Physics 1916

On the Gravitational Field of a Mass Point according to Einstein's Theory

Karl Schwarzschild

The first exact solution of Einstein's equations — and the radius where space-time tears.

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In depth · the introduction

Take any object, squeeze it small enough, and Einstein's equations say space-time will close around it like a trapdoor — and Schwarzschild found exactly where the trapdoor lies.

The big idea

Late in 1915 Einstein finished general relativity: gravity is not a force but the bending of space and time around mass. But his equations were ferociously hard, and he could only solve them approximately. Schwarzschild solved them exactly for the simplest case — the empty space around a single round, still mass, like an idealised star.

Out of the answer fell a single number tied to the mass, today called the Schwarzschild radius. For most objects it is tiny and means nothing: the Sun's is about 3 kilometres, buried deep inside the Sun. But it marks a threshold. If you could crush a mass down inside that radius, the bending of space-time becomes total: not even light can climb back out. That surface — the point of no return — is what we now call the event horizon, the edge of a black hole.

How it came about

The timing is almost unbelievable. Einstein presented his final field equations to the Berlin Academy in November 1915. By December, Karl Schwarzschild — director of the Potsdam observatory, but at that moment a 42-year-old artillery officer serving on the Russian front in the First World War — had read them and worked out the exact solution between his duties calculating shell trajectories.

He mailed it to Einstein, who was astonished and presented it to the Academy on Schwarzschild's behalf in January 1916. 'I had not expected that one could formulate the exact solution of the problem so simply,' Einstein wrote back. Schwarzschild was already gravely ill with an autoimmune disease he had contracted at the front. He sent a second paper, then was invalided home, and died in May 1916. He never knew that the strange radius in his equations would become one of the most famous objects in physics.

Why it mattered

This was the first exact, complete solution of Einstein's theory — proof that the new physics produced not just corrections to Newton but a clean, calculable geometry. It nailed down the orbit of Mercury exactly, and it became the testing ground for everything relativity predicted: light bending, time running slow in gravity, the precise paths of orbits.

And hidden inside it was something neither Schwarzschild nor Einstein believed was real — the black hole. For half a century most physicists treated the horizon as a mathematical glitch. Only later did it turn out to describe actual objects out in the universe, places where stars have collapsed so far that they have vanished behind their own Schwarzschild radius.

A way to picture it

Think of a swimmer in a river that flows toward a waterfall and speeds up as it nears the edge. Far upstream the current is gentle; the swimmer can always turn and stroke back to shore. But somewhere there is a line where the river finally moves faster than the swimmer can ever swim. Cross it, and no matter how hard you try, you are carried over the falls. Space-time near a heavy mass flows inward like that current, and the event horizon is exactly that line — the place where 'fast enough to escape' would mean faster than light, which nothing can manage.

Where it sits

Schwarzschild's solution is the first child of Einstein's general relativity, and the parent of all black-hole science. Newton had explained gravity as a force pulling across empty space; Einstein recast it as curved geometry; Schwarzschild gave that geometry its first exact face. From here the story runs forward to the rotating black holes described by Roy Kerr in 1963, to the ripples in space-time that LIGO caught as gravitational waves in 2015, and to the first photograph of a black hole's shadow in 2019. Every one of those begins with the radius he found on the Eastern Front.

The original document

Original source text

§1 — The problem Einstein posed

K. Schwarzschild · Sitzungsber. Preuss. Akad. Wiss. Berlin (Math.-Phys.) 1916: 189–196 · communicated January 13th, 1916

In his work on the motion of the perihelion of Mercury (see Sitzungsberichte of November 18th, 1915) Mr. Einstein has posed the following problem:

Let a point move according to the prescription δ∫ds = 0, where ds = √(Σ g_µν dx_µ dx_ν), µ, ν = 1, 2, 3, 4 … In short, the point shall move along a geodesic line in the manifold characterised by the line element ds.

§1 — The four conditions on the field

Let x1, x2, x3 stand for rectangular co-ordinates, x4 for the time; furthermore, the mass at the origin shall not change with time, and the motion at infinity shall be rectilinear and uniform. Then … the following conditions must be fulfilled too:

1. All the components are independent of the time x4.

2. The equations g_ρ4 = g_4ρ = 0 hold exactly for ρ = 1, 2, 3.

3. The solution is spatially symmetric with respect to the origin of the co-ordinate system in the sense that one finds again the same solution when x1, x2, x3 are subjected to an orthogonal transformation (rotation).

4. The g_µν vanish at infinity, with the exception of the following four limits different from zero: g44 = 1, g11 = g22 = g33 = −1.

§4 — The exact line element

When one introduces these values of the functions f in the expression (9) of the line element and goes back to the usual polar co-ordinates one gets the line element that forms the exact solution of Einstein's problem:

ds² = (1 − α/R) dt² − dR² / (1 − α/R) − R² (dϑ² + sin²ϑ dφ²), R = (r³ + α³)^(1/3). (14)

The latter contains only the constant α that depends on the value of the mass at the origin.

§4–§6 — Where it tears, the link ρ = α³, and the orbits

Therefore all the conditions are satisfied apart from the condition of continuity. f1 will be discontinuous when 1 = α(3x1 + ρ)^(−1/3), 3x1 = α³ − ρ. In order that this discontinuity coincides with the origin, it must be ρ = α³. Therefore the condition of continuity relates in this way the two integration constants ρ and α.

[ … ]

For an ideal mass point, however, it follows that the angular velocity does not, as with Newton's law, grow without limit when the radius of the orbit gets smaller and smaller, but it approaches a determined limit n0 = 1 / (α√2). (For a point with the solar mass the limit frequency will be around 10⁴ per second.)

Communicated to the Prussian Academy of Sciences, Berlin · January 13, 1916