100 years ago, the Russian mathematician and physicist Alexander Friedmann (1888-1925) published some equations that he had found while studying Albert Einstein’s relativistic theory. The so-called Friedmann equations were the first to describe a universe in motion; Until now, no one – not even Einstein himself – had conceived of a universe – with matter – in expansion. That was a theoretical rupture similar to that of Charles Darwin’s theory of evolution or Alfred Wegener’s theory of continental drift.
Friedmann stood out as a mathematician at the institute: already then he published, in a specialized journal, an article on the so-called Bernoulli numbers, which was endorsed by David Hilbert, one of the most prominent figures in international mathematics. In that period Friedmann was also one of the main leaders of high school students in the general strike that followed the first Russian revolution of 1905.
At university he continued to publish results on mathematics and physics applied to hydrodynamics, aerodynamics, geophysics, meteorology or aerology – which studies the upper layers of the atmosphere. However, his most outstanding work was in cosmology, that is, in the study of the entire universe as a physical system.
A few years earlier, in 1915, Albert Einstein had established the theoretical basis of modern cosmology: general relativity, which combined various ideas about matter, space and time with the theory of gravity. Einstein initially developed his theory by thinking of stars, planetary systems, and other isolated systems. But in 1917, he set out to apply his ideas to the universe as a whole, to try to arrive at a theoretically consistent model.
He looked for the simplest solution, assuming that the density of the universe and its geometry were the same everywhere and also that all directions were equivalent, that is, he considered a homogeneous and isotropic cosmos. In addition, he established – since it seemed indisputable to him – that the state of the universe, on a global level, was invariable. Until now, science had always considered it this way and observations did not seem to indicate otherwise.
When analyzing his equations on this static model, Einstein observed, surprised, that there were no solutions: there were no constant density values or geometries in space, which also remained constant in time, which fulfilled his equations. To remedy this, he modified the original equations, adding what he called a cosmological term, which did allow static solutions to be obtained.
The same year, the Dutch astronomer Willem De Sitter found another possible solution to Einstein’s equations: an empty universe, devoid of all matter. This result alerted the German: according to his conception, the geometry of the cosmos was created by the distribution of matter and, in the absence of this, the equations should not make any sense.
In 1922 Friedmann began to work on the problem and although he did assume, like his colleagues, that the universe was homogeneous and isotropic, he did not consider it to be static. In an orderly fashion, he analyzed the equations under these hypotheses and found, on the one hand, Einstein’s static solution and De Sitter’s empty universe solution and, on the other, solutions of a universe with matter in motion. Among them, there was a great variety of cases: those in which, with the passage of time, the radius of curvature increases indefinitely, or those in which it does so periodically – the universe contracts to a point and then returns. to increase its radius to a certain value, to then contract back to a point, and so on–.
That meant a radical change in the conception of the cosmos: evolution, which had already been assumed in the species or the formation of the Earth, also affected the universe as a whole. And he even obtained a first approximation of the age of the universe, of ten billion years – only three billion below the currently accepted value. However, as Friedmann recognized, at the time his models were only theoretical constructs and were not supported by available experimental observations.
Three months after Friedmann’s papers were published, Einstein responded with an article in the same journal, claiming that the Russian’s main result was wrong. But, after discussing in person with Yuri Alexandrovich Krutkov, who knew Friedmann’s work in detail, Einstein understood that the solutions were correct and that, indeed, they represented other possible dynamics of the universe. He acknowledged his error in an article published in May 1923.
In the following years, better observations of the speeds of the galaxies were obtained and it was concluded that the vast majority seemed to be moving away from ours, as published by the astronomer Edwin Hubble in 1929. The Belgian priest and scientist Georges Lemaître connected this distance of the galaxies with the Friedmann equations, concluding that the universe is expanding.
In 1931 Einstein was finally convinced of the great value of Friedmann’s work. Moreover, he considered the introduction of the cosmological term, which he had needed to obtain solutions of a static universe, his greatest scientific blunder: proof of how his prejudices prevented him from seeing the expansion that was deduced from the equations of the.
Friedmann was able to get rid of that idea of a static universe, perhaps, in part, because of the society and the historical moment in which he participated, highly changing and convulsive. He was born in Saint Petersburg, was a professor of Mathematics and Physics at the University of Petrograd and died in Leningrad: the same city with three different names, due to the great political changes of the time. On the other hand, his deep knowledge of meteorology – which deals with physical systems with many sudden changes – could also make him consider other types of dynamics to describe the cosmos, and accept that, as the Greek philosopher Heraclitus said, “everything flows, nothing it remains.
Ernest Nungesser He is a professor at the Polytechnic University of Madrid
Agate Timon G. Longoria is coordinator of the ICMAT Mathematical Culture Unit.
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems”.
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