3b58f8099aafcf7d089.pdf

2) We missed the promised galactic fireworks of a life time on May 19, 2018 predicted through independent analyses by the GRAVITY collaboration (led by Reinhard Genzel) and the KECK/UCLA Galactic Center Group (led by Andrea Ghez); when the star S2 was predicted to be at its pericenter near Sagittarius A* (BH of Milky-way) and would plunge into or be gobbled up by the the Monster; producing a spectacular fireworks visible from the Earth! . But it turned out that the supermassive monster was not hungry at all, and the poor S2 was able to blithely get around the very nose of the monster unscathed, without losing even one hair!

Fortunately, this exciting and highly anticipated firework do not merit even a little mention in preset day Wikipedia! https://en.wikipedia.org/wiki/S2_(star) and Einstein's GR continues its highly precise predictive power!

Fantasy, thy name is Einsteinian physics!