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May/June 1922: Friedmann Imagines a Changing Universe — to Einstein’s Chagrin

Alexander Friedmann used Einstein’s general theory of relativity to propose a universe that could expand or contract.

Published May 15, 2024
Alexander Friedmann and Albert Einstein
Alexander Friedmann (left), a Russian mathematician and meteorologist, took a keen interest in Albert Einstein’s 1917 paper applying the general theory of relativity to cosmology.
Credit: Leningrad Physico-Technical Institute, courtesy AIP Emilio Segrè Visual Archives (1925)/Ferdinand Schmutzer (1921)

In May 1922, a relatively unknown Russian mathematician and meteorologist sent a casual note to a friend outlining ideas that would rearrange our understanding of the universe.

Along with the letter, Alexander Friedmann typed up a manuscript detailing how Albert Einstein’s general theory of relativity showed that the universe could expand or shrink. He sent the manuscript and note to his friend, the Austrian physicist Paul Ehrenfest.

“I'm sending you a brief note regarding the question about the possible shape of the universe more general than the cylindrical world of Einstein, and the spherical world of de Sitter,” the correspondence reads, referring also to another solution proposed by Dutch astronomer Willem de Sitter. “Aside from these two cases there appears also a world, the space of which possesses a curvature radius varying with time.”

In other words, Friedmann thought the universe might not be static after all — that it could expand or contract at a calculable rate. Several years before Georges Lemaître theorized an expanding universe and Edwin Hubble measured it, Friedmann had unearthed the foundations of what would become the Big Bang theory.

Einstein first derived the general theory of relativity in 1915, publishing equations that describe how mass curves spacetime as gravity. Then, in a 1917 paper titled “Cosmological Considerations in the General Theory of Relativity,” he applied his field equations to the whole universe.

But a mathematical problem had emerged. The accepted paradigm of the time was that the universe was static, and Einstein thought that any alternative was impossible — but his field equations, he and later de Sitter realized, would allow for a universe that could change.

“From the standpoint of astronomy … I have erected but a lofty castle in the air,” Einstein, then in Berlin, wrote in a letter to de Sitter, who was also coming up with cosmological solutions to the general relativity equations. “For me, though, it was a burning question whether the relativity concept can be followed through to the finish, or whether it leads to contradictions.”

He devised a parameter for the equations called the cosmological constant that he thought would keep the universe closed and static. The constant was merely a “hypothetical term,” he admitted. Still, “I am satisfied now that I was able to think the idea through to completion without encountering contradictions,” he wrote in a letter to de Sitter.

Along came Friedmann to provide contradictions aplenty. Born in St. Petersburg in 1888, Alexander Friedmann studied math and physics at university and applied these studies to aeronautics and meteorology research. He flew in missions for the Russian Imperial Army in World War I and spent down time modeling the trajectory of dropped bombs, which he verified during flights over the Austrian front.

By 1922, Friedmann was back in St. Petersburg, then called Petrograd. There, while he worked as a professor, he began to tackle Einstein’s 1917 paper in earnest. In what would later be dubbed the Friedmann equations, he showed that by varying a few assumptions, including the value of the cosmological constant, Einstein’s equations could describe a universe that expands over time, contracts over time, or contracts and then expands again.

Friedmann wrote up his conclusions in a paper, “On the Curvature of Space,” which was accepted for publication by the respected German journal Zeitschrift für Physik on June 29, 1922.

Einstein swiftly wrote to the journal with a refutation. “The results concerning the non-stationary world … appear to me suspicious,” he wrote. “In reality it turns out that the solution given in it does not satisfy the field equations.”

Friedmann, undaunted, responded to Einstein with his precise calculations. And, Friedmann asked, if the esteemed physicist verified independently that they were indeed correct, could he please inform the editors of Zeitschrift für Physik and amend his earlier criticisms?

Einstein did eventually retract his statement after meeting with a colleague of Friedmann’s in May 1923. “My criticism … was based on an error in my calculations. I consider that Mr. Friedmann’s results are correct and shed new light,” he wrote in a letter to Zeitschrift für Physik. (Einstein would later call the cosmological constant his “greatest blunder.”)

New light was an understatement. In 1923, Friedman published a book, The World as Space and Time, elaborating on his findings and considering how his equations might actually map to the physical world. He published a second paper in 1924, outlining more possibilities for an infinite static or non-static universe.

He was aware that no observational evidence existed. “All these scenarios must be considered as curiosities which cannot be presently supported by solid astronomical experimental data,” he wrote.

Early data arrived that very same decade. In 1929, Edwin Hubble observed that the redshift of a galaxy was directly proportional to the galaxy’s distance from Earth. This implied that the further the galaxy, the faster it was moving away — proof of Friedmann’s theoretical work on a changing universe. In the 1960s, the discovery of the cosmic microwave background provided yet more evidence of the universe’s expansion, and by the 1970s, the term “Big Bang” had caught on to describe what was by then the reigning cosmological theory.

Tragically, Friedmann did not live to see any of this evidence. He contracted typhoid and died in August 1925, four years before Hubble’s observational evidence of the universe’s expansion. He was just 37 years old. For many years, both during his life and after his death, his contributions were largely overlooked.

Nevertheless, his work represents an important piece of physics history. A 1993 biography of Friedmann describes his contributions succinctly: “As Copernicus made the Earth go round the Sun, so Friedmann made the Universe expand.”

Tess Joosse

Tess Joosse is a science writer based in the Midwest.

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