An undated picture of the mathematician Alexander Friedmann.
100 years ago, the Russian mathematician and physicist Alexander Friedmann (1888-1925) published some equations he found while studying Albert Einstein’s relativistic theory. The so-called Friedmann equations describe for the first time a universe in motion; Until now nobody – not even Einstein himself – had imagined a universe – with matter – in expansion. This was a theoretical break 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: at that time he was already publishing an article in a specialist journal on the so-called Bernoulli numbers, which was supported by David Hilbert, one of the most prominent personalities in international mathematics. During this period Friedmann was also one of the main leaders of high school students in the general strike after 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, ie the study of the entire universe as a physical system.
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A few years earlier, in 1915, Albert Einstein had created the theoretical basis of modern cosmology: the general theory of relativity, which combined various concepts of matter, space and time with the theory of gravitation. Einstein initially developed his theory with a view to stars, planetary systems and other isolated systems. But in 1917 he set about applying his ideas to the universe as a whole to try to arrive at a theoretically consistent model.
He looked for the simplest solution by assuming that the density of the universe and its geometry are the same everywhere and all directions are the same, ie he considered a homogeneous and isotropic cosmos. He also found – since it seemed undeniable to him – that the state of the universe was immutable on a global scale. Until now, science had always seen it that way, and observations seemed to show otherwise.
When analyzing his equations on this static model, Einstein was surprised to find that there were no solutions: there were no constant density values or geometries in space that also remained constant over time that satisfied his equations. To fix this, he modified the original equations and added what he called a cosmological notion that allowed static solutions to be obtained.
In the same year, Dutch astronomer Willem De Sitter found another possible solution to Einstein’s equations: an empty universe devoid of any matter. This result alarmed the German: according to his idea, the geometry of the cosmos was created by the distribution of matter and without it the equations should not make sense.
In 1922 Friedmann began working on the problem, and although he assumed, like his colleagues, that the universe was homogeneous and isotropic, he did not consider it static. In an orderly manner, he analyzed the equations under these hypotheses and found Einstein’s static solution and De Sitter’s solution for the empty universe on the one hand, and solutions for a universe with moving matter on the other. Among them there were a variety of cases: those in which the radius of curvature increases infinitely over time, 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, then contract back to a point, and so on-.
This meant a radical change in the conception of the cosmos: the evolution assumed in the nature or formation of the earth also affected the universe as a whole. And he even got a first approximation of the age of the universe of 10 billion years – only 3 billion short of 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 the publication of Friedmann’s work, Einstein responded with an article in the same journal, claiming that the Russian’s main finding was wrong. But after a personal conversation with Yuri Alexandrovich Krutkov, who knew Friedmann’s work in detail, Einstein understood that the solutions were correct and that they actually represented other possible dynamics of the universe. He admitted his error in an article published in May 1923.
In the years that followed, better observations of the velocities of galaxies were obtained and it was concluded that the vast majority seemed to be moving away from ours, as astronomer Edwin Hubble published in 1929. The Belgian priest and scientist Georges Lemaître connected this distance of the galaxies to the Friedmann equations and concluded that the universe is expanding.
In 1931, Einstein was finally convinced of the great value of Friedmann’s work. Furthermore, he considered the introduction of the cosmological notion, which he had needed to obtain solutions for a static universe, to be his greatest scientific error: a demonstration of how his prejudices prevented him from seeing the expansion derived from his equations.
Friedmann was able to break away from this notion of a static universe, highly changeable and convulsive perhaps in part because of the society and historical moment in which he was a part. 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 major political changes of the time. On the other hand, his deep knowledge of meteorology – which deals with physical systems with many sudden changes – might also lead him to consider other types of dynamics to describe the cosmos and to accept that, as the Greek philosopher Heraclitus said, ” everything flows, nothing stays.
Ernest Nungesser He is a professor at the Polytechnic University of Madrid
Agate Timon G. Longoria is the coordinator of the ICMAT Mathematical Culture Unit.
coffee and theorems is a section dedicated to mathematics and the environment in which it arises, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline and meeting points between mathematics and others share social networks and cultural expressions and remember those who shaped its development and knew how to turn coffee into theorems. The name recalls the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that converts coffee into theorems”.
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