What theorem states that any spherically symmetric vacuum solution in general relativity must be static?

The theorem stating that any spherically symmetric vacuum solution in general relativity must be static is Birkhoff's theorem.

More precisely, the result is often called the Birkhoff–Jebsen theorem. It says that a spherically symmetric solution of the vacuum Einstein field equations is not only static but also asymptotically flat. Outside a spherical, nonrotating body, the spacetime therefore has the Schwarzschild geometry.

Jørg Tofte Jebsen first proved the result in 1921, and George David Birkhoff independently rediscovered it in 1923. A striking consequence is that a perfectly spherical star can pulsate without producing gravitational waves: spherical pulsations change the star's surface but do not create the changing quadrupole structure needed for gravitational radiation.

A common mix-up is to assume that “spherically symmetric” automatically means “static.” In general relativity, the theorem is powerful because it establishes that restriction for vacuum regions. It applies outside the matter, not necessarily inside a dynamic or collapsing star.