Birkhoff's theorem (relativity)

The Birkhoff theorem ( according to George David Birkhoff 1923) states:

• If there is a test mass in the external gravitational field of a spherically symmetric mass distribution, so does the mass distribution as a point mass.

The Birkhoff theorem is the generalization of Newton 's theorem for the general - relativistic case Represents the Newtonian theorem itself applies only in the non - relativistic limit.

The exact formulation of Birkhoff 's theorem in the context of general relativity is:

• A spherically symmetric vacuum solution of Einstein's field equations outside a mass distribution must be static, and this solution must be the Schwarzschild solution.

An immediate consequence of the Birkhoff theorem is that a spherically symmetric mass distribution, performs the spherically symmetric vibrations, nevertheless acts as a point mass. The vibrations have no effect on the spacetime and in particular can not emit gravitational waves.

The Birkhoff theorem corresponds in electrodynamics the facts that the electric field outside a spherically symmetric charge distribution is the same as an equivalent point charge at the center of the charge distribution box. Accordingly, the field is still static, even if the charge distribution (spherical symmetrical ) oscillates. An electromagnetic wave is not emitted.