Every relativistic particle has 4-speed equal to $c$, since $g_{ \mu \nu} \frac{ dx^\mu}{ d\tau} \frac{ dx^\nu}{ d\tau} = c^2$. With the choice of $k = +1$ in the FRW metric, the cosmological scale factor $a(t)$ has the natural interpretation of the radius of the sphere $S^3_a = \{ x \in \mathbb{ R}^4 : (x, x) = a^2\}$. Thus, a particle at rest in the cosmological frame has 4-speed equal to $\frac{ da}{ dt}$. This leads us to infer that $\dot a = c$, which respresents a simple kinematic constraint linking the speed of light to the cosmological scale factor. This drastically changes the $k=+1$ picture from a closed deaccelerating universe to an open accelerating universe, settles the horizon problem, and provides for a new cosmological model more appealing to our natural intuition. In this paper we shall consider ramifications of this model.
Categories: gr-qc astro-ph