Birkhoff's theorem (relativity)


In general relativity, Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. This means that the exterior solution must be given by the Schwarzschild metric.
The theorem was proven in 1923 by G. D. Birkhoff. However, Stanley Deser recently pointed out that it was published two years earlier by a little-known Norwegian physicist, Jørg Tofte Jebsen.

Intuitive rationale

The intuitive idea of Birkhoff's theorem is that a spherically symmetric gravitational field should be produced by some massive object at the origin; if there were another concentration of mass-energy somewhere else, this would disturb the spherical symmetry, so we can expect the solution to represent an isolated object. That is, the field should vanish at large distances, which is what we mean by saying the solution is asymptotically flat. Thus, this part of the theorem is just what we would expect from the fact that general relativity reduces to Newtonian gravitation in the Newtonian limit.

Implications

The conclusion that the exterior field must also be stationary is more surprising, and has an interesting consequence. Suppose we have a spherically symmetric star of fixed mass which is experiencing spherical pulsations. Then Birkhoff's theorem says that the exterior geometry must be Schwarzschild; the only effect of the pulsation is to change the location of the stellar surface. This means that a spherically pulsating star cannot emit gravitational waves.

Common Misunderstang

A common misunderstanding of the Birkhoff's theorem is that for a spherically symmetric thin shell, the interior solution must be given by the Minkowski metric; in other words, that the gravitational field must vanish inside a spherically symmetric shell. This is therefore not true also if mentioned in a lot of books. https://arxiv.org/abs/1203.4428

Generalizations

Birkhoff's theorem can be generalized: any spherically symmetric solution of the Einstein/Maxwell field equations, without Λ, must be stationary and asymptotically flat, so the exterior geometry of a spherically symmetric charged star must be given by the Reissner–Nordström electrovacuum.