In the bimetric scalar-tensor gravitational theory there are two frames associated with the two metrics ĝ_μν and g_μν, which are linked by the gradients of a scalar field ϕ. The choice of a comoving frame for the metric ĝ_μν or g_μν has fundamental consequences for local observers in either metric spacetimes, while maintaining diffeomorphism invariance. When the metric g_μν is chosen to be associated with comoving coordinates, then the speed of light varies in the frame with the metric ĝ_μν.

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... ervers 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 ϕ describes a dark energy component in the Friedmann equation for the cosmic scale without acceleration. If we choose ĝ_μν to be associated with comoving coordinates, then an observer in the g_μν 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 α=e^2/ħ c does not vary. The problem of cosmological horizons as viewed in the two frames is discussed.