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.