What Geometrical Symmetry Do FRW Metrics Exhibit?

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Discussion Overview

The discussion centers on the geometrical symmetry of Friedmann-Robertson-Walker (FRW) metrics, exploring concepts such as isotropy, homogeneity, and the implications of different coordinate charts. Participants examine the nature of symmetry in the context of both spatial and spacetime manifolds.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants assert that FRW metrics exhibit isotropy and homogeneity, indicating no special direction or location.
  • Others question whether the symmetry of a manifold is dependent on the choice of coordinate chart, suggesting that the FRW metric's properties may vary based on this choice.
  • One participant clarifies that homogeneity requires uniformity in the gravitational field and matter density across space.
  • There is a discussion about whether the FRW metric is one of the maximally symmetric metrics in 4D spacetime.
  • Some participants note that while the spatial components may exhibit certain symmetries, the entire spacetime manifold does not necessarily share these properties.

Areas of Agreement / Disagreement

Participants express differing views on the nature of symmetry in FRW metrics, with no consensus reached regarding the implications of coordinate choice or the characterization of the entire manifold.

Contextual Notes

Limitations include the dependence on the definitions of isotropy and homogeneity, as well as the unresolved nature of how these concepts apply to the entire spacetime manifold versus spatial components.

TrickyDicky
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Does anybody know what kind of geometrical symmetry FRW metrics present? I know it's not spherically symmetric, but I think I recall having read it shows radial symmetry.
 
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TrickyDicky said:
Does anybody know what kind of geometrical symmetry FRW metrics present? I know it's not spherically symmetric, but I think I recall having read it shows radial symmetry.

Isotropy and homogeneity. No special direction, and no special place. Geometrically it depends which coordinate chart you use.
 
Mentz114 said:
Geometrically it depends which coordinate chart you use.

I thought the symmetry in a manifold (like spherical symmetry in Schwarzschild manifold for instance) was not dependent on the coordinate chart choice. Maybe you mean what FRW specifically I refer to, let's say the one with flat 3-space.
 
Mentz114 said:
Isotropy and homogeneity.

Mentz, that would be the spatial component only, right? I meant the whole spacetime FRW manifold.
 
TrickyDicky said:
Mentz, that would be the spatial component only, right? I meant the whole spacetime FRW manifold.
Homogeneity requires that the gravitational field should be the same everywhere, so g00 cannot depend on the position. Also the matter density should not depend on position. The (only?) metrics that satisfy this have the form given here
http://scienceworld.wolfram.com/physics/Robertson-WalkerLineElement.html.

Also see
http://www.phys.washington.edu/users/dbkaplan/555/lecture_03.pdf.
 
TrickyDicky said:
Does anybody know what kind of geometrical symmetry FRW metrics present? I know it's not spherically symmetric, but I think I recall having read it shows radial symmetry.

Why isn't the FRW metric spherically symmetric?
Is it one of the maximally symmetric metric of 4D space-time?

thx
 
micomaco86572 said:
Why isn't the FRW metric spherically symmetric?
The space - like hypersurfaces are, just not the entire manifold.
 

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