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| In ], '''Birkhoff's theorem''' states that any ] solution of the ] must be ] and ]. This means that the ] must be given by the ]. | In ], '''Birkhoff's theorem''' states that any ] solution of the ] must be ] and ]. This means that the ] must be given by the ]. | ||
| 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 ] 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 ] ] in the ]. | 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 ] 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 ] ] in the ]. | ||
| The conclusion that the exterior field must also be ''static'' 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 ]. This means that a spherically pulsating star cannot emit ]. | The conclusion that the exterior field must also be ''static'' 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 ]. This means that a spherically pulsating star cannot emit ]. | ||
| Another interesting consequence of Birkhoff's theorem is that for a spherically symmetric ], the ] must be given by the ]; in other words, the gravitational field must vanish inside a spherically symmetric shell. This agrees with what happens in Newtonian gravitation. | |||
| Birkhoff's theorem can be generalized: any spherically symmetric solution of the ] must be static and asympotically flat, so the exterior geometry of a spherically symmetric charged star must be given by the ]. | Birkhoff's theorem can be generalized: any spherically symmetric solution of the ] must be static and asympotically flat, so the exterior geometry of a spherically symmetric charged star must be given by the ]. | ||
Revision as of 20:01, 29 May 2005
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 solution.
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.
The conclusion that the exterior field must also be static 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.
Another interesting consequence of Birkhoff's theorem is that for a spherically symmetric thin shell, the interior solution must be given by the Minkowski metric; in other words, the gravitational field must vanish inside a spherically symmetric shell. This agrees with what happens in Newtonian gravitation.
Birkhoff's theorem can be generalized: any spherically symmetric solution of the Einstein/Maxwell field equations must be static and asympotically flat, so the exterior geometry of a spherically symmetric charged star must be given by the Reissner/Nordstrom electrovacuum.
The theorem was proven in 1927 by G. D. Birkhoff (author of an even more famous Birkhoff theorem, the pointwise ergodic theorem which lies at the foundation of ergodic theory).
Reference
- . ISBN 0-19-859686-3.
{{cite book}}: Missing or empty|title=(help); Unknown parameter|Author=ignored (|author=suggested) (help); Unknown parameter|Publisher=ignored (|publisher=suggested) (help); Unknown parameter|Title=ignored (|title=suggested) (help); Unknown parameter|Year=ignored (|year=suggested) (help) See section 14.6 for a proof of the Birkhoff theorem, and see section 18.1 for the generalized Birkhoff theorem.
External link
- ScienceWorld page on Birkhoff's Theorem