# Maximal symmetry at the speed of light

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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.

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

asked Oct 9, 2019
edited Jan 6
I assume that you mean by 'the orbits are lightlike' that the metric restricts to each group orbit to be degenerate?

This post imported from StackExchange MathOverflow at 2020-01-06 16:19 (UTC), posted by SE-user Robert Bryant
You should use stars *stars*, not $math mode$ $math mode$, for italics. I have edited accordingly.

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

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

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

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