Friedmann (1922): The Equations Behind an Expanding Universe

The Friedmann equations come from a 1922 paper by the Russian mathematician and meteorologist Alexander Friedmann, who showed that Einstein’s general relativity allows universes whose size changes with time. Einstein had assumed a static cosmos. Friedmann proved it need not be, laying the mathematical ground for the expanding universe.

The question before the paper

When Einstein applied general relativity to the universe as a whole, he took for granted that it was static, unchanging in size forever. Friedmann asked a mathematician’s question: is a static universe the only solution the equations permit, or just the one Einstein chose?

What the paper found

• Dynamic universes are allowed. Einstein’s field equations admit solutions in which the radius of curvature of space changes over time, growing or shrinking.
• Static is a special case. Einstein’s universe is one possibility among many, not a requirement of the theory.
• A family of models. A follow-up paper in 1924 extended the analysis to universes with negative spatial curvature, completing the set of geometries cosmologists still use.

How he did it

The work is pure theory. Friedmann assumed a universe filled uniformly with matter and solved Einstein’s equations for how its scale evolves with time. The result is a pair of equations that link the expansion rate of space to what the universe contains and to its curvature. Zeitschrift für Physik accepted the paper on June 29, 1922.

Why it mattered

Einstein first published a note claiming the solution contained a mathematical error. After Friedmann’s colleague Yuri Krutkov met him in Leiden in May 1923, Einstein retracted, writing that Friedmann’s results were correct. The episode shows how unwelcome a changing universe was, even to the theory’s author.

Friedmann’s equations became the backbone of cosmology. Georges Lemaître rediscovered an expanding solution in 1927 and linked it to the recession of galaxies, which Edwin Hubble measured in 1929.

What we know now

Every modern analysis of the universe’s history runs through the Friedmann equations, from the Planck 2018 parameters to the dark energy measurements of DESI. Without them there is no Big Bang model, no estimate of the age of the universe and no forecast of its far future.

Who wrote it

Alexander Friedmann was a professor in Petrograd (today St. Petersburg) who had applied mathematics to meteorology and aeronautics before turning to Einstein’s theory. He wrote this paper alone, in German. He died of typhoid in August 1925, aged 37, four years before Hubble’s observations confirmed that the universe expands.

Read the original

Friedmann, A. (1922). “Über die Krümmung des Raumes.” Zeitschrift für Physik. doi:10.1007/BF01332580. The journal version is behind a paywall; a historical account of Friedmann’s work is free from the American Physical Society.