The question of whether the sun revolves around the earth, or vice versa, has nothing to do with the Theory of Relativity. It has everything to do with gravity, which allows only one conclusion. I will discuss relativity below, and begin by explaining the correct answer.
Newton’s Law of Gravity, one of the most famous discoveries in scientific history, is F=Gm1m2/d2. Here m1 and m2 are two masses, d is the distance between them, F is a force which acts on each mass, directed toward the other, and G is a constant. In the case of the earth m1 and the sun m2, the sun experiences a force F pulling toward the earth. From Newton’s law of motion, F=ma, the sun accelerates toward the earth by F/m2. The earth experiences the same force, pulling toward the sun, so it accelerates by F/m1. The sun has a mass of about 1030 kg, about 333,000 times the mass of the earth. That means that F/m1 is 333,000 times larger than F/m2. So the earth’s acceleration toward the sun is far greater than the sun’s toward the earth. The situation is not symmetrical. It is not a matter of opinion, despite Schneerson’s “firm belief.”
To produce a stable orbit requires a precise relative velocity of the two bodies. This is the actual case for the sun and earth, as evidenced by the fact that the earth has not fallen into the sun, which would happen if the earth’s orbital velocity was too small. Nor has the earth flown off into space, which would happen if the earth’s orbital velocity was too large. The earth is in a stable orbit, in accordance with Newton’s laws. Its orbital velocity, which can be measured, is about 30 km/s. https://en.wikipedia.org/wiki/Earth%27s_orbit
In general, it is approximately true that the lighter body orbits the heavier one. The strictly correct answer, which can be worked out mathematically from Newton’s laws, is that the two bodies rotate around their mutual center of gravity, called the barycenter. https://en.wikipedia.org/wiki/Barycenter When one body is much, much heavier than the other, the mutual center of gravity is often somewhere inside the larger body. That is the case with the earth/sun system: the center of gravity is 449 km from the center of the sun, which has a radius of 696,000 km. The verbal simplification, that the light body is the one doing the rotating while the heavy body remains still, is almost exactly correct.
The basic mathematics of planetary motion, along with a significant amount of empirical evidence, emerged from the work of Copernicus, Galileo, Brahe and Kepler by 1630. https://www.physics.unlv.edu/~weistrop/ast104s05/ch4nic.htm Newton’s Laws of motion, including gravitation, were published in 1686. It is true that Newton could not explain why gravitational attraction occurred, but his law nevertheless described how it worked with great accuracy. That provided sufficient insight to prove that the earth, and all the other planets, revolve around the sun and not the reverse. [https://education.seattlepi.com/newton-explain-planetary-motion-6264.html] When Schneerson claimed that relativity theory settled the question of planetary motion, “half a century ago” as he was writing in 1975, his time line was off by about a factor of seven. The issue was actually settled before 1700, long before Einstein was born. Schneerson ignores all the previous developments in the field.
Schneerson’s claims about relativity can be refuted in more than one way, the simplest of which, perhaps, is reducto ad absurdum. If one thinks that relativity justifies a belief that the sun revolves around the earth, then exactly the same argument can be used to justify a belief that the earth revolves around the moon. As Schneerson says, both views “have equal validity.” The same logic leads to a belief that the sun revolves around Halley’s Comet. The earth revolves around the International Space Station, or any other satellite of your choice. An airplane in flight is stationary while the entire earth moves rapidly beneath it, since both plane and earth are equally valid vantage points. The earth conveniently slows down when the pilot wants to land.
These are obvious absurdities that no one believes. Yet Schneerson suggests, in the case of the sun and earth, that it is a matter of “personal choice and belief” how the orbital dynamics actually function. The reducto ad absurdum tells us that something is seriously amiss with his reasoning.
In Einstein’s book “Relativity,” he states carefully what he calls the “principle of relativity.” The word “principle,” as opposed to “theory,” is very important. They are not the same thing. In fact the principle is one of the postulates on which the theory is based. Here are some excerpts from chapter V, page 13 [Crown Publishers, New York 1961].
“If K is a Galilean co-ordinate system, then every other co-ordinate system K’ is a Galilean one, when, in relation to K, it is in a condition of uniform motion of translation.”
In preceding paragraphs, he defined a “Galilean” coordinate system as one whose motion is that of uniform translation, i.e., its motion is in a straight line and it is neither accelerating nor rotating.
“Relative to K’ the mechanical laws of Galileo-Newton hold good exactly as they do with respect to K.” The italics are mine. What is invariant here is the laws of physics, not the verbal descriptions made by observers in two different locations.
“We advance a step farther in our generalization when we express the tenet thus: If, relative to K, K’ is a uniformly moving co-ordinate system devoid of rotation, then natural phenomena run their course with respect to K’ according to exactly the same laws as with respect to K. This statement is called the principle of relativity.”
Again the point is very clear. What the principle asserts is the invariance of the laws that govern the motion of objects. This principle does not offer the slightest justification for saying that the statement “the earth revolves around the sun” is equivalent to the statement that “the sun revolves around the earth.” In fact, just the opposite: the principle indicates that Newton’s Law of Gravity is equally true on the earth as it is on the sun. The first statement – the earth revolves around the sun - is consistent with Newton’s Law of Gravitation. The second statement is not.
Before leaving the subject of relativity, I would like to mention the next paragraph, which begins as follows:
“As long as one was convinced that all natural phenomena were capable of representation with the help of classical mechanics, there was no need to doubt the validity of the principle of relativity.” Again, italics mine. Einstein is saying here that the principle of relativity was accepted by physics long before his own time. He did not invent it; not even close. It was already a principle of classical mechanics, at least as old as Newton.
The nineteenth century saw the development of electrodynamics and optics, and the deduction by James Clerk Maxwell that the vacuum speed of light was a constant. This, too, was established science before Einstein. But the two principles appeared to contradict each other; as he says, in that same paragraph, “it did not appear impossible” that the principle of relativity was, in fact, wrong. It was Einstein who removed this doubt. The Theory of Relativity explains what sort of universe permits both the principle of relativity and a constant speed of light. The resulting picture, explained by Einstein with compelling logic, is the one unversally held by physicists today.
None of that has anything to do with the motions of the planets. To be very precise, relativity makes a small correction to Newton’s laws of motion. For objects that move much slower than the speed of light, such as planets, this correction is insignificant.
I will acknowledge one truth about Schneerson’s claims: if the only thing you know about physics is the principle of relativity, it would be quite difficult to understand how the solar system works. It would be especially true if the only visible celestial bodies were the earth and the sun, and one did not have observations of the other planets. Galileo, in Dialog Concerning Two World Systems, offers empirical evidence that all the other planets orbit the sun. From this knowledge, even though he was lacking a theory of gravity, he made a strong case in favor of a heliocentric solar system. He reasoned that placing the earth at the center of the solar system, and requiring all the other celestial objects to revolve around it while somehow also revolving around the sun, did not make any geometric sense. He also found evidence that Jupiter had moons of its own, further reinforcing the idea that not everything revolves around the earth. Nevertheless, it was Newton’s Law of Gravity that formed the strongest proof that the solar system revolves around its center of mass, which is of course its heaviest object, the Sun.
All of Schneerson’s analysis in his second letter ignores the law of gravity and all its implications. It also, as I have said, misinterprets Einstein and the principles on which his theories are based. It uses simple sentences as a substitute for accurate scientific reasoning. He considers the sentence “the earth revolves around the sun,” and then uses his cartoonish view of relativity to justify exchanging the words “earth” and “sun.” That’s not how science works, at least not for about the last four hundred years. As I pointed out, continuing such a line of reasoning quickly leads into absurdities.
That Schneerson’s misinterpretations are theologically convenient, given his stated belief in the scientific accuracy of an ancient religious text, should not escape our attention.