Bimetric MOND gravity

A new relativistic formulation of MOND is advanced, involving two metrics as independent degrees of freedom: the MOND metric $g_{\mu \nu }$, to which alone matter couples, and an auxiliary metric ${\widehat{g}}_{\mu \nu }$. The main idea hinges on the fact that we can form tensors from the difference of the Levi-Civita connections of the two metrics, $C_{\beta \gamma }^{\alpha }={\Gamma }_{\beta \gamma }^{\alpha }-{\widehat{\Gamma }}_{\beta \gamma }^{\alpha }$, 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 $a_0^{-1}{C}_{\beta \gamma }^{\alpha }$. I look at a subclass of bimetric MOND theories governed by the action $I=-(16\pi G{)}^{-1}\int [\beta g^{1/2}R+\alpha {\widehat{g}}^{1/2}\widehat{R}-2(g\widehat{g}{)}^{1/4}f(\kappa )a_0^2\mathcal{M}(\widetilde{{\rm Y}}/a_0^2)]d^{4}x+I_M(g_{\mu \nu },{\psi }_i)+{\widehat{I}}_M({\widehat{g}}_{\mu \nu },{\chi }_i)$, with $\widetilde{{\rm Y}}$ as a scalar quadratic in the $C_{\beta \gamma }^{\alpha }$, $\kappa =(g/\widehat{g}{)}^{1/4}$, $I_M$ as the matter action, and allow for the existence of twin matter that couples to ${\widehat{g}}_{\mu \nu }$ 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 $\widetilde{{\rm Y}}\propto g^{\mu \nu }(C_{\mu \lambda }^{\gamma }{C}_{\nu \gamma }^{\lambda }-C_{\mu \nu }^{\gamma }{C}_{\lambda \gamma }^{\lambda })$. This theory introduces only one new constant, $a_0$; it tends simply to general relativity (GR) in the limit $a_0\to 0$ 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 $\widetilde{{\rm Y}}$, but lensing is still enhanced and MOND-like, with an effective logarithmic potential.