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

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.

6 pages and 5 figures Added a minor change to the abstract. Corrected the typo in equation (1). Corrected the factor of 2 in equation (2)

References in corpus (7)

• Relativistic gravitation theory for the MOND paradigm
• Modified Newtonian Dynamics in the Milky Way
• X-ray Group and cluster mass profiles in MOND: Unexplained mass on the group scale
• MOND effects in the inner solar system
• Milky Way potentials in CDM and MOND. Is the Large Magellanic Cloud on a bound orbit?
• Escaping from MOND
• Loss of mass and stability of galaxies in MOND

Cited by in corpus (10)

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• Einstein's Theory of Gravity and the Problem of Missing Mass
• Comparing different realizations of modified Newtonian dynamics: virial theorem and elliptical shells
• Modified Kepler's Law, Escape Speed and Two-body Problem in MOND-like Theories
• Modifying Gravity: You Can't Always Get What You Want
• The Geometry Of Modified Newtonian Dynamics
• Equivalence Principle Violation in Weakly Vainshtein-Screened Systems
• Cosmology of the Galileon extension of Bekenstein's theory of relativistic Modified Newtonian Dynamics
• Modified gravitational collapse, or the wonders of the MOND
• Machian MOND: a variable $a_0$ in galaxy clusters