In this chapter we take up the problems outlined at the end of the last chapter: How are we to justify the fundamental assumptions needed for non-equilibrium statistical mechanics? How are we to show their consistency with the underlying dynamics? Although most of the important answers to these questions are covered in this chapter, one approach, the approach that invokes cosmological features of the universe “in the large,” is reserved for Chapter 8.

Because the organizational structure of this long chapter is a bit complex, it might be useful to outline here the route we shall follow through the issues. Part I deals with two important preliminaries from experimental science and from the use of computers to model the evolution of dynamical systems. These are results that later will be referred to a number of times in different contexts. Part II outlines the programs that have been offered to “back up” the three approaches to the derivation of a kinetic equation we examined in Chapter 6. Part III then takes up the various basic approaches that have been suggested to provide a physical explanatory context for the success of statistical mechanics in the non-equilibrium situation. First, a number of “unorthodox” approaches are discussed.

Philosophers on explanation

Causation and the Humean critique

We wish to know not only what happens in the world, but why it happens. We are interested, we say, in explanations of the phenomena of the world. We want explanations of both particular occurrences (“why did this rock dissolve when I placed it in the water?”) and of general kinds of happenings (“why does salt dissolve when placed in water?”). But what is it to give an explanation of a happening or a class of happenings?

Aristotle is famous for offering a first philosophical account of the nature of explanation. It consists, he says, in giving the “causes” of the phenomenon in question. Here, he includes what is called the “efficient cause,” corresponding most closely to our present idea that an event is produced by continuous earlier events that produce the phenomenon. Influenced by his experience in biology, he thinks of all events as governed by function, end, or purpose as well, so that a full account of why they occur must bring out their place in the accomplishment of some end or purpose. That is, a full explanatory account must offer the “final cause” of the phenomenon as well as its proximate efficient cause.

Director of the Main Geophysical Observatory

In February 1925, A. A. Friedmann was appointed Acting Director of the Main Geophysical Observatory, and in June was confirmed in this post. It was another example of what N. M. Gunter was to mention in his speech in memory of Friedmann: “His practical activity was such that it could not fail to be noticed by everybody, no matter under whom he worked; we know that everywhere Alexander Alexandrovich very quickly rose to the highest administrative positions.” Friedmann himself somewhat regretted the high appreciation of his organizational talent. In his letter to Steklov from Perm (of June 9, 1918) he complained: “My quick grasp of practical matters often renders me ill service, because my colleagues try to give me some responsible practical task; I flatly rejected administrative university posts, as to other kinds of work, I found it embarrassing to reject them at the very beginning …” As in Perm, in Leningrad Friedmann's resistance was broken and he plunged into public and administrative work. According to Gunter, Friedmann resurrected the Leningrad Physico-Mathematical Society: “The society began to work properly when A. A. became its secretary. He drafted the first charter of the society and got it adopted. Extremely busy, to be found only in his observatory study, he found time to attend the sessions of the board, he took upon himself the supervision of abstracts contributed to the Zeitschrift, advocated the publication of a journal during his visits to Moscow and was going to take part in editing it.” In February 1925, Friedmann became the editor of the geophysics section in the Great Soviet Encyclopedia. And he never interrupted his research and teaching.

The first months

When the war began, Vladimir Steklov was in Britain with his wife. They did not manage to return to Russia the usual way – via Germany – and set off for home by a long way: by sea to Norway, then via Sweden and Finland with a stop at Viborg. In the evening of August 5 (Old Style), the Steklovs arrived at the Finland Station in St. Petersburg and suddenly stumbled across Friedmann. The encounter proved to be quite timely, because Steklov had no money at all with him, so Friedmann gave him some and helped to hire a cab. The next day Steklov wrote in his diary: “Friedmann turned up unexpectedly … Had volunteered to join up and serve in an aviation company; sent by the Main Physical Observatory. He helped fix the lighting for the fuses had blown, otherwise we would have been sitting in the dark. So he put things right, Olga had come across some spare fuses. Had a lot of tea with rolls bought at Viborg, Friedmann left at 1 a.m.”

Friedmann writes in his autobiography that “in order to introduce aerological observations into aviation practice and thus, on the one hand, provide a service to aviation, and, on the other hand, increase the number of aerological stations, he joined, with the permission and approval of B. B. Golitsyn, Director of the Observatory, the volunteer aviation detachment; in which he worked, first on the northern front, near the towns of Osovets and Lyk, and later on other fronts, to organize aerological observations and aeronautical services in general.”

Thus began Friedmann's war odyssey, which lasted for some three years.

Friedmann's theory discovered the most grandiose natural phenomenon, the cosmological expansion. Having passed through a rigorous checkup in the discussion with Einstein, it very soon found proof and confirmation in astronomical observations.

This concluding chapter of the book deals with the astronomical investigations that set up an observational basis for the science of the Universe; it discusses the further development of Friedmann's legacy and the ideas of modern cosmology, a living and rapidly developing science that eagerly absorbs the latest findings provided by astronomical observations and the brightest ideas produced by fundamental physics.

Galaxies and the Universe

Speaking of stars in his 1917 cosmological article (see Chapter 9), Einstein referred to them as the basic elements constituting the Universe. That was the period when the astronomical concept that our stellar system was unique enjoyed wide popularity. It was assumed that the Milky Way with all its stars, the Sun being one of them, was the whole Universe. Einstein too may have believed this. In the early 1920s, however, the astronomical picture of the world was altered drastically. The leading role in cosmology turned to be played by another class of astronomical objects – the nebulae.

Nebulae had long been known to astronomers, and they were regarded, within the concept of the unique stellar system, as comparatively small gas clouds floating among the stars. The true nature of the nebulae was established when the first major telescopes appeared. Three hundred years before, Galileo had invented a telescope and used it to see individual stars in the Milky Way. He proved therefore that the whitish strip encompassing the whole sky consists of stars that the naked eye cannot distinguish.

Return to the Observatory

In the early 1920s, Friedmann gets the chance to return to Petrograd. In the winter he gets an invitation to take up a position as a senior research worker – a member of the Atomic Commission at the State Optical Institute – and in the spring he returns to the Main Geophysical Observatory as a senior physicist, and organizes a mathematical bureau there.

Six years before, at the very beginning of the war, Friedmann had been thinking of setting up a theoretical department in the Observatory. In his letters to B. B. Golitsyn we read: “If there is anything that gives me strength now, it's my scientific work and thinking about the Observatory and my future work in it” (May 20, 1915). “We could process computations for this problem using the Ritz method: it will be quite possible to do it in a mathematical department… By August I will probably submit to you a reasoned, at least as a first approximation, report on the mathematical department… Trying to sort out the details of the organization of a mathematical department I come across various questions of a mathematical nature” (June 25, 1915). He writes about the same to V. A. Steklov: “B. B. Golitsyn promised me, after the war is over, of course, a position as a senior physicist; he wants to put me in charge of the mathematical department which I discussed with you in the spring, and which is likely to be set up at the Observatory. It's hard to think of a better place for me” (July 25, 1915)

Arrival and first months

According to various documents and reminiscences about Friedmann, he came back from Perm to Petrograd in the spring or summer of 1920. V. A. Steklov's diaries enabled us to fix the date. In his diary for 1919–20, there is an entry dated May 20, 1920: “At 11.30 p.m. Tamarkin and Friedmann turned up unexpectedly. They came from Perm in a special stove-heated freight car, the ride took them twelve days. In Perm they were detained and searched, were about to have their provisions taken [let us recall that the Civil War was not yet over, and profiteers were severely dealt with – V. F.], but everything was sorted out. Friedmann wants to settle here for good and, of course, at the University … He is very talented! Of course, he is asking for assistance. I said I would talk to [A. A.] Ivanov, but I will not make any arrangements. Let them act on their own. I may see Ivanov on Sunday.”

Steklov's biographers note that behind the outward sternness there was a sympathetic soul. Thus, as soon as May 23, he writes: “At 1 p.m. there was an emergency session of the Pulkovo Committee. I told Ivanov that tomorrow Friedmann would submit an application to get a transfer here [i.e. to the University – Y. F.] as a lecturer. He promised to arrange everything.”

How did Petrograd in the latter half of May 1920 strike Friedmann? Let us recall that the Civil War was going on. The newspapers were publishing dispatches from the battle fronts. On the northern front, close to Petrograd, there was shooting in the Sestra River area.

Do we know much about our ancestors? How little we know about them! Moscow schoolchildren have been reported to remember at best the names of their grandfathers. Adults who care to think about their roots know the names of their great-grandparents and the names and patronymics of their grandparents.

It is different with people who have left their mark in politics, science or culture. So, what is known about the hero of this book – Alexander Alexandrovich Friedmann? A sample opinion poll among physicists showed that his works of 1922–24 on relativistic cosmology are included among the two or three most outstanding achievements of Soviet physicists. It would seem that interest in his personality would be heightened and therefore satisfied. Yet, it is not the case. His biography was published only once, in a thin brochure put out by Znanie (Knowledge) Publishers. The massive volume of Classics of Science, devoted to Friedmann and published in 1966, contains his short autobiography – “Curriculum Vitae.” It also contains Friedmann's major works in hydromechanics, dynamic meteorology, atmospheric physics and relativistic cosmology. The “Addenda” to the volume have a few reminiscences of his contemporaries about him, reprinted mainly from journals and magazines of the 1920s. There are also Friedmann's extremely interesting letters to Vladimir Steklov, his teacher, whom he so much revered, and to Boris Golitsyn – two outstanding Russian scientists with whom he was associated for many years. These materials are a source for biographical notes on Friedmann. No biographical dictionary of science or Soviet encyclopaedia fails to mention his name, starting with the first edition of the Great Soviet Encyclopaedia, in the preparation of which he himself took part.

Undergraduate studies

Friedmann was lucky: in 1906, the year he entered the University, Professor Vladimir Andreyevich Steklov (1864–1926) was transferred there from Kharkov. He was a Corresponding Member and later a Full Member of the Academy of Sciences, and in the Soviet period its Vice-President: one of the organizers of science in the USSR, a brilliant mathematician who continued the best traditions of the St. Petersburg mathematical school, glorified by the names of P. L. Chebyshev, A. A. Markov, A. N. Korkin, A. M. Lyapunov and many others.

Vladimir Steklov played an extremely important role in Alexander Friedmann's life. He was not only an outstanding mathematician, but also had an unusually bright personality. Nature had endowed him with excellent genetics. His uncle on his mother's side was Nikolai Alexandrovich Dobrolyubov, the famous Russian literary critic. His father, Andrei Ivanovich Steklov, a highly educated person, taught history and Hebrew at the Nizhni Novgorod Theological Seminary. Vladimir Steklov had unquestionable literary and musical talents – he could have been a successful opera singer. Steklov has bequeathed to us not only classical works in mathematics and mathematical physics, but also works of some literary merit. These include his book To America and Back (Leningrad, 1925), based on his impressions of his trip to the United States, as well as his books about Lomonosov and Galileo. Yet his vast literary legacy is still waiting to be published and commented upon. For almost two decades Vladimir Steklov kept a diary, and the archives of the USSR Academy of Sciences hold colorful notebooks filled with his daily jottings.

The Universe we live in can be described by a theory created by Alexander A. Friedmann. This theory is a triumph of scientific thought. For the first time ever the human mind embraced the Universe as a whole in its dynamics and development.

What was it that brought him such extraordinary success?

The young school mathematician was noticed by Markov; Steklov became his teacher at university; and in his last days one of his concerns was to publish a posthumous memoir of Lyapunov's. Friedmann was educated by the famous St. Petersburg school of mathematicians that had been founded by Chebyshev and made famous by those four outstanding scientists. His research displayed the excellent inherent features of the school: the ability to link mathematical problems with the fundamental issues of natural science, concrete choice of the object of research, sufficient generality in the problem set-up and a tendency to reduce the problem “to a number,” making it possible to apply practically and verify experimentally the elaborated theory. Just as in mechanics and meteorology, so in cosmology he “worked it out to a number” and gave an estimate for the age of the Universe: 10 billion years as the order of magnitude.

He also absorbed the ideas of modern physics while studying it under the guidance of great scientists, e.g. Ehrenfest, one of the major theoretical physicists of the 20th century. Friedmann was able to receive from him both concrete knowledge and an independent, critical approach to any scientific problem, however well it might be supported by the authority of the classics.

At the turn of the 20th century, St. Petersburg had 15 secondary educational establishments – classical, modern and military schools. Among the teachers of physics and mathematics we see names which became famous far beyond the history of secondary education. In the 4th Gymnasium, physics was taught by Apollon Pavlovich Afanasiev, who later became a university professor; another university professor, Karl Karlovich Baumgardt, taught physics and cosmology at the 8th Gymnasium named for Karl Mai, together with F. N. Indrikson, the author of the school physics textbook, who presented in his lectures a famous university general physics course of Professor O. D. Khvolson. Why did Friedmann's parents choose the 2nd Gymnasium as the place for Friedmann, Jr. to study for nine years? There appear to be two reasons. The first is that the gymnasium was relatively close to the house where they were living at the time. Friedmann's father occupied Apt. 4 in 35 Moika Street. The house exists today, under the same number; furthermore, as the cast-iron plaque on it says, it is protected by the state as an architectural monument of the early 19th century. Its corner looks onto the Zimnyaya Kanavka (canal, or, literally, ditch) and is next to the Naval Archives building, started in 1883 and completed one year before Friedmann's birth – in 1887; an inscription to that effect appears on the impressive facade looking onto Khalturin Street, formerly Millionnaya Street.

With Friedmann's return to Petrograd, the last and exceptionally eventful and productive period of his scientific activity began. His talent reached maturity, gained confidence in itself and full strength; yet his desire for knowledge, his hunger for creative effort seemed to be insatiable. Friedmann would tell his relatives and friends: “No, I'm an ignoramus, I know nothing, I should sleep less and should not do anything outside science, because all this so-called ‘life’ is a mere waste of time.” He was on the threshold of the most important accomplishment in his life – he was soon to develop his cosmological theory.

Friedmann's long-time and close friend, the mathematician (and later Professor) Alexander Felixovich Gavrilov, wrote: “The last five years in the life of this amazing person were full of virtually selfless work in new areas, with tremendous achievements.”

Studying relativity

Friedmann set out to study the general theory of relativity with unusual diligence and at the same time with great interest and zeal. At that time this theory was referred to as the strong relativity principle. That was by no means his only passion at that time, but the theory of relativity undoubtedly overwhelmed him with its broad scope, its simple and clear theoretical basis, its elegant mathematical apparatus. The structure of the Universe as a whole had for the first time become the object of exact scientific study. The nature of space and time was linked in the new theory with the distribution and motion of gravitating masses in the Universe.

The general theory of relativity was developed by Albert Einstein in 1914–16.

This book came out in Russian in 1988, the centenary of the birth of Alexander Friedmann, an outstanding Soviet scientist. The anniversary was widely marked by the scientific community with national and international conferences and symposia dedicated to this event as a tribute to the enormous contribution A. A. Friedmann made to the development of hydrodynamics, meteorology and, particularly, relativistic cosmology. Very little has been written about Friedmann. The present book is the first detailed biography of the scientist. The material is generally given chronologically and is based on the scientific works of Friedmann, documentary records and reminiscences by his contemporaries published in the late 1920s. The book also includes unpublished reminiscences about Alexander Friedmann.

The division of the work between the authors was as follows: Chapters 1–3, 5, 6 and 10, as well as the last section of Chapter 9, have been written by V. Ya. Frenkel; Chapters 4, 7 and 11 by E. A. Tropp; Chapters 8, 9 and 12 by A. D. Chernin. Given this clear division, in some chapters the material is presented in the first person singular. As to the method of citation, given that the book is popular science, the authors decided not to overburden it with footnotes. Quotations are either from archives or from the literature given at the end of the book.

The authors would like to thank the staff workers of the Leningrad archives for their assistance. We are grateful to Professor G. A. Grinberg and Professor L. G. Loitsyansky, and also to S. Ye. Malinina, O. N. Trapeznikova and A. B. Shekhter who shared their reminiscences with us.

Master's studies

Preparations for Master's examinations left almost no time for research: the theoretical courses which were to be studied for the examinations were quite extensive, and the examination requirements were extremely challenging. In July 1911, in the letter in which Friedmann informed his professor about his forthcoming marriage, he also gave a brief “graduate student's” report: “Our studies with Yak. Dav. seem to be going quite well. They have naturally been confined to reading the courses you recommended and articles for the Master's examinations. We are through with hydrodynamics and are going on to study the theory of electricity.” Therefore, during his Master's studies Friedmann published relatively few papers. In one article – ‘On finding particular solutions of the Laplace equation”, which appeared in 1911 in the Bulletin of the Kharkov Mathematical Society – he solved, as any Master's degree student was supposed to do, the problem set in his teacher's doctoral thesis: find all orthogonal coordinates in which Laplace's three-dimensional equation admits of partial separation of the variables. In solving this problem, according to V. A. Steklov's testimonial, Friedmann ‘displayed the required ingenuity and a good knowledge of analysis.” Another paper on the theory of partial differential equations, “On finding isodynamic surfaces,” came out in 1912. Friedmann learned about the problem of isodynamic surfaces from A. E. H. Love's textbook on the theory of elasticity. In Lamé's definition, isodynamic surfaces are surfaces in an elastic body, on which there are no oblique tensions, and each normal tension depends on only one coordinate.

“The surest and deepest way to study the geometry of the world and the structure of our Universe with the help of Einstein's theory consists in the application of this theory to the whole world and in the use of astronomical research.” These words are from the last page of The World as Space and Time. Friedmann is speaking here about the cosmological problem, about the application of the general theory of relativity to the study of the world as a whole, of the world considered as a single physical system. By September 1922, when these words were being written, three different attempts had been made to solve the cosmological problem: one by Einstein himself, another by the Dutch astronomer W. de Sitter, and the third by Friedmann in the first of his two cosmological works. Cosmology was taking its very first steps, and the results so far did not look very encouraging. As Friedmann says after the above-quoted phrase, “mathematical analysis lays down its arms, faced with the difficulties of the question, and astronomical investigations do not yet give a sufficiently reliable basis for experimental studies of the Universe.” And nevertheless: “these circumstances cannot be seen as more than temporary difficulties; our descendants will undoubtedly come to know the character of the Universe in which we are fated to live … ”

However, events developed much faster than could be expected from the initial predictions. And first of all thanks to Friedmann's work. His first study, as soon became clear, was already a major step in the right direction. Cosmology was to develop rapidly even in the 1920s.