# Birkhoff's theorem (relativity)

General relativity :
$G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }$ :
Introduction
HistoryTimelineTests
Mathematical formulation :
Fundamental concepts
Equivalence principle
Special relativity
World line
Pseudo-Riemannian manifold :
Phenomena
Kepler problem
Gravitational lensing
Gravitational redshift
Gravitational time dilation
Gravitational waves
Frame-dragging
Geodetic effect
Event horizon
Singularity
Black hole
Spacetime
Spacetime diagrams
Minkowski spacetime
Einstein–Rosen bridge :
EquationsFormalisms
Equations
Linearized gravity
Einstein field equations
Friedmann
Geodesics
Mathisson–Papapetrou–Dixon
Hamilton–Jacobi–Einstein
Formalisms
ADM
BSSN
Post-Newtonian
Advanced theory
Kaluza–Klein theory
Quantum gravity :
Solutions
Schwarzschild (interior)
Reissner–Nordström
Einstein–Rosen waves
Wormhole
Gödel
Kerr
Kerr–Newman
Kerr–Newman–de Sitter
Kasner
Lemaître–Tolman
Taub–NUT
Milne
Robertson–Walker
Oppenheimer–Snyder
pp-wave
van Stockum dust
Weyl−Lewis−Papapetrou
Hartle–Thorne :
Scientists
Einstein
Lorentz
Hilbert
Poincaré
Schwarzschild
de Sitter
Reissner
Nordström
Weyl
Eddington
Friedmann
Milne
Zwicky
Lemaître
Oppenheimer
Gödel
Wheeler
Robertson
Bardeen
Walker
Kerr
Chandrasekhar
Ehlers
Penrose
Hawking
Raychaudhuri
Taylor
Hulse
van Stockum
Taub
Newman
Yau
Thorne
others :
Physics portal
 Category :
vte :
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 (i.e. the spacetime outside of a spherical, nonrotating, gravitating body) must be given by the Schwarzschild metric. The converse of the theorem is true and is called Israel's theorem. The converse is not true in Newtonian gravity.
The theorem was proven in 1923 by George David Birkhoff (author of another famous Birkhoff theorem, the pointwise ergodic theorem which lies at the foundation of ergodic theory).

## 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 (partly) 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, which requires at least a mass quadrupole structure.

## Generalizations

Birkhoff's theorem can be generalized: any spherically symmetric and asymptotically flat solution of the Einstein/Maxwell field equations, without $\Lambda$, must be static, so the exterior geometry of a spherically symmetric charged star must be given by the Reissner–Nordström electrovacuum. In the Einstein-Maxwell theory, there exist spherically symmetric but not asymptotically flat solutions, such as the Bertotti-Robinson universe.