Noncovariance at low accelerations as a route to MOND

Abstract

MOND has limelighted the fact that Newtonian dynamics (ND) and general relativity (GR) have not been verified at accelerations below MOND's $a_0$. In particular, we do not know that all the principles underlying ND or GR apply below $a_0$. I discuss possible breakdown of general covariance (GC) in this limit. This resonates well with MOND, which hinges on accelerations. Relaxing GC affords more freedom in constructing MOND theories. I exemplify this with a simplified theory whose gravitational Lagrangian is $\mathcal{L}_M\propto \ell_M^{-2}\mathcal{F}(\ell_M^{2}\mathcal{R})$, where $\mathcal{R}= g^{\mu\nu} (\Gamma^\gamma_{\mu\nu}\Gamma^\lambda_{\lambda\gamma}-\Gamma^\gamma_{\mu\lambda} \Gamma^\lambda_{\nu\gamma})/2$. $\Gamma^\gamma_{\mu\nu}$ is the Levi-Civita connection of a metric, $g_{\mu\nu}$, and $\ell_M=c^2/a_0$ is the MOND length. Requiring $\mathcal{F}(z)\rightarrow z+\zeta$, for $z\gg 1$ gives GR with a cosmological constant $\zeta c^{-4}a_0^2$ for high accelerations. In the MOND limit $\mathcal{F}'(z\ll 1)\propto z^{1/2}$. In the nonrelativistic limit the metric is of the form $g_{\mu\nu}\approx \eta_{\mu\nu}-2\phi\delta_{\mu\nu}$, as in GR, but the potential $\phi$ solves a MOND, nonlinear Poisson analog. This form of $g_{\mu\nu}$ also produces gravitational lensing as in GR only with the MOND potential. I show that this theory is a fixed-gauge expression of BIMOND, with the auxiliary metric constrained to be flat. The latter theory is thus a covariantized version of the former a-la St\"{u}ckelberg. This theory is also a special case of so-called $f(\mathcal{Q})$ theories -- aquadratic generalizations of `symmetric, teleparallel GR', which are, in turn, also equivalent to constrained BIMOND-type theories. (Abridged.)