Does Birkhoff's theorem still apply to an ultra-dense spherical shell of matter falling into a black hole? In previous posts I discovered that the effects on space-time geometry of an ultra-dense stationary spherical shell yielded pretty simple equations but I imagine some sort of secondary non-linear effects start happening in the dynamic case...maybe something analogous to linear frame dragging. Or does Birkhoff's Theorem still apply?
 A: The Brinkhoff theorem was recently challenged in this 2017 paper:
Does Birkhoff’s theorem really hold?
The author claims that:

Birkhoff’s theorem is inconsistent with the post-Newtonian solution to Einstein feld equations for a dynamic spherically-symmetric system in the weak-feld limit. The time-spatial component of metric, which is missed by Birkhff’s theorem, is indispensable for characterizing the external gravitational feld of the dynamic spherically-symmetric system. The radial motion of the spherically-symmetric system can produce gravitomagnetic effects, which may play important roles in the post-Newtonian dynamics.

And concludes:

According to the covariance of Einstein feld equations, when introducing a coordinate transformation, both the components of the metric tensor and the matter energy-momentum tensor will simultaneously change according to the rules of tensor transformations, and this may change the problem we are solving, since in relativity the coordinate transformations include the changes of reference frame. This important issue is overlooked not only in the derivation of Birkhoff’s theorem, but also in many literatures, due to the misinterpretation and misuse of the covariance of general relativity.

The jury is still out...
