What symmetries define FRW spacetime and their impact on expanding universes?

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SUMMARY

The discussion centers on the symmetries defining Friedmann-Robertson-Walker (FRW) spacetime, highlighting Lorentz symmetry, rotational symmetry, and translational symmetry, while clarifying the absence of time symmetry. The FRW metric is established as the unique representation of an isotropic and homogeneous universe, characterized by six Killing vectors corresponding to three translations and three rotations. The conversation emphasizes that Lorentz transformations are local rather than global, and discusses the implications of conformal flatness on the FRW metric, particularly in relation to tidal forces and local Lorentz invariance.

PREREQUISITES
  • Understanding of Friedmann-Robertson-Walker (FRW) metric
  • Familiarity with Killing vectors in differential geometry
  • Knowledge of Lorentz transformations and their local properties
  • Concept of isotropy and homogeneity in cosmology
NEXT STEPS
  • Research the implications of Killing vectors in general relativity
  • Study the concept of conformal flatness in cosmological models
  • Explore the relationship between cosmological redshift and expanding universes
  • Investigate local versus global properties of Lorentz invariance in spacetime
USEFUL FOR

Cosmologists, theoretical physicists, and students of general relativity seeking to deepen their understanding of the symmetries in expanding universes and the implications of the FRW metric.

TrickyDicky
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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
 
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TrickyDicky said:
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?
 
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.
 
bcrowell said:
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.
 
3 translations (homogeneity)
A little addition: this translation vector generally has a different proper length in each spatial slice. What's constant is its comoving length.
This property is intimately tied to the "loss of momentum" in expanding universes, including cosmological redshift.
 

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