The man who taught mathematics at Einstein

GenevaThe young aspiring mathematician took notes with impeccable care. He knew his German friend would also appreciate it. He didn't know it yet, but that classmate sitting next to him in geometry class would eventually become one of the most brilliant minds of the 20th century: Albert Einstein. Sherlock Holmes had John Watson, Don Quixote had Sancho Panza, and Einstein always had Marcel Grossmann by his side, the faithful companion who not only shared his notes but also became a key figure in the development of a theory that would change our understanding of the universe. "Those notes were my salvation," Einstein explained.

Marcel Grossmann was born in 1878 in Budapest, where he lived until he was fifteen before moving to Zurich to complete his secondary education. It was in the Swiss city that he met Einstein at the Zurich Polytechnic University, ETH Zurich, one of the most prestigious universities in the world. There were only five or six students in the class. All of them brilliant. Some were more inclined towards mathematics and others towards physics. "Grossmann had more of a predilection for mathematics than Einstein," comments Tilman Sauer, a theoretical physicist at the University of Mainz and a historian of science specializing in the development of general relativity. Among the small, select group of students was also Mileva Marić, who would later become Einstein's wife. The three formed a close-knit group of friends that would become pivotal in the history of modern science. "During their studies, Grossmann, Einstein, and Marić spent a lot of time together, both in class and in the cafeteria," explains Sauer.

Einstein's career began after graduating in 1900, but his early career was not meteoric. "While Grossmann found success in academia from the start, Einstein had a much harder time finding work." It was thanks to the indispensable intervention of Grossmann's father that the young German physicist secured a position at the Swiss patent office in Bern. In 1905, in the Swiss capital, between patent applications, Einstein conducted the research that would catapult him to fame. It was in this annus mirabilis who published the theory of special relativity as well as some of the principles of quantum mechanics, for which he received the Nobel Prize in Physics in 1921. "The job at the patent office was the greatest that Grossmann ever did as a friend," Einstein declared.

From fame in Zurich

By the early 1910s, Einstein was already world-renowned. The German physicist was a figure every university wanted to boast about. Einstein turned down several offers, including positions at Columbia University, the University of Prague, and Utrecht University. But at his heart was the idea of returning to the city where he grew up and where the foundations of his thinking were laid. Einstein longed to return to Zurich, where Grossmann had secured a professorship in geometry at ETH Zurich in 1907 and where, three years later, he became head of the mathematical physics department. "Grossmann already had a long career as a professor and had published numerous articles and books on geometry," explains Sauer. His influence was crucial in enabling Einstein's return to the city. Furthermore, Einstein obtained letters of recommendation from Marie Skłodowska Curie herself and from Henri Poincaré, one of the most important mathematicians of the early 20th century. Thus, in August 1912, the Jewish physicist and his family arrived back in Zurich, settling at number 116 Hofstrasse.

Upon returning to that Swiss city, Einstein had already developed a principle destined to change our understanding of the universe: the equivalence principle, which states that the effects of a gravitational field are indistinguishable from the effects of uniform acceleration. Despite the power of this assertion, the creator of the theory of relativity encountered a seemingly insurmountable obstacle when it came to articulating the theory in an appropriate mathematical language.

"Grossmann, you have to help me or I'll go crazy."

Einstein worked for more than eight years on a formulation of his theory, but he couldn't quite find the mathematical framework that accurately described his ideas about gravitation. After several failed attempts, in the summer of 1912, just after settling back in Zurich, that companion appeared again, to save his life once more and establish Einstein as one of the most important figures in 20th-century human thought.

Grossmann suggested using a type of geometry called Riemann geometry, a branch of mathematics that studies curved surfaces. This was the mathematical framework Einstein needed to describe the geometry of spacetime, which, through its deformations, gives rise to the force of gravity. "Grossmann was the architect of the mathematical architecture of general relativity," says Sauer.

Grossmann and Einstein agreed to share authorship of the two articles they co-wrote in 1912 and 1913, which laid the foundations of general relativity. Grossmann was happy to take responsibility for the mathematical aspects, but not for the physical interpretation, which he left entirely to the German physicist. "Grossmann would never go so far as to claim co-authorship of the theory of general relativity," Sauer remarks.

Without Grossmann, Einstein would have struggled to develop a complete theory. "Einstein would have easily lost his way, and the theory would have come later and probably in a very different form," Sauer states. The collaboration with Grossmann had a significant impact on the physicist's view of mathematics. Einstein even declared: "Thanks to the help of a friend of mine, I am learning about subtleties in mathematics which, in my naiveté, I had always thought were useless."

A genius not entirely solitary

In 1914, Einstein accepted an offer from the University of Berlin. He thus returned to his country to publish the work that had taken him a decade to complete and that would forever change the understanding of the cosmos. The friendship between Grossmann and Einstein continued for decades, although distance and the multiple sclerosis that would later end Grossmann's life strained their relationship. Furthermore, spurred on by the rise of German fascism and the persecution of Jews, Einstein had to emigrate to the United States in 1933 to work at the Institute for Advanced Study in Princeton, where he would spend the last years of his life until his death in 1955.

Far from the classic image of a solitary genius, Einstein was surrounded throughout his life by friends and collaborators with whom he worked on theories that would go down in history bearing his name, but which also contained the efforts of other equally exceptional minds. No one doubts Einstein's individual genius, but his friendship with Grossmann exemplifies how the life and work of a genius are also, in essence, a product of the relationships with those around him.

A malleable space-time

Einstein succeeded in relating mass and energy to the deformation they cause in the surrounding region of space. Unlike Newton's theory, spacetime ceased to be a static framework in which objects moved and became a malleable entity with its own properties. Space and time are no longer flat, but warp when under the influence of a massive body like a planet or a star. It is this deformation of the fabric of spacetime that generates the effect of gravity. This relationship is expressed by what are known as Einstein's field equations, often considered the most elegant mathematical equations ever created by human ingenuity.

Given a distribution of mass and energy, these equations result in the curvature of spacetime. The physicist John Wheeler expressed this relationship as follows: "Mass tells spacetime how to deform, and spacetime tells mass how to move."

This relationship stems from mathematical entities that Einstein, before developing the theory of general relativity, was unfamiliar with: tensors. "Einstein sought to make measurements independent of the observer and the frame of reference," explains José Francisco Domínguez, a doctoral candidate at the Polytechnic University of Catalonia who studies the impact of this paradigm shift on mathematics. Grossman introduced Einstein to the concept of the metric tensor. This mathematical element serves as a rule for measuring distances on non-flat surfaces. "Following Einstein's general relativity, these types of mathematical tools, which were largely unknown at the time, became widespread," Domínguez explains.

The first observational test that Einstein's equations passed was during the solar eclipse of 1919.Einstein's equations predicted that, due to the curvature of spacetime, the light from stars behind the Sun should be deflected as it passes near the star. This deflection, although small, should be measurable. An observation led by the English astronomer Arthur Eddington confirmed that, indeed, starlight was deflected by exactly the angle predicted by Einstein's equations. This was the first experimental verification of the theory, which, more than a hundred years later, continues to yield breakthroughs such as the description of black holes and the gravitational waves emitted by the collision of two black holes.