Bimetric MOND gravity

A new relativistic formulation of MOND is advanced, involving two metrics as independent degrees of freedom: the MOND metric ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$, to which alone matter couples, and an auxiliary metric ${\stackrel{^}{g}}_{\ensuremath{\mu}\ensuremath{\nu}}$. The main idea hinges on the fact that we can form tensors from the difference of the Levi-Civita connections of the two metrics, ${C}_{\ensuremath{\beta}\ensuremath{\gamma}}^{\ensuremath{\alpha}}={\ensuremath{\Gamma}}_{\ensuremath{\beta}\ensuremath{\gamma}}^{\ensuremath{\alpha}}\ensuremath{-}{\stackrel{^}{\ensuremath{\Gamma}}}_{\ensuremath{\beta}\ensuremath{\gamma}}^{\ensuremath{\alpha}}$, and these act like gravitational accelerations. In the context of MOND, we can form dimensionless ``acceleration'' scalars and functions thereof (containing only first derivatives) from contractions of ${a}_{0}^{\ensuremath{-}1}{C}_{\ensuremath{\beta}\ensuremath{\gamma}}^{\ensuremath{\alpha}}$. I look at a subclass of bimetric MOND theories governed by the action $I=\ensuremath{-}(16\ensuremath{\pi}G{)}^{\ensuremath{-}1}\ensuremath{\int}[\ensuremath{\beta}{g}^{1/2}R+\ensuremath{\alpha}{\stackrel{^}{g}}^{1/2}\stackrel{^}{R}\ensuremath{-}2(g\stackrel{^}{g}{)}^{1/4}f(\ensuremath{\kappa}){a}_{0}^{2}\mathcal{M}(\stackrel{\texttildelow{}}{\ensuremath{\Upsilon}}/{a}_{0}^{2})]{d}^{4}x+{I}_{M}({g}_{\ensuremath{\mu}\ensuremath{\nu}},{\ensuremath{\psi}}_{i})+{\stackrel{^}{I}}_{M}({\stackrel{^}{g}}_{\ensuremath{\mu}\ensuremath{\nu}},{\ensuremath{\chi}}_{i})$, with $\stackrel{\texttildelow{}}{\ensuremath{\Upsilon}}$ as a scalar quadratic in the ${C}_{\ensuremath{\beta}\ensuremath{\gamma}}^{\ensuremath{\alpha}}$, $\ensuremath{\kappa}=(g/\stackrel{^}{g}{)}^{1/4}$, ${I}_{M}$ as the matter action, and allow for the existence of twin matter that couples to ${\stackrel{^}{g}}_{\ensuremath{\mu}\ensuremath{\nu}}$ alone. Thus, gravity is modified not by modifying the elasticity of the space-time in which matter lives, but by the interaction between that space-time and the auxiliary one. In particular, I concentrate on the interesting and simple choice $\stackrel{\texttildelow{}}{\ensuremath{\Upsilon}}\ensuremath{\propto}{g}^{\ensuremath{\mu}\ensuremath{\nu}}({C}_{\ensuremath{\mu}\ensuremath{\lambda}}^{\ensuremath{\gamma}}{C}_{\ensuremath{\nu}\ensuremath{\gamma}}^{\ensuremath{\lambda}}\ensuremath{-}{C}_{\ensuremath{\mu}\ensuremath{\nu}}^{\ensuremath{\gamma}}{C}_{\ensuremath{\lambda}\ensuremath{\gamma}}^{\ensuremath{\lambda}})$. This theory introduces only one new constant, ${a}_{0}$; it tends simply to general relativity (GR) in the limit ${a}_{0}\ensuremath{\rightarrow}0$ and to a phenomenologically valid MOND theory in the nonrelativistic limit. The theory naturally gives MOND and ``dark energy'' effects from the same term in the action, both controlled by the MOND constant ${a}_{0}$. In regards to gravitational lensing by nonrelativistic systems--a holy grail for relativistic MOND theories--the theory predicts that the same potential that controls massive-particle motion also dictates lensing in the same way as in GR: Lensing and massive-particle probing of galactic fields will require the same ``halo'' of dark matter to explain the departure of the present theory from GR. This last result can be modified with other choices of $\stackrel{\texttildelow{}}{\ensuremath{\Upsilon}}$, but lensing is still enhanced and MOND-like, with an effective logarithmic potential.