# Maximal symmetry at the speed of light

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.

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.
• You should use stars *stars*, not $math mode$ $math mode$`, for italics. I have edited accordingly. – LSpice Oct 24 '19 at 16:14
• How about taking $\chi = t-z$ in your example? That leaves 3 translations and 3 Lorentz transformations and the orbits are certainly light-like, right? – Timothy Budd Oct 27 '19 at 11:25
• Nice example, indeed the isometry group is 6-dimensional with three translations (in the $x$, the $y$ and the lightlike $t-z$ direction), one rotation around the $z$ axis and two Lorentz boosts (in the $x$ and the $y$ direction). The orbits are 3-dimensional indexed by constant $\chi=t-z$ and they are all of the same type, $(0,-,-)$. For cosmological applications, I would like furthermore the isometry group to contain the entire rotation group. – Thomas Schucker Oct 28 '19 at 14:27