Tensors and Relativity: Chapter 8

The vacuum field equations

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Outside the field producing mass the energy- momentum tensor vanishes i.e. tex2html_wrap_inline1248 . The field equations are therefore

equation1021

It follows that all the components of tex2html_wrap_inline1245 vanish.

From tex2html_wrap_inline1252 we have immediately that tex2html_wrap_inline1254 ; thus tex2html_wrap_inline1256 depend only on the radial coordinate r. It follows that tex2html_wrap_inline1260 can then only be satisfied if tex2html_wrap_inline1262 is also independent of time, i.e.

equation1023

Since tex2html_wrap_inline1264 occurs in the line element in the combination tex2html_wrap_inline1266 , one can always make the term involving f(t) vanish by the coordinate transformation

equation1025

so that in the new coordinates tex2html_wrap_inline1270 and tex2html_wrap_inline1272 . That is if the metric components no longer depend on time. We have proved Birkhoff's theorem:  every spherically symmetric vacuum solution is independent of time, i.e. the solution is static.

If one considers the vacuum gravitational field produced by a spherically symmetric star, then the field remains static even if the material in the star experiences a spherically symmetric radial displacement [ explosion ]. Thus Birkhoff's theorem is the analogue of the statement in electrodynamics that a spherically symmetric distribution of charges and currents does not radiate.