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

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Discussion Overview

The discussion centers on the symmetries associated with Friedmann-Robertson-Walker (FRW) spacetime and their implications for expanding universes. Participants explore the nature of these symmetries, including isotropy and homogeneity, and the role of Lorentz transformations and conformal flatness.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that FRW spacetime includes Lorentz symmetry, rotationally and translationally symmetries, but not time symmetry.
  • Others argue that the FRW metric is unique in representing an isotropic and homogeneous universe in spatial dimensions, while time symmetry is absent to allow for evolution.
  • A participant questions whether Lorentz invariance of the metric arises from being conformally flat.
  • It is noted that classifying symmetries typically involves listing Killing vectors, with FRW metrics having 6 Killing vectors corresponding to 3 translations (homogeneity) and 3 rotations (isotropy).
  • Some participants assert that Lorentz transformations are defined locally and not globally, suggesting that a global equivalent of a Lorentz boost may not exist in FRW metrics.
  • There is a discussion on the relationship between conformal flatness and Lorentz transformations, with some participants stating that conformal flatness does not connect to Lorentz transformations.
  • One participant adds that the translation vector has a different proper length in each spatial slice, with constant comoving length, linking this property to the "loss of momentum" in expanding universes and cosmological redshift.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between Lorentz transformations and FRW spacetime, as well as the implications of conformal flatness. There is no consensus on these points, indicating ongoing debate and exploration of the topic.

Contextual Notes

Participants mention limitations regarding the definitions of symmetries and the context in which local versus global properties are discussed. The discussion does not resolve these complexities.

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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