In the Newtonian theory of gravity, the shell theorem says that the gravitational field inside a massive and spherically symmetric shell is zero. The conclusion is the same in general relativity, as Birkhoff's theorem readily proves.
Whereas the Newtonian shell theorem generalizes to higher dimensions, it is not clear to me that Birkhoff's theorem would as well. After all, additional assumptions are needed in proving the uniqueness of Schwarzschild-Tangherlini spacetime (the higher dimensional analogue of Schwarzschild spacetime) -- namely, a topologically spherical and nondegenerate horizon. I therefore suspect that Birkhoff's theorem does not generalize, unless you impose additional assumptions.
Does Birkhoff's theorem generalize to higher dimensions without additional assumptions? If not, what are the additional assumptions?
