General relativity
${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
Introduction History Timeline Tests 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
Equations Formalisms 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
v t e

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.^[1][2] The converse is not true in Newtonian gravity.^[3][4]

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).

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.

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.^[5]

Birkhoff's theorem can be generalized: any spherically symmetric and asymptotically flat solution of the Einstein/Maxwell field equations, without ${\displaystyle \Lambda }$, 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.

• Birkhoff's theorem (electromagnetism)
• Newman–Janis algorithm, a complexification technique for finding exact solutions to the Einstein field equations
• Shell theorem in Newtonian gravity
• Quadrupole formula

• Deser, S & Franklin, J (2005). "Schwarzschild and Birkhoff a la Weyl". American Journal of Physics. 73 (3): 261–264. arXiv:gr-qc/0408067. Bibcode:2005AmJPh..73..261D. doi:10.1119/1.1830505. S2CID 119454232.
• D'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford: Clarendon Press. ISBN 0-19-859686-3. See section 14.6 for a proof of the Birkhoff theorem, and see section 18.1 for the generalized Birkhoff theorem.
• Birkhoff, G. D. (1923). Relativity and Modern Physics. Cambridge, Massachusetts: Harvard University Press. LCCN 23008297.
• Jebsen, J. T. (1921). "Über die allgemeinen kugelsymmetrischen Lösungen der Einsteinschen Gravitationsgleichungen im Vakuum (On the General Spherically Symmetric Solutions of Einstein's Gravitational Equations in Vacuo)". Arkiv för Matematik, Astronomi och Fysik. 15: 1–9.

• Birkhoff's Theorem on ScienceWorld