Birkhoff's theorem (relativity)

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.{{Cite journal |last=Penrose |first=Roger |date=1965-01-18 |title=Gravitational Collapse and Space-Time Singularities |journal=Physical Review Letters |volume=14 |issue=3 |pages=57-59 |doi=10.1103/PhysRevLett.14.57 |bibcode=1965PhRvL..14...57P| s2cid=116755736| url=https://doi.org/10.1103/PhysRevLett.14.57}}

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

{{reflist}}

• {{cite journal |author1=Deser, S |author2=Franklin, J |name-list-style=amp |title=Schwarzschild and Birkhoff a la Weyl|journal= American Journal of Physics|volume=73|issue=3|year=2005|pages=261–264|arxiv=gr-qc/0408067|doi=10.1119/1.1830505|bibcode = 2005AmJPh..73..261D |s2cid=119454232 }}
• {{cite book | author=D'Inverno, Ray | title=Introducing Einstein's Relativity | publisher=Clarendon Press | location=Oxford | year=1992 | isbn=0-19-859686-3 | url-access=registration | url=https://archive.org/details/introducingeinst0000dinv }} See section 14.6 for a proof of the Birkhoff theorem, and see section 18.1 for the generalized Birkhoff theorem.
• {{cite book | author=Birkhoff, G. D. | title=Relativity and Modern Physics | publisher=Harvard University Press | location=Cambridge, Massachusetts | year=1923 | lccn=23008297}}
• {{cite journal | author=Jebsen, J. T. | title=Über die allgemeinen kugelsymmetrischen Lösungen der Einsteinschen Gravitationsgleichungen im Vakuum (On the General Spherically Symmetric Solutions of Einstein's Gravitational Equations in Vacuo) | journal=Arkiv för Matematik, Astronomi och Fysik | year=1921 | volume=15 | pages=1–9}}

• [http://scienceworld.wolfram.com/physics/BirkhoffsTheorem.html Birkhoff's Theorem] on ScienceWorld

Category:Theorems in general relativity