Einstein is legendary for his “gedankenexperiments” (thought experiments) ability. However, even the brilliant genius discovered between 1905 and 1908 that his thought experiments alone could not produce a mathematical method of his concepts of gravity. In 1905, Einstein put forth three papers, each a masterful work of science. One in particular, “On the Electrodynamics of Moving Bodies” became the basis for Einstein’s Special Theory of Relativity (SR) . The reason for the word special in SR is derived from the theory requiring inertial frames. This is a special case in which there is no acceleration occurring between two observers. Special Relativity works for many scenarios in physics, but where there is acceleration, it is not permissible. In the author’s opinion, it is often abused, as with the experiment entitled Around-the-World Atomic Clocks: Predicted Relativistic Time Gains . This special case scenario excludes any gravitational field (a lab in the presence of gravity), and since gravity produces acceleration, almost all, if not all, physical locations are excluded from using SR. If you are on Earth reading this, your body is accelerating toward the ground at approximately 9.8 meters per second per second; that is why you don’t “float” away from a chair when you sit it in; you are constantly accelerated toward the ground. Now, imagine a place where this doesn’t exist. That is the limitation of Special Relativity.
Einstein understood this restriction and, shortly after 1905, pursued a general theory of relativity, which included acceleration. In his quest for the theory, he performed many thought experiments, but the one that may be the most notable was his windowless elevator thought experiment , a thought experiment often visited in this book. The year was 1907, and Einstein had imagined a closed elevator. In the elevator is a lab. Within the lab, Einstein conjectured that it is impossible to tell if the elevator was undergoing gravity (the force acting on the lab, pulling everything to the floor) or if the elevator was undergoing upward acceleration, forcing everything to the floor. This became known as the principle of equivalence. General Relativity (GR) was based on this single thought experiment . Einstein often referred to this thought as the “The happiest thought of my life” . Unfortunately for Einstein, this day would be followed by years of struggle to finish his theory.
Though Einstein had the conceptual aspect of GR formed, he struggled with math and how to develop the concept formally. Between 1896 and 1900, Einstein attended the Swiss Federal Polytechnical School in Zurich, later known as the Swiss Federal Institute of Technology . Here, one of Einstein’s math classes was taught by Professor Hermann Minkowski. A class Einstein shared with Marcel Grossmann. Einstein was known for not taking notes and paying attention only to the part of math that showed their physical properties and behaviors instead of the mathematical rules ( a curse this author suffers from also, late in life having returned to college to “undo” much of the teaching previously received to relearn the proper/formal methods). This latter appeared to cripple Einstein’s ability to formally develop the math needed for General Relativity. In his class with Marcel, Marcel allowed Einstein to use his notes before exams. Thus, Einstein had no notes to look back upon nor a full picture of the mathematical structure. Einstein, in a class of seven students, barely made 4th place with a 4.91 average out of 6 . If one does a web search asking if Einstein was an excellent mathematician, they will notice quickly that it is difficult to distinguish between Einstein, the legend, and Einstein, the human. In the author’s opinion, Einstein was a good (not great) mathematician. Mathematical logic did not always flow easily with Einstein as it did with his thought experiments. To under girt this opinion, one only needs to realize Einstein took more time between 1905 and 1912 (before he sought help from a college friend) trying to form the math than one does if one went to college and obtained formal training. Math was not a natural subject for Einstein; history supports this belief.
Marcel was a very organized individual, while Einstein maintained a more fanciful imaginary personality. Amazingly, the two hit it off very well. Both were quick-witted and had a strong sense of humor. After college, Marcel’s father was instrumental in helping Einstein secure his first position out of college as a patent clerk in Berne in 1902 . It was not Einstein’s first choice, but because of his behavior of not taking classes seriously (like math), he was unable to find a position at a college . Fast forward to 1907 when Einstein had his great elevator thought experiment, discussed later, Einstein found himself failing to formalize the theory. By 1909 Einstein had dedicated the entirety of his resources to formalizing his general theory. Unfortunately, Einstein’s lackadaisical treatment of math within his education was about to catch up with him .
In his 1905 paper on SR, Einstein provided a genius-level insight into space and time. He showed time is not universal but relative between observers. This arguably led to his most famous equation, E=mc2. Yet his math skills left the theory in an incomplete state. His former teacher, Hermann Minkowski, formulated his theory of SR into a mathematical structure. In 1909 Minkowski famously stated, “From this hour on, space by itself, and time by itself, shall be doomed to fade away in the shadows, and only a kind of union of the two shall preserve an independent reality” and introduced a mathematical structure of SR known today as Minkowski’s space . Einstein later used this space as a key component in his theory of General Relativity. It appears beyond coincidence that Minkowski’s paper came out in 1909, and in that same year, Einstein committed himself full-time to work on his theory. The author has found no commentary on these two events happening relative to each other. Though subtle, they should not be glossed over. Einstein understood his mathematical shortcomings, and to have his college professor advance his work must have troubled him, troubled him to the point that he dedicated all his efforts to completing his theory. In addition, Einstein was known for giving witty and profound sayings. Having Minkowski’s work produce a famous quote (from Einstein’s thoughts) must have played on his mind. Even with this level of inspiration, Einstein could not produce a working solution by himself, even after years of trying.
In Minkowski space, there is a Lorentz invariance that is maintained, a type of transformation between two observers in Special Relativity, which preserves the universal speed of light between observers. Even in Einstein’s working through the Lorentz invariance, there were assumptions made that to this day are controversial in scientific society . There is no question Einstein was working at a genius level in his ability to view the world differently than those before him. However, his math understanding and skill level did not operate at the same level as his insight. It is here Einstein, in the author’s opinion, formed an Achilles’s heel. Einstein heavily depended on others to bridge the mathematical gap between insight and formulation. By 1909, Einstein was in the depths of trying to formulate his general theory (GR) into a mathematical structure. He struggled for a minimum of three years.
It should be noted during this time of struggling, Einstein followed many concepts, even considering abandoning the universal speed of light . His ability to maintain an open mind and pursue the many different “rabbit holes” is, in the author’s belief, the genius of Einstein. Later in his life, when confronted with the workings of quantum mechanics and probability, Einstein lost this ability and closed his mind to the probability concept when talking with Niels Bohr . It is from this conversation Einstein’s famous quote, “God does not play dice with the universe,” started to percolate and was eventually written in a 1945 correspondence . A sad reality when looking back at his work on General Relativity and his ability to maintain an open, pursuing mindset. The world can only speculate what great insight Einstein may have been able to give to quantum if he had approached it with the same open mind and fervor he approached General Relativity. All-be-it Einstein had areas in his studies in which he showed much stubbornness and even rude correspondence. Einstein’s treatment of ether over the course of his career shows this. A wonderful read, for those interested, is “Einstein and the Ether” by Ludwik Kostro , where Einstein’s treatment of ether was absolute between 1905 and 1916 . Einstein, in his 1905 paper on Special Relativity, never mentioned the Michelson-Morley experiment, which showed that experimentally there was no evidence of a “velocity” in ether (what makes up empty space). After 1916, Einstein relaxed his stance on ether, saying it was too radical . Thus, it is not outside the realm of possibility that Einstein was mono-focused or formed a mono-vision on certain aspects of his theory, blinding him to considering other possibilities.
Moving forward three years to 1912, Einstein had arrived at a point of true desperation in his quest for the mathematical component of GR. Feeling this desperation on returning from Prague to Zurich, Einstein reached out to his college friend Marcel Grossmann at the Zurich Polytechnic and said, “Grossmann, you’ve got to help me or I will go crazy” . It should be noted the word crazy during this time of history was not a word to be spoken lightly, nor a word that means what it does today. It was not until the roaring twenties the word crazy took on a new meaning, matching more closely to today’s slang use of it . The word, when Einstein used it, represented a mental state worthy of commitment to an asylum. It is not out of context to believe Einstein was in a desperate mental state to find a solution. The author emphasizes it is important to consider the desperation experienced by Einstein during this time. The race was on, and the pressure was tremendous. He had already witnessed his college professor, Minkowski, advance his work with much success. What else did he feel others would take off his work and advance under their own name? To imagine Einstein leaped from mountain peak to mountain peak (thus overlooking much of the connecting valleys) is within reason. This race may have very well pushed both Einstein and Grossmann to make educated assumptions that later present problematic acceptance of the framework .
During this time, many scholars were interested in Einstein’s work, both to advance it and destroy it. Einstein did not wish to lose his work (or ideas) to someone else. Einstein was quite sincere in his speaking with Grossmann that mentally he was extremely stressed in trying to formulate the math and desperately needed help. To give credence to this level of sincerity, Einstein left his position at Charles University in Prague to take a position at the Swiss Federal Polytechnical School in Zurich so he could enlist Grossmann’s help .
There Grossmann and Einstein formulated the mathematical structure of GR utilizing Gauss’s theory of curved surfaces (recall Riemann was a student of Gauss). Their work resulted in a joint paper in June 1913 entitled the Entwurf paper , in English: Outline of a Generalized Theory of Relativity and of a Theory of Gravitation. After this paper, a famous mathematician, David Hilbert, started working on a solution to Einstein’s gravity. A work that would later form a General Relativity priority dispute . Thus, another area of stress that pushed Einstein to finish his theory quickly. The 1913 joint paper with Grossmann was, in many ways, a complete theory, but they struggled with how to balance the theoretical equation with a mathematical structure called the Ricci tensor. To them, it did not correspond to Newtonian mechanics in a low gravity field . This stalled the completion by approximately two years.
After Grossmann’s and Einstein’s paper in 1913, the two stayed friends but never did any other work together. Grossmann is given much credit for the work but never felt as though any of Einstein’s theory was his work . Nor did he pursue credit for it. Einstein and Grossmann stayed in contact until Marcel Grossmann died in 1936. Marcel suffered from Multiple Sclerosis and contended with a continual decline in health until it ultimately took his life . In 1935, Marcel was presented as an Honorary Member of the Swiss Mathematical Society (SMS). Einstein considered Grossmann a friend and one who helped him overcome his mathematical shortcomings. On October 29, 1912, after returning to Zurich to work with Grossmann, Einstein wrote to Arnold Sommerfield (a mathematical physicist), “I am now working exclusively on the gravitation problem, and believe that I can overcome all difficulties with the help of a mathematician friend of mine here. But one thing is certain, that I have never before in my life troubled myself over anything so much, and that I have gained enormous respect for mathematics, whose more subtle parts I had considered until now in my ignorance to be a pure luxury! ” . In April 1914, Einstein corresponded with his friend Paul Ehrenfest. “Grossmann wrote to me that he is now also succeeding in deriving the gravitational equations from the general theory of covariance. This would be a nice addition to our investigation ” .
While Einstein worked on his theory between 1905 and 1915, on several occasions, he came close but was not completely able to obtain his theory. One major step was Einstein’s elevator thought experiment, an experiment in which it is postulated a person in a closed elevator cannot tell if the elevator is undergoing acceleration (moving up) or if it is under the influence of gravity (stationary). This equivalence principle inspired Einstein to calculate the perihelion shift of Mercury early in his pursuit of GR. There was even a trip set up by a friend of Einstein to picture the stars around the sun during a solar eclipse. Thankfully for Einstein, during the eclipse, clouds obstructed the ability to get any pictures of the stars about the sun. Why was it thankfully? Because Einstein’s original calculations were off by one-half! This could have been such an embarrassment to Einstein he may not have been able to recover and finish his theory (or had the motivation to continue). But, later, with the help of skilled mathematicians, he was able to arrive at the correct calculation, and in 1919, Sir Eddington captured images of the stars about the sun during an eclipse and brought physical proof to Einstein’s theory .
Even with the future confirmation by Sir Eddington, Einstein was not aware of this success during the first ellipse attempt. Einstein, feeling confident the first time, only to discover new math showed it was off by 1/2, must have always been on his mind when considering any work. His math had failed him for the first time in his solitude and individual ability. There is sound evidence to form an opinion his inability to wield the math at the level he needed frustrated Einstein and left him feeling vulnerable. If one has ever worked under this type of frustration and vulnerability, one understands it is easy to overreach or make untested assumptions. Einstein, though a genius, was still human and susceptible to all the inadequacies of being human. It is more likely that, during this time, he (and Grossmann, who was under pressure from Einstein) did not deep dive into the many rabbit holes found in the theory by using Riemannian Geometry. Rabbit holes like gravitational waves, black holes, time travel, and zero-point energy. These concepts were most likely foreign to them, and they had never considered them before rushing to publish the work.
Another area of Einstein’s theory that has not been proved is the assumption made that all local spaces (remember Riemannian geometry and tangent spaces, which behave like Euclidean geometry) maintain the workings of Special Relativity (SR). One of those findings is Einstein’s equation E=mc2. This basically implies any energy can take the form of mass (the m in the equation), and mass can be transformed into any other energy (the E in the equation) with a factor of the square of the speed of light (the c in the equation). Thus, at every tangent location on the manifold, an observer will measure the energy released from mass obeying this equation. The problem arises when carrying this from one tangent space to another tangent space on the manifold. There is no mathematical or experimental proof showing energy obeys this equation between spaces. Actually, on the contrary, there is physical evidence this equation is not invariant when energy is transferred between spaces.
There is a well-known gravitational effect called the “red shift” or “blue shift” of light when it is traveling away from or into a gravitational well respectfully (a place where there is a gravitational potential) . If a photon travels into the well where there is an increasing gravitational potential, the photon gains energy (blue shifts). When the photon travels out of the well where there is a decreasing gravitational potential, the photon loses energy (red shifts). Therefore, when two observers are located at different tangent spaces (local spaces) on the manifold, they will observe a different energy from the photon as it is transferred from location to location. This is not what happens to mass; it maintains its energy following Einstein’s E=mc2 equation. This can be compared to Euclid’s treatment of parallel lines; they never touch in “special” circumstances, but Euclid proclaimed it an absolute, much in the same manner Einstein’s work proclaims E=mc2 absolute, not a special case. Later, Gauss and Riemann were able to demonstrate Euclid was correct, but only in special cases (the manifold is flat). Does Einstein’s Special Relativity E=mc2 fall under the same treatment?
If so, how is it demonstrated? Imagine if one observer transforms the photon to mass and sends the mass back to the other observer. The other observer transforms the mass back to a photon and sends it to the first observer. They repeat this process indefinitely. Eventually, the mass will completely disappear (because of the loss of energy of the photon) or the mass will become infinite (because of the gain of energy). The outcome depends on whether the photon is blue-shifting or red-shifting when transferred between the two observers. This is but one area of Einstein’s choosing of Riemannian geometry in which assumptions had to be made. In the author’s opinion, these types of assumptions have left gaps that need addressing within Einstein’s theory. Were these types of un-revisited assumptions because of the stress Einstein was under to complete his theory or possibly a case of mono-vision, as with his stance on the ether?