Birkhoff's theorem fails to save MOND from non-local physics

TL;DR

This paper examines the limitations of Birkhoff's theorem in both general relativity and MOND, revealing that in MOND, non-zero accelerations occur inside spherical shells, challenging the assumption of gravitational shielding.

Contribution

It demonstrates that Birkhoff's theorem does not hold in MOND, showing significant accelerations inside spherical shells through analytical and numerical methods.

Findings

In GR, acceleration inside shells is negligible.

In MOND, acceleration inside shells can be substantial.

Numerical simulations confirm analytical results for galaxy-cluster systems.

Abstract

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 non-vanishing 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.