View source for Birkhoff's theorem (relativity)

{{short description|Statement of spherically symmetric spacetimes}}
{{general relativity|expanded=theorems}}

[[File:Jebsen-Arkiv 1921.jpg|thumb|Front page of ''Arkiv för Matematik, Astronomi och Fysik'' where Jebsen's work was published]]
In [[general relativity]], '''Birkhoff's theorem''' states that any [[spherically symmetric spacetime|spherically symmetric solution]] of the [[vacuum field equations]] must be [[static spacetime|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.<ref>{{Cite journal |last=Israel |first=Werner |date=25 December 1967 |title=Event Horizons in Static Vacuum Space-Times |url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.164.1776 |journal=Physical Review |volume=164 |issue=5 |pages=1776–1779 |doi=10.1103/PhysRev.164.1776 |bibcode=1967PhRv..164.1776I |via=American Physical Society}}</ref><ref>{{Cite book |last=Straumann |first=Norbert |title=General Relativity |series=Graduate Texts in Physics |year=2013 |publisher=Springer Graduate texts in Physics |isbn=978-94-007-5409-6 |edition=2nd |pages=429 |doi=10.1007/978-94-007-5410-2|bibcode=2013gere.book.....S }}</ref> The converse is not true in Newtonian gravity.<ref>{{Cite book |last=Padmanabhan |first=Thanu |title=Cosmology and Astrophysics through problems |publisher=Cambridge University Press |year=1996 |isbn=0-521-46783-7 |pages=8,150}}</ref><ref>{{Cite book |last=Padmanabhan |first=Thanu |title=Sleeping beauties in theoretical physics: 26 Surprising insights |publisher=Springer Lecture notes in Physics |year=2015 |isbn=978-3-319-13442-0 |volume=895 |pages=57–63 |chapter=5 |series=Lecture Notes in Physics |doi=10.1007/978-3-319-13443-7 |bibcode=2015sbtp.book.....P |issn=0075-8450}}</ref>

The theorem was proven in 1923 by [[George David Birkhoff]] (author of another famous ''[[Ergodic theory#Probabilistic formulation: Birkhoff–Khinchin theorem|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 equivalence|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 mechanics|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.<ref>{{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}}</ref>

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

==See also==
* [[Birkhoff's theorem (electromagnetism)]]
* [[Newman–Janis algorithm]], a [[complexification]] technique for finding [[Exact solutions in general relativity|exact solutions]] to the [[Einstein field equations]]
* [[Shell theorem]] in Newtonian gravity
* [[Quadrupole formula]]

==References==
{{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}}

==External links==
*[http://scienceworld.wolfram.com/physics/BirkhoffsTheorem.html ''Birkhoff's Theorem''] on ''[[ScienceWorld]]''

[[Category:Theorems in general relativity]]