General Relativity

The Happiest Thought

Special relativity left out gravity. In 1907, still working at the patent office in Bern, Einstein had what he later called the happiest thought of his life: a person falling freely does not feel their own weight.

From that he built the equivalence principle. Inside a closed room you cannot tell whether you are standing on the Earth or being pushed upward by a rocket, and you cannot tell floating in deep space from falling freely. Gravity and acceleration are, locally, the same thing.

That has consequences. If a beam of light crosses an accelerating room, it appears to bend. So light must bend in a gravitational field too. And clocks deeper in a gravitational field must run slower.

Eight Years of Work

Turning the insight into a theory needed mathematics Einstein did not yet have. In 1912 he turned to his old classmate Marcel Grossmann, who introduced him to the geometry of curved spaces developed by Bernhard Riemann. Their joint paper of 1913 came close but was not right.

In November 1915 Einstein presented four papers to the Prussian Academy of Sciences in Berlin, one each week. On 18 November he showed that the new theory explained a small, long-standing anomaly in the orbit of Mercury. On 25 November he presented the field equations in their final form. The mathematician David Hilbert, working in parallel, submitted closely related equations five days earlier. Surviving printer's proofs of Hilbert's paper, dated December, do not contain the final equations, which is part of why historians credit the theory to Einstein.

What the Equation Says

The left side of the field equations describes the curvature of spacetime. The right side describes the matter and energy present. The equals sign is the theory: matter tells spacetime how to curve, and curved spacetime tells matter how to move.

A planet orbiting the Sun is not being pulled by a force. It is following the straightest possible path through a spacetime that the Sun's mass has curved. The term with $\mathrm{\Lambda }$, the cosmological constant, was added in 1917; it has its own story in Cosmology.

What It Predicts

• Light is deflected by massive bodies, by 1.75 arcseconds at the edge of the Sun.
• Time runs slower where gravity is stronger.
• Orbits slowly rotate, as Mercury's does.
• Accelerating masses send ripples through spacetime: gravitational waves.
• Enough mass in a small enough region closes itself off entirely. Einstein doubted such objects could exist. We now photograph them and call them black holes.

Where It Stands

More than a century on, general relativity has passed every test set for it. It is also known to be incomplete: it does not fit together with quantum theory, and it breaks down at the center of a black hole and at the first instant of the universe. Finding the theory that contains both is the largest unsolved problem in physics.