  In the bimetric scalar-tensor gravitational theory there are two frames
associated with the two metrics {\hat g}_{\mu\nu} and g_{\mu\nu}, which are
linked by the gradients of a scalar field \phi. The choice of a comoving frame
for the metric {\hat g}_{\mu\nu} or g_{\mu\nu} has fundamental consequences for
local observers in either metric spacetimes, while maintaining diffeomorphism
invariance. When the metric g_{\mu\nu} is chosen to be associated with comoving
coordinates, then the speed of light varies in the frame with the metric {\hat
g}_{\mu\nu}. Observers in this frame see the dimming of supernovae because of
the increase of the luminosity distance versus red shift, due to an increasing
speed of light in the early universe. Moreover, in this frame the scalar field
\phi describes a dark energy component in the Friedmann equation for the cosmic
scale without acceleration. If we choose {\hat g}_{\mu\nu} to be associated
with comoving coordinates, then an observer in the g_{\mu\nu} metric frame will
observe the universe to be accelerating and the supernovae will appear to be
farther away. The theory predicts that the gravitational constant G can vary in
spacetime, while the fine-structure constant \alpha=e^2/\hbar c does not vary.
The problem of cosmological horizons as viewed in the two frames is discussed.
