The idea that gravity and light are one geometry
In 1921 Albert Einstein presented a paper to the Prussian Academy of Sciences by Theodor Kaluza, titled Zum Unitätsproblem der Physik (“On the Unity Problem of Physics”). Einstein had known the idea since 1919 and had written to Kaluza about it with approval. Two years separate Einstein's first knowledge of the idea from its presentation.
Einstein had turned gravity into geometry: matter tells spacetime how to curve, and curved spacetime tells matter how to move. Maxwell's electromagnetism sat beside it as a separate field living on that geometry. Kaluza asked what would happen if there were one more direction of space, so that spacetime had five dimensions instead of four, and the only thing in the universe were the geometry of that five-dimensional world.
He kept the simplest possible law, the five-dimensional version of Einstein's vacuum equations. He added one restriction, now called the cylinder condition: nothing depends on the fifth coordinate. The universe is the same at every point along the extra direction, like a long cylinder whose shape never changes along its length.
Then he wrote the five-by-five metric in blocks. The familiar 4×4 block is the metric of ordinary spacetime. The extra row and column hold four new numbers, one for each ordinary direction, and Kaluza identified them with the electromagnetic potential, scaled by a constant fixed by Newton's gravitational constant. There is also one extra number, a scalar field that Kaluza had to carry along.
The result: the five-dimensional vacuum equations split into the four-dimensional Einstein equations, and Maxwell's equations without sources, plus an equation for the scalar. Gravity and electromagnetism come out of a single object.
Because nothing depends on the fifth coordinate, motion along it is conserved, by the same logic that makes momentum conserved when space is uniform. Kaluza saw that the conserved momentum in the fifth direction behaves like electric charge, and that a particle's path in five dimensions, seen in four, obeys the Lorentz force law of a charged particle. This is the part that has survived every later rewriting. Electric charge is momentum in a direction you cannot see.
The theory did not work cleanly for the electron. Seen in four dimensions, the five-dimensional motion gives the Lorentz force only when the particle's charge-to-mass ratio is small; for the electron it is enormous, and the scalar field's pull would then dominate. Einstein pointed this out to Kaluza, and Kaluza counted it a real shortcoming. He also introduced the particle's source terms by hand, outside the vacuum theory. Einstein and Jakob Grommer later showed that Kaluza's vacuum equations cannot give a non-singular, spherically symmetric particle. In a 1922 letter to Weyl, Einstein still wrote that Kaluza had come closest to reality.
Kaluza's paper ends on a note of doubt. A mathematician who proposes that the universe has an unseen direction does not ask you to believe it. He asks whether the numbers fall out of it, and they nearly did.