Recent measurements of the microwave radiation remaining from the big bang suggest that our 3D space is flat.
This note considers a metric for space time with flat time slices and that also is natural for a universe with Hubble expansion.
For coordinate system (t, x, y, z) the covariant metric is:
| −t−2 | 0 | 0 | 0 |
| 0 | t−2| 0 | 0
| | 0 | 0 | t−2
| 0 | | 0 | 0 | 0 | t−2 | | |
The manifold is for t<0.
Points with constant x, y & z recede from each other just as Hubble discovered.
Since gtt depends on t, coordinate time t differs from a real clock which will read T = −log(−t) + c.
As t runs from −∞ to 0, T runs from −∞ to +∞.
There is no ‘big bang’ in this metric for all real values of T are included.
The distance between a pair of points with constant coordinates is proportional to eT.
The distance between the points is accelerating, as in our universe.
See this.
The peculiar time coordinate makes 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
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 clock time T goes from −∞ to ∞ as coordinate time t goes from −∞ to 0.
This is clearly not the standard big bang model.
The distance to galaxies (with constant spatial coordinates) increases exponentially.
Following is Wrong as it stands:
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 negative 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.
??
Indeed <t, x, y, z> → <t+a, eax, eay, eaz> is an isometry.