Given a spherically symmetric mass distribution
the shell theorem states that a test particle at radius $r$ experiences no net force from shells at a radius $R > r$;
the Birkhoff theorem states that (with the additional condition of asymptotic flatness) the metric is static, and any vacuum shell corresponds to a radial branch of the Schwarzschild solution.
But the Birkhoff theorem does not explicitly say that geodesics of test particles at radius $r$ follow entirely from an „inner solution“ and are not affected by the mass distribution at $R > r$.
Question: Under which conditions is it possible to derive a similar statement like the shell theorem? Or do I overlook anything, and this „shell theorem“ is implicitly included in the Birkhoff theorem?
