Birkhoff theorem

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The Birkhoff theorem (named after George David Birkhoff in 1923, whereby a derivation was published in a Norwegian physics journal by Jørg Tofte Jebsen as early as 1921 ):

"The external gravitational field of a spherically symmetrical mass distribution is equal to the field of a point-like accumulation of the entire mass in the center."

The Birkhoff theorem represents the generalization of the non-relativistic Newtonian shell theorem for the general theory of relativity .

The exact formulation of the Birkhoff theorem in the context of general relativity is:

"A spherically symmetrical vacuum solution of Einstein's field equations outside of a spherically symmetrical mass distribution must be static and this solution must be the Schwarzschild solution ."

A direct consequence of the Birkhoff theorem is that a spherically symmetrical mass distribution that executes spherically symmetrical oscillations nevertheless acts like a constant point mass in the outside area . The vibrations have no effects on spacetime and, in particular, cannot send out gravitational waves.

In electrodynamics, the Birkhoff theorem corresponds to the fact that the electric field outside of a spherically symmetrical charge distribution is identical to the field of an equivalent point charge in the center of the charge distribution. As a result, the field is always static, even if the charge distribution carries out (spherically symmetrical) oscillations. An electromagnetic wave is not emitted.

See also

literature

  • Ray D'Inverno: Introducing Einstein's Relativity . Clarendon Press, Oxford 1992, ISBN 0-19-859686-3 ( Section 14.6 contains a proof of the Birkhoff theorem. Section 18.1 deals with the generalized Birkhoff theorem).
  • GD Birkhoff: Relativity and Modern Physics . Harvard University Press, Cambridge, MA 1923.

Individual evidence

  1. ↑ About the general spherical symmetric solutions of Einstein's gravitational equations in a vacuum Translation into English (2005), from Arkiv för Matematik, Astronomi och Fysik, 15, No. 18 (1921) pp. 1-9