Are there examples of 1 + 3 dimensional pseudo-Riemannian manifolds with 6 dimensional isometry group whose orbits are light-like (i.e., the metric restricted to each orbit is degenerate)?

Here is a (spatially flat) example in local coordinates $t$, $x$, $y$, $z$: $$ d\tau^2= b(\chi) ^2[dt^2-dx^2-dy^2-dz^2], \quad \chi :=t^2-x^2-y^2-z^2, $$ with a positive, monotonic function $b$, the `scale factor'. The isometry group is the Lorentz group (at least its connected component of the identity) generated by 6 Killing vectors: 3 infinitesimal rotations $R_x=y\,\partial_z- z\,\partial_y$, $R_y$, $R_z$ and 3 infinitesimal boosts $L_x=x\,\partial_t+t\,\partial_x$, $L_y$, $L_z$. The orbits are 3-dimensional and indexed by $\chi $: the light cone, $\chi =0$, is light-like $(0,-,-)$; the family of orbits with positive $\chi$, they are space-like $(-,-,-)$ and lie in the interior of the light cone; the family of orbits with negative $\chi$, they have signature $(+,-,-)$ and lie outside.

I would be happy to see an example with an entire family of light-like orbits.

About my motivation:

The cosmological principle postulates spacetimes with maximal symmetry on hyper-surfaces of *simultaneity*. A spatially flat example of such a spacetime is the one above but with the scale factor $b(\chi)$ replaced by $a(t)$. Simultaneity is ill defined in relativity and cosmological models using the cosmological principle together with Einstein's equation suffer from the horizon problem. The proposed cures involve infinitely many parameters and resemble epicycles.

This post imported from StackExchange MathOverflow at 2020-01-06 16:19 (UTC), posted by SE-user Thomas Schucker