Recently I attempted to compute the stress-energy tensor for the space that I guessed was shaped like an expansionary phase of our universe.
That attempt is stuck.
Before that I came across a modification to the Poincaré’s conformal map from Hyperbolic plane to the Euclidean half plane.
The modification was merely to change the metric from:

In both spaces the model half plane was {(x, y) | y>0}.
I wrote a program to plot geodesics based on the 2nd metric.
Here is the hyperbolic pseudo metric picture explained here.

I want to understand the symmetries of the space with metric diag(1, −e^{t}, −e^{t}, −e^{t}) for −∞<t<∞.
I think I show here that the metric diag(1/T^{2}, −1/T^{2}, −1/T^{2}, −1/T^{2}) for −∞<t<0 is isometric to the former.

This explores a generalization of a technique to model uniformly curved spaces.