The covariant metric is

–t–2	0	0	0
0	t–2	0	0
0	0	t–2	0
0	0	0	t–2

Points with constant x, y & z recede from each other just as Hubble prescribes. The manifold is for t>0. Since g_tt is not constant, coordinate time t differs from a real clock which will read T = log(–t) + c. Indeed the distance between the points points is accelerating in fact, for which there is some evidence. The distance between a pair of points is proportional to e^T.

That the coefficients for the space dimensions diminish with time is just a convenient way to insure mutual recession for particles whose spatial coordinates are constant. I must first justify the peculiar metric which might more naturally use T instead t as the time coordinate. The time coordinate is shrinking to make the metric formally similar to Poincaré’s metric that I have met in other circumstances and where I have learned how to compute geodesics from the literature!

For the 2D space-time with metric

–t–2	0
0	t–2

the null geodesics are the 45 degree lines x = c+t and x = c–t. The map of geodesics extended in time are just hyperbolæ with both foci at t=0 and with asymptotes at 45 degrees, together with lines of constant x. (The map of geodesics of tachyons are hyperbolæ with foci (t, x, y, z) and (–t, x, y, z).)

Note that physics time goes back to –∞ as coordinate time t goes back to 0. This is clearly not the standard big bang model. Also the distance to galaxies increases exponentially. As far as I know this has not been falsified. It would require a non-zero cosmological constant.

I think that this is the deSitter space with novel parameterization. It needs a ‘one point compatification’ however which can be had by adding neighborhoods of the point at infinity of the form {(t, x, y, z) | t>a} for some unbounded set of positive a’s.

It is curious that a space slice for constant t yields a flat 3D space. Such subspaces do not emerge naturally from the other metrics.

This is the same space described here, I think.