Limited utility of Birkhoff’s theorem in modified Newtonian dynamics: Nonzero accelerations inside a shell

We investigate the consequences of Birkhoff's theorem in general relativity (GR) and in modified Newtonian dynamics (MOND). We study, in particular, the system of a finite-mass test particle inside a spherical shell. In both GR and MOND, we find nonvanishing acceleration for that test particle. The direction of the acceleration is such that it pushes the test particle toward the center of the shell. In GR, the acceleration is found analytically in the case of a small test mass with a small displacement from the center of the shell. In MOND, the acceleration is found analytically in the limit of large test mass and small displacement, and a comparison to numerical values is made. Numerical simulations are done for more general cases with parameters that mimic the system of a galaxy in a cluster. In GR, the acceleration is highly suppressed and physically insignificant. In MOND, on the contrary, the acceleration of the point particle can be a significant fraction of the field just outside of the spherical shell.