Mathematics and Albert Einstein

One of the many urban legends about the Relativity genius claims that Einstein failed mathematics at school. Nothing could be further from the truth: in fact, his grades in Algebra and Geometry were even better than in Physics. This false rumor, which has been repeated over and over, comes from a wrong interpretation of the grading scales. Moreover, in his memoirs he himself recounts his passion for one of the works most celebrated by mathematicians, Euclid’s Elements.

However, during his first years as a researcher he was not sure whether mathematics were so essential for physics. This is what led him to choose the latter, as he himself says:

I saw that mathematics were divided into many specialties and each, on its own, could absorb an entire life. Consequently, I saw myself as Buridan’s ass, which was incapable of deciding between two bundles of hay. This was presumably due to the fact that my intuition in mathematics was not strong enough to clearly define what was basic… Moreover, my interest in the study of nature was no doubt stronger; and when I was a student I was still not sure that having access to in-depth knowledge of the basic principles of physics depended on the most intricate mathematical methods. I only understood this little by little, after years of independent scientific work.

In fact, despite having opted for physics, he ended up appreciating mathematics as the basis for his own creation, and he even asserted:

Of course, experience retains its quality as the ultimate criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics.

In fact, mathematical creativity was fundamental in Einstein’s contributions. At the time he was conceiving the General Theory of Relativity, he needed knowledge of more modern mathematicss: tensor calculus and Riemannian geometry, the latter developed by the mathematical genius Bernhard Riemann, a professor in Göttingen. These were the essential tools for shaping Einstein’s thought.

Specifically, non-Euclidean geometries seemed almost tailored to Relativity. Discovered shortly before, in a completely abstract way, they revolutionized geometry. This type of model emerged when thinking differently about Euclid’s fifth postulate. This principle, taken as an axiom by Euclid, establishes that given a straight line and a point outside it, only one parallel line can cross that point. Later, many mathematicians tried to prove it as a consequence of the rest of the axioms, which were more intuitive. After centuries of failures, the negation of this postulate led to hyperbolic geometry (there is an infinite number of parallel lines) and spherical geometry (there is none). The geniuses Lobachevski and Bolyai, and later Beltarmi and Félix Klein, opened a paradise for the creators of universe models.

Later, the tensors and connections studied by Christoffel (1829-1900), Gregorio Ricci (1853-1925) and Tullio Levi-Civita (1873-1941), and the geometric theory developed by Riemann, completed the toolbox Einstein needed for his theories. To be able to handle this sophisticated creation, Einstein corresponded with some mathematicians, including Levi-Civita, who helped him correct some errors in his writings. In an excerpt of these letters, Einstein praises his colleague’s mathematics:

“I admire the elegance of his calculation method; it must be great to ride those fields on the horse of genuine mathematics while we have to do our hard work on foot”.

The influence of Hermann Minkowski, David Hilbert and Felix Klein was noticeable and Albert Einstein soon considered mathematics as the essence of his work. To round off his theory, Einstein sought the support of his friend Marcel Grossmann, also a mathematician, who even though we warned him of the cumbersome mathematical course he was about to embark on, put him on the right track.

And this is how Einstein, using his intuition and knowledge of physics, and resorting to mathematics, created an extraordinary theory that no one has been able to match.

Manuel de León

ICMAT/ Royal Academy of Science