Noncovariance at low accelerations as a route to MOND
Abstract
MOND has limelighted the fact that Newtonian dynamics and general relativity (GR) have not been verified to any accuracy at very low accelerations—at or below the MOND acceleration a_{0}: Without invoking madetomeasure "dark matter," Newtonian dynamics (and hence general relativity) fail in accounting for galactic dynamics at such low accelerations. In particular, we do nont have evidence that all the cherished, underlying principles of Newtonian dynamics or GR, such as locality or Lorentz invariance, still apply in the MOND limit. I discuss the possibility that the principle of general covariance might not apply in this limit. This would be in line with suggestions that general covariance, where it does hold, is only an emergent, and hence approximate, property of relativistic dynamics. This idea also resonates well with MOND, which hinges on accelerations, for example, because the existence of an effective absolute inertial frame is natural in MOND. Relaxing general covariance affords more freedom in constructing candidate MOND theories. For example, it may permit constructing puremetric, local MOND theories, which is thought impossible with general covariance. I exemplify this with a MONDoriented, oversimplified, noncovariant theory, where the gravitational Lagrangian is L_{M}∝ℓ_{M}^{2}F (ℓ_{M}^{2}R ) , where R =g^{μ ν}(Γ^{γ}_{μ ν}Γ^{λ}_{λ γ}Γ^{γ}_{μ λ}Γ^{λ}_{ν γ})/2 is the (nonscalar) firstderivative part of the Ricci scalar R . Γ^{γ}μ_{ν} is the LeviCivita connection of a metric g_{μ ν}, which couples minimally to matter, and ℓ_{M}=c^{2}/a_{0} is the MOND length, which is of cosmological magnitude, being, e.g., of the order of the de Sitter radius of our Universe. This L_{M} gives a covariant theory in the highacceleration limit by requiring that F (z )→z +ζ , for z ≫1 , which gives GR with a cosmological constant ζ c^{4}a_{0}^{2} . In the MOND limit, F^{'}(z ≪1 )∝z^{1 /2}. In the nonrelativistic limit, the metric has a solution of the form g_{μ ν}≈η_{μ ν}2 ϕ δ_{μ ν} , as in GR, but the potential ϕ solves a MOND, nonlinear Poisson analog. This form of g_{μ ν} also produces gravitational lensing as in GR, only with the MOND potential. I show that this theory is a fixedgauge expression of bimetric MOND (BIMOND), with the auxiliary metric constrained to be flat. The latter theory is thus a covariantized version of the former à la Stückelberg. This theory is also a special case of socalled f (Q ) theories—aquadratic generalizations of "symmetric, teleparallel GR," which are, in turn, also equivalent to constrained BIMONDtype theories.
 Publication:

Physical Review D
 Pub Date:
 October 2019
 DOI:
 10.1103/PhysRevD.100.084039
 arXiv:
 arXiv:1908.01691
 Bibcode:
 2019PhRvD.100h4039M
 Keywords:

 General Relativity and Quantum Cosmology;
 Astrophysics  Cosmology and Nongalactic Astrophysics;
 Astrophysics  Astrophysics of Galaxies;
 High Energy Physics  Phenomenology
 EPrint:
 13 pages, version accepted for publication in Phys. Rev. D