# FRW metric symmetries

## Main Question or Discussion Point

What are the symmetries determined by FRW spacetime? I guess they include Lorentz symmetry, rotationally and translationally symmetries, but not time symmetry. Is this right?
Thanks

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What are the symmetries determined by FRW spacetime? I guess they include Lorentz symmetry, rotationally and translationally symmetries, but not time symmetry. Is this right?
Thanks
FRW metric is the unique metric to present a universe which is isotropic and homogeneous in spatial part not in time. By rotational symmetry we mean isotropy and by translational symmetry we mean homogeneity of the universe. However the universe is not symmetry respect to time, if so, I think it couldn't evolve.

And Lorentz invariance of the metric comes from being conformally flat?

bcrowell
Staff Emeritus
Gold Member
When one talks about classifying the symmetries of a spacetime, usually that means listing its Killing vectors. Killing vectors are global. The FRW metrics have 6 Killing vectors, which I believe correspond to the 3 translations (homogeneity) and 3 rotations (isotropy).

Lorentz transformations are defined locally, not globally. I don't think it is even possible to define the global equivalent of a Lorentz boost in an FRW metric. Even if you could, it definitely wouldn't be a Killing vector of the metric, because FRW metrics have a preferred frame that is at rest with respect to the local matter.

I don't think conformal flatness connects in any way to Lorentz transformations. The FRW metrics have to be conformally flat because any deviation from conformal flatness would imply the existence of tidal forces at a given point, but due to isotropy there is no preferred direction for the tidal forces.

Aside from isotropy and homogeneity, general relativity requires that locally
(eg, near the origin) the line element be invariant under Lorentz transformations.

Lorentz transformations are defined locally, not globally.

I don't think conformal flatness connects in any way to Lorentz transformations.
I should have specified that I was referring to local Lorentz invariance. And conformally flat simply means that FRW metric can be obtained from Minkowski metric by an angle-preserving transformation so I figured it might also preserve Lorentz invariance locally. But I guess talking about a local property is not the right context for symmetries.

Ich