The Birkhoff theorem states that the isotropic metric is time-independent. What is the exact meaning of this? Does it mean that for the isotropic source, the metric in any coordinate system is time-independent? Or Does it mean that there exists a specific coordinate system in which the metric is time-independent? It is very confusing.
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It's the latter. I can always redefine the time coordinate in some weird way (such as $t \to 2t + t^2 + \sin(t)$) so that everything is formally dependent on $t$. – knzhou Feb 05 '18 at 12:48
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The latter. There exists a specific coordinate system in which the metric is time-independent. Obviously it is possible to introduce time-dependence in the metric via a change of coordinates. For further details and the exact meaning of the Birkhoff theorem, see my Phys.SE answer here.
Also note that Birkhoff theorem states that a spherically symmetric solution of the vacuum EFE must be static, i.e. it admits a global, non-vanishing, timelike Killing vector field that is irrotational. Note that this is a coordinate-independent statement.
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