A solution written in a trench
In November 1915 Einstein finally published the field equations of general relativity — a tangled knot of equations that nobody, including Einstein, expected to solve exactly any time soon. He guessed that only slow, approximate answers would ever be possible. He was wrong within weeks. A German astronomer named Karl Schwarzschild, then over fifty and serving in the German army on the Russian front of the First World War, read Einstein's paper between artillery calculations and found an *exact* solution. He mailed it to Einstein in December 1915, from the trenches.
Einstein was astonished. "I had not expected," he wrote back, "that one could formulate the exact solution of the problem in such a simple way." He presented Schwarzschild's result to the Prussian Academy on his behalf. The story has a tragic coda: Schwarzschild caught a rare autoimmune disease at the front and died in May 1916, only months after his discovery. He never lived to see how strange — and how famous — his solution would become.
What it describes: the space around a star
To make the equations crackable, Schwarzschild made the simplest honest assumption: a single lump of matter that is perfectly round, not spinning, and sitting alone in otherwise empty space — no other stars, no nearby planets. The Schwarzschild solution then gives the exact geometry of the *empty* spacetime *outside* that mass. The Sun, the Earth, an isolated star — to very good approximation, the curved spacetime surrounding each of them is Schwarzschild's.
Remarkably, the whole geometry depends on just one number: the total mass M of the central body. Nothing else. Two stars of the same mass — one made of hydrogen, one of pure iron — bend the spacetime around them in exactly the same way. This single solution quietly delivered general relativity's first real predictions: the tiny, anomalous wobble in Mercury's orbit that had puzzled astronomers for decades, and the precise bending of starlight grazing the Sun. The geometry below describes it in plain words.
Schwarzschild geometry (every symbol in plain words):
ds^2 = -(1 - rs/r) c^2 dt^2 + dr^2 / (1 - rs/r) + (angle terms)
\__________/ \____________/
time part space part
ds = the spacetime "distance" between two nearby events
r = how far you are from the center of the star
rs = the Schwarzschild radius (rs = 2GM / c^2) -- set by mass M alone
c = the speed of light; G = Newton's gravity constant
Far away (r much bigger than rs): 1 - rs/r is almost 1
-> spacetime looks flat, ordinary, Newtonian.
Closer in (r shrinks toward rs): 1 - rs/r shrinks toward 0
-> time slows, space stretches: gravity's grip tightens.The number that wouldn't go away
Look hard at the factor (1 - rs/r) sitting in both the time part and the space part. Here rs is the Schwarzschild radius, a special distance fixed entirely by the mass: rs = 2GM/c². For the Sun it works out to about 3 kilometers; for the Earth, about 9 millimeters — a marble. As long as the star's actual surface lies *outside* this radius (and the Sun's does, hugely), nothing odd happens; the marble-sized rs is buried deep inside ordinary matter and the geometry is perfectly tame everywhere you can stand.
But the algebra refuses to look away. *If* you could squeeze the entire mass inside its own Schwarzschild radius — crush the Sun down below 3 km — then at exactly r = rs the factor (1 - rs/r) hits zero, and the geometry behaves wildly: in these coordinates the time part collapses and the space part blows up. To Schwarzschild and his contemporaries this looked like a mathematical glitch in an unphysical fantasy. Surely no real object could ever be packed that tightly. So they set it aside.
The black hole hiding inside
What everyone dismissed as a glitch turned out to be a doorway. That surface at r = rs is what we now call the event horizon — a one-way boundary in spacetime. From the (1 - rs/r) factor you can read its meaning straight off: as you approach rs, time (the dt part) is dialed down toward a standstill for a distant watcher, while the escape it takes to climb back out grows without limit. Cross inward, and *every* path that points 'outward' actually leads deeper in. Not even light can climb back. The mass, sealed below, is a black hole in all but name.
- 1915–16: Schwarzschild writes the exact geometry around a round mass; the strange surface at r = rs is treated as a meaningless coordinate quirk.
- 1930s–60s: physicists slowly realize a heavy enough collapsing star *can* shrink past rs, and that the surface is a real, one-way horizon — not a glitch.
- 1967 onward: the object earns its name — the *black hole* — and astronomers begin finding real ones, from star-mass remnants to the giants at the hearts of galaxies.