# The vacuum field equations

## The vacuum field equations

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

It follows that all the components of vanish.

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

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

so that in the new coordinates and . 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.