On the Curvature of Space
The universe need not stand still: Einstein's own equations let space expand — or expand and fall back.
Einstein had quietly assumed the universe just sits there, the same forever. A young Russian opened the same equations and found they say something far stranger — the whole cosmos can grow.
The big idea
When Einstein finished general relativity, he applied it to the whole universe and got an answer he disliked: the cosmos wanted to move. To keep it still — as everyone then assumed it must be — he added a fudge factor, the ‘cosmological constant’, tuned to hold space frozen. Alexander Friedmann, a mathematician and meteorologist in Petrograd, simply asked: what if we don't force it to be still?
Solving the equations without that assumption, he found not one universe but a family of them, all changing in time. Some expand forever. One expands, slows, stops, and then falls back together — what he called the ‘periodic world’. Which fate a universe meets depends on a single thing: how much matter it contains. Pack in enough, and gravity eventually reverses the expansion; too little, and it grows without end.
How it came about
Friedmann was not an astronomer chasing the sky; he was a brilliant applied mathematician who had flown reconnaissance and computed bombing tables in the First World War. Reading the new relativity, he saw a freedom others had missed and published his expanding solutions in 1922.
The reaction was humbling, then vindicating. Einstein himself read the paper and rejected it, printing a note that said Friedmann had made a mistake. Friedmann wrote back with his calculations laid out line by line. A year later Einstein published a retraction: the error, he admitted, had been his own, and Friedmann's results were correct. Tragically, Friedmann died of typhoid in 1925 at just thirty-seven — four years before Edwin Hubble looked through a telescope and saw distant galaxies actually flying apart, exactly the moving universe Friedmann had found on paper.
Why it mattered
This is the moment the universe got a history. Before Friedmann, ‘the cosmos’ meant a fixed, eternal stage. After him, it became a thing with a past and a future — something that could have begun, that is changing now, and that will end in one of a few definite ways. Rewind any expanding solution and everything rushes together to a single instant: the seed of the Big Bang. Every modern map of cosmic history — the afterglow of that beginning, the forging of the first elements, the fate billions of years hence — is read off the equations Friedmann wrote.
A way to picture it
Throw a ball straight up. Whether it falls back depends entirely on how fast you threw it against how strong gravity is. Throw it gently and it rises, stops, and comes down — that's the ‘periodic’ universe, expansion reversed into a collapse. Throw it at escape velocity or faster and it never returns — that's a universe that expands forever. The amount of matter in the cosmos plays the role of gravity's strength, deciding which throw you've made. In the tool below, dial the density and watch the universe's whole life draw itself out, either climbing away forever or arcing over and crashing back.
Where it sits
Friedmann's equations grew straight out of Einstein's 1915 general relativity, but turned its picture of the cosmos upside down. Five years later, Georges Lemaître independently rediscovered the expanding solution and added the physical idea of a ‘primeval atom’ — the universe born from a single dense point. Edwin Hubble's 1929 measurement (in this Library) supplied the missing evidence. Together their names live on in the FLRW model that all of cosmology still uses, and the thread runs on to the collapsing stars of Oppenheimer and Snyder (1939) and the dark energy that now governs the universe's future.
The radius of curvature varies between 0 and x₀. We shall call this universe the periodic world.