The Foundation of the General Theory of Relativity
Gravity is not a force but the curving of spacetime by matter and energy.
Einstein realised that gravity is not a force pulling on you, but the shape of space and time itself — and that even a beam of starlight must follow its curves.
The big idea
Newton had said gravity was a force reaching invisibly across empty space, holding the planets in their orbits. Einstein saw it differently. Mass and energy bend the space and time around them, and everything else — planets, falling apples, beams of light — simply takes the straightest available path through that bent geometry. There is no pulling rope; there is curved spacetime, and motion through it.
A famous one-line summary, from the physicist John Wheeler, captures it: spacetime tells matter how to move, and matter tells spacetime how to curve. Einstein's equations are the exact statement of that two-way conversation.
How it came about
Einstein finished special relativity in 1905, but it described only observers gliding steadily in straight lines, and it had no room for gravity. For the next eight years he wrestled with how to fold gravity in, guided by a single image he later called the happiest thought of his life: a person in free fall feels no weight at all. From this he reasoned that gravity and acceleration are the same thing seen from different points of view.
To turn the idea into equations he needed the geometry of curved surfaces — mathematics he did not know. His old classmate, the mathematician Marcel Grossmann, taught it to him. After years of wrong turns, in November 1915 — in a final sprint that overlapped with the mathematician David Hilbert working on the same problem — Einstein reached the field equations, and set them out in full in this 1916 paper.
Why it mattered
It replaced a 230-year-old picture of the universe. Newton's gravity had been almost perfect, but it left a tiny error in Mercury's orbit unexplained and treated gravity as an instant pull across any distance. Einstein's theory cured the Mercury error exactly, made gravity a local effect that travels at the speed of light, and predicted entirely new things — that starlight bends near the Sun, that clocks slow in gravity, that the universe expands, and that spacetime itself can ripple. Each of these has since been seen.
A way to picture it
Picture a heavy ball resting on a stretched rubber sheet. It makes a dip, and a marble rolled nearby curves toward it — not because the ball pulls the marble, but because the sheet beneath the marble is bent. Spacetime is that sheet, only in four dimensions: the Sun makes a dip, the Earth circles within it, and even a ray of light passing the rim has to bend. The picture is imperfect — real spacetime is not a sheet seen from outside — but it holds the heart of it: gravity is geometry.
Where it sits
This is the second of Einstein's two relativities, built on the first (special relativity, 1905) and completing Newton's account of gravity (the Principia, 1687) rather than discarding it — Newton survives wherever gravity is weak. The line runs onward through the ripples in spacetime it predicted, confirmed when LIGO detected gravitational waves in 2016, a hundred years later. Its open ends — black-hole singularities, the Big Bang, dark energy — are where today's search for a quantum theory of gravity begins.
The special theory of relativity is based on the following postulate, which is also satisfied by the mechanics of Galileo and Newton.
The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant).