The Line Elements in the Hubble Expansion

In this lecture I present the line elements that express the Hubble expansion. The coordinates here are the spatial coordinates $x$, $y$, $z$, and the velocity coordinate $v$, which are actually what astronomers use in their measurements. Two such line elements are presented: the first is the empty space (no matter exists), and the second with matter filling up the Universe. These line elements are the comparable to the standard ones, the Minkowski and the FRW line elements in ordinary general relativity theory.