General Relativity and Curved Spacetime

Einstein took gravity, the most familiar force in the universe, and dissolved it — replacing it with geometry. Apples, planets, and light all follow the same warped paths through a spacetime that bends in the presence of mass.

For two centuries Newton's gravity stood as the model of what a physical theory should be: a single tidy equation, a force pulling masses toward each other across the void, accurate enough to land probes on planets. And yet Newton himself never quite believed it. The idea that the Sun could reach across a hundred and fifty million kilometres of empty space and yank on the Earth — instantly, without anything in between — struck him as absurd. In a 1693 letter he called action at a distance “so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it.” He published the equation anyway. It worked. He left the question of how for someone else.

It took two hundred and twenty-two years for that someone else to arrive.

The cracks in Newton's universe

By the early 1900s, three cracks had appeared in Newton's picture. First, the orbit of Mercury did not quite close. Each loop, its closest approach to the Sun drifted forward by 43 arcseconds per century — a tiny but stubborn anomaly. Astronomers hypothesised an unseen inner planet, “Vulcan,” to perturb it. Vulcan was never found.

Second, Newton's gravity propagated instantly. Move the Sun, and the Earth would feel it the same instant. But Einstein's 1905 special relativity had just established a strict cosmic speed limit: nothing — no signal, no influence, no information — could travel faster than light. Newtonian gravity broke this rule.

Third, there was a strange coincidence buried in the equations. The inertial mass of an object — how hard it resists being pushed — was numerically identical to its gravitational mass — how strongly gravity pulled on it. There was no reason these had to be the same number. They just were, to every decimal place anyone could measure. In Newton, this was a coincidence. Einstein decided it was a clue.

The happiest thought

In 1907, Einstein later recalled, he was sitting in the patent office in Bern when he had “the happiest thought of my life.” The thought was this: a person in free fall feels no gravity. A painter falling from a roof, an astronaut in orbit, a coin dropped inside a falling lift — none of them feel weight. Inside that freely falling frame, the laws of physics look exactly like the laws of physics in deep space, far from any mass.

The reverse is also true. Stand in a windowless rocket accelerating at 9.8 m/s² in deep space, and you cannot, by any local experiment, distinguish your situation from standing in a windowless room on the surface of the Earth. A dropped ball falls the same way. A beam of light bends the same way. There is no measurement — none — that can tell the two situations apart.

This is the equivalence principle, and Einstein took it absolutely seriously. If gravity is locally indistinguishable from acceleration, then gravity is not really a force at all. It is a feature of the reference frame — of the geometry of spacetime itself. The reason inertial and gravitational mass are identical is that they are the same thing, looked at two ways.

Gravity is geometry

It took Einstein eight more years of relentless effort — with crucial mathematical scaffolding from his friend Marcel Grossmann, drawing on the differential geometry of Riemann — to turn the happiest thought into equations. In November 1915 he published the result: the Einstein field equations. Stripped of its indices, the heart of them is a single sentence.

Spacetime tells matter how to move; matter tells spacetime how to curve.

That is John Wheeler's famous compression, and it is genuinely the whole theory. Spacetime is a four-dimensional fabric of events — three of space, one of time. A massive object — a star, a planet, a glass of water — warps the fabric around it. Other objects, drifting through that region, follow the straightest possible paths available in the warped geometry. These paths are called geodesics. On a flat plane, geodesics are straight lines. On a sphere, they are great circles. In curved spacetime near the Sun, they are ellipses.

From this picture, several consequences fall out almost immediately. Light, being massless, still travels along spacetime geodesics — so light bends when it passes a massive object. A clock deeper in a gravitational well ticks more slowly than one higher up, because time itself is part of the warped fabric. Two orbiting masses ought to stir the geometry around them, sending ripples outward at the speed of light. Pack enough mass into a small enough volume, and the curvature becomes so extreme that nothing — not even light — can escape it.

The verdict

Within a month of publishing, Einstein computed the orbit of Mercury from his new equations. The drift was 43 arcseconds per century. Exactly. Decades of patient astronomy, settled by a sheet of paper.

In May 1919, Arthur Eddington led an expedition to the island of Príncipe to photograph a total solar eclipse. With the Sun's glare blocked by the Moon, stars visible near the Sun's edge appeared shifted from their usual positions by precisely the amount Einstein had predicted — their light bent by the Sun's curvature of space. The result was announced in London that November. Einstein became, overnight, the most famous scientist alive.

The verdicts have kept coming. Gravitational time dilation is now a daily engineering correction: GPS satellites tick slightly faster than ground clocks, and without compensating for that, your phone's location would drift by kilometres a day. The Hulse–Taylor binary pulsar, observed since 1974, spirals inward at exactly the rate Einstein's equations predict for two neutron stars radiating gravitational waves. In September 2015, the LIGO detectors caught one of those waves directly — a pair of black holes, more than a billion light-years away, merging into one and shaking the fabric of spacetime by a thousandth the width of a proton. The signal arrived after a hundred years in transit. It matched the prediction.

What Einstein gave us is stranger than a new equation for gravity. He removed gravity from the list of forces entirely. The apple does not fall because the Earth pulls on it. The apple, drifting freely, is moving along the straightest line available through the local geometry — and that geometry, dimpled by the Earth's mass, points its straight lines downward. The fall is not a tug. It is a path. The universe, it turns out, is not a stage on which gravity acts. The stage itself is what gravity is.