Bimetric MOND gravity
  A new relativistic formulation of MOND is advanced, involving two metrics as
independent degrees of freedom: the MOND metric g_mn, to which alone matter
couples, and an auxiliary metric g*_mn. The main idea hinges on the fact that
we can form tensors from the difference, C^a_bc, of the Levi-Civita connections
of the two metrics, and these act like gravitational accelerations. In the
context of MOND we can form dimensionless `acceleration' scalars, and functions
thereof, from contractions of C^a_bc/a0. I look at a class of bimetric MOND
theories governed by an action with the gravitational Lagrangian density b
sqrt(g)R+a sqrt(g*) R* -2(gg*)^{1/4}f(k)a0^2M(U/a0^2), and with matter actions
I(g_mn,psi)+I*(g*_mn,chi), with U a scalar quadratic in the C^a_bc,
k=(g/g*)^{1/4}, and allowing for the existence of twin matter, chi, that
couples to g*_mn alone. In particular, I concentrate on one interesting and
simple choice of the scalar U. This theory introduces only one new constant,
a0; it tends simply to General Relativity in the limit a0 goes 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 a0. As regards 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. This
last result can be modified with other choices of U, but lensing is still
enhanced and MOND-like, with an effective logarithmic potential.
