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 $\mathcal{R}=g^{\mu \nu }({\Gamma }_{\mu \nu }^{\gamma }{\Gamma }_{\lambda \gamma }^{\lambda }-{\Gamma }_{\mu \lambda }^{\gamma }{\Gamma }_{\nu \gamma }^{\lambda })/2$. ${\Gamma }_{\mu \nu }^{\gamma }$ is the Levi-Civita connection of a metric, $g_{\mu \nu }$, and ${\ell }_M=c^2/a_0$ is the MOND length. Requiring $\mathcal{F}(z)\to z+\zeta$, for $z\gg 1$ gives GR with a cosmological constant $\zeta c^{-4}{a}_0^2$ for high accelerations. In the MOND limit ${\mathcal{F}}^{\prime }(z\ll 1)\propto z^{1/2}$. In the nonrelativistic limit the metric is of the form $g_{\mu \nu }\approx {\eta }_{\mu \nu }-2\varphi {\delta }_{\mu \nu }$, as in GR, but the potential $\varphi$ 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 Stückelberg. 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.)

Comments: 13 pages, version accepted for publication in Phys. Rev. D