Birkhoff's theorem

Birkhoff's theorem states that any spherically symmetric solution of the vacuum Einstein field equations is necessarily static and is necessarily the Schwarzschild metric. Spherical symmetry alone forces the exterior geometry to be Schwarzschild, with no remaining freedom and no time dependence — the staticity assumed in Schwarzschild's original derivation turns out to be a consequence rather than an extra hypothesis.

The physical consequences are far-reaching. A spherically symmetric star may pulsate, collapse, or explode, and as long as the motion stays exactly spherical, the gravitational field outside it does not change at all. There is therefore no monopole gravitational radiation: a breathing sphere emits no gravitational waves, the exact analogue of the electromagnetic result that a spherically symmetric charge distribution does not radiate. The theorem also implies that spacetime inside an empty spherical cavity at the center of a spherical mass is perfectly flat — the relativistic version of Newton's shell theorem.

Birkhoff's theorem is why the spacetime around the Sun can be treated as exactly Schwarzschild despite the Sun's convection and magnetic activity: those motions are very nearly spherical, and the spherical part of the field is frozen by the theorem. It guarantees that the classical tests of general relativity all probe one and the same exterior geometry.

The result is named for the American mathematician George David Birkhoff, who proved it in his 1923 textbook Relativity and Modern Physics. It was, however, stated earlier by the Norwegian physicist Jørg Tofte Jebsen in 1921, and so is sometimes called the Jebsen–Birkhoff theorem. It remains a cornerstone argument for why the Schwarzschild geometry is so universal in general relativity.