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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. . 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.