Einstein's Equations vs Precession of Mercury

When Einstein in 1916 presented a new model of gravitation in the form of  Einstein's equations, he desperately needed evidence that his model was better than Newton's. He found this in a back-of-the-envelope computation adding precisely the missing 43 arcseconds in the extra precession of the perihelion of Mercury (slight perturbation of the rotation of the elliptical orbit of Mercury of 5600 arcseconds per century from other planets), to make the prediction by Newton's model of 531 arcseconds/century made by Le Verrier in 1859 to fit exactly with the observed 574 arcseconds/century. 

Einstein thus started from the known results of 531 by Newton and observed 574 and miraculously obtained exactly the missing 43 by a very simple computation which he claimed captured the difference between his and Newton's model.  

If we ask chatGPT about this apparent miracle we get the following information:

1. Le Verrier's 531 has been confirmed to within 1 arcsecond by modern high tech computation and precise data. This is in itself a miracle. No improvement since 1859!
2. It is impossible to directly compute the observed 574 by solving Einstein's equations including in particular the other planets, because the equations are impossible to solve both analytically and computationally. 
3. Computation always start from 531 with Newton's equations and 571 is obtained as a correction of 43 claimed to be based on Einstein's equations, even if they are impossible to solve. This is also a miracle.

To get perspective on these miracles as prime evidence that Einstein's equations are more precise than Newton's, let us note 

• 1 arcsecond/century corresponds to a relative accuracy of about $2\times 10^{-9}$ per revolution of Mercury (415 revolutions/century) if errors add up linearly. 
• Data errors cannot be expected to cancel and so may well add up linearly. 
• A precision of $2\times 10^{-9}$ is thus needed in data such as planet masses, gravitational constant G, initial data, composition of the Sun, tidal motions, other celestial objects than planets, to get the precession right to say 1 arcsecond over 100 year. 
• The gravitational constant is known to at best 6 decimal places, other data with less. 

The conclusion that the desired precision cannot be reached, cannot be avoided. There is factor of 1000 between required and available precision. Yet, the computation of the precession of the perihelion of Mercury by Einstein is still presented as prime evidence that Einstein's equations describe gravitation better than Newton and thus as an undeniable major victory of modern physics over classical physics. 

Check if this is confirmed by chatGPT after careful reading of the literature!

But at his deathbed Einstein begged: "Newton, forgive me!"

Perihelion Precession of Mercury vs Modern Physics vs Pataphysics

Science of Imaginary Solutions: Pataphysics

Einstein's General Theory of Relativity GR (1915) is viewed to be a crown jewel of modern physics replacing classical concepts of space, time and motion under gravitational force expressed in Newtonian mechanics, by an entirely new geometric world of "curved space-time" without gravitational force. 

Newton's mechanics fostered the scientific revolution in the 18th century, while GR opened to the revolution of modern physics of the 20th century. 

At least, this is what (most) modern physicists tell us: Newton's world of mechanics has to be replaced by Einstein's GR world of geometry. More precisely, Newton's mechanics has to be replaced by GR only for extreme speeds or gravitational force/curvature, while GR and Newton agree in most cases. 

The acceptance of GR has grown only slowly over the 20th century, since evidence of superiority of GR over Newton has shown to be evasive, as expressed by fact that the first Nobel Prize directly connected to GR was given only in 2020 to Roger Penrose:

•  for the discovery that black hole formation is a robust prediction of the general theory of relativity. 

The Prize is thus given to the "discovery that GR predicts" the existence black holes, which however cannot be verified. Is that evidence that GR is correct? That GR gives a prediction, the correctness of which cannot be tested? So the Prize in Physics has been awarded to the discovery of an aspect of GR as mathematical fiction regardless of any actual real truth value of GR.  It is like discovering that a certain mythological tradition admits the existence of Unicorns, regardless of the existence of any real ones. This looks like a Nobel Prize in Pataphysics as a branch of philosophy or science that examines imaginary phenomena that exist in a world beyond metaphysics. 

The first evidence of GR was presented by Einstein in a computation using GR to correct a Newton prediction of the precession of the perihelion of Mercury (very slow rotation of the elliptic orbit around the Sun) to exactly fit with observation. But the Newton prediction was made without a computer and so could not account for the full complexity of the problem involving all other planets and unknown inner motion of the Sun and more. 

So it is not clear that Newton fails as concerns Mercury. An example of the correction brought by viewing Sun-Mercury as a true two-body problem still within Newtonian mechanics, with Mercury influencing the motion of the Sun, instead of a one-body problem with fixed Sun, is given in

• The secret of planets’ perihelion between Newton and Einstein by Christian Corda. 

This shows that the correction captured by GR can also be captured by Newton. This is not surprising since the orbit of Mercury is not extreme at all, and so Newton and GR should agree. 

If then Mercury can be taken off the list of evidence of superiority of GR, what remains are extreme cases, so extreme that not even GR can be expected to work, such as black holes, so extreme that they cannot be observed, or even predicted by GR to be honest?  

Why is it important to normalise modern physics back to Newton's mechanics? Because, Newton's mechanics works very well together with quantum mechanics, where speeds are low and gravitation weak. Hopefully this can take modern physics out of its permanent crisis since 100 years caused by an unresolvable conflict between Einstein's mechanics and quantum mechanics: From pataphysics to real physics! In particular, quantum mechanics can be relieved of relativistic mechanics since speeds are low. 

PS1 The Nobel Prize to Penrose/GR is more precisely motivated as follows:

• A black hole is a supermassive compact object with a gravitational force so large that nothing, not even light, can escape from it. 
• In 1964, Roger Penrose proposed critical mathematical tools to describe black holes. 
• He showed that Einstein’s general theory of relativity means the formation of black holes must be seen as a natural process in the development of the universe. 
• He was also able to describe black holes in detail: at their farthest depths is a singularity where all known laws of nature dissolve.

Every word here triggers questions: Supermassive? Nothing can escape? Proposed? Critical mathematical tools? Must be seen? Natural process? In detail? Farthest depths? Singularity? All known laws of nature dissolve? 

Compare with Ethan Siegel: 

• This Is Why Scientists Will Never Exactly Solve General Relativity

Returning to Newton could offer a great relief from a 100 year spell. 

It is illuminating to inspect the picture presented by the Nobel Committee in its description of the scientific work of Penrose as concerns the nature of the interior of a black hole: 

Sometimes a picture tells more than 1000 words...

PS2 GR was initially met with deepest skepticism and was counted down and out by the 1950s. Then miraculously GR was revived in the 1960s and on, until the great triumph of detection in 2015 of utterly faint gravitational waves emitted by the most violent event thinkable in the from of collision of two black holes 1.3 billion years ago. Credible?  How much of the present crisis of physics can be blamed on GR?