Why Are Off-Diagonal Ricci Tensors Zero in Symmetric FRW Metrics?

  • Context: Graduate 
  • Thread starter Thread starter pleasehelpmeno
  • Start date Start date
  • Tags Tags
    Ricci tensor Tensor
Click For Summary

Discussion Overview

The discussion revolves around the conditions under which off-diagonal Ricci tensors are zero in symmetric Friedmann-Robertson-Walker (FRW) metrics. It explores the implications of symmetry and isotropy in the context of specific coordinate systems.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant seeks a formal definition regarding the zero value of off-diagonal Ricci tensors in symmetric FRW metrics.
  • Another participant questions whether the discussion pertains to a specific set of coordinates, indicating a potential dependency on the coordinate choice.
  • A participant confirms the coordinates as (t, r, θ, φ) and suggests that the symmetry of the metric leads to the conclusion that off-diagonal components must be zero.
  • It is proposed that the off-diagonal component R_{θt} changes sign under a transformation that preserves isotropy, implying that if it were nonzero, it would contradict the isotropic nature of the metric.
  • The same participant notes that the diagonal component R_{θθ} remains unchanged under this transformation, reinforcing the argument about symmetry.

Areas of Agreement / Disagreement

Participants express differing views on the implications of symmetry and coordinate choice, indicating that the discussion remains unresolved regarding the formal justification for the zero values of off-diagonal Ricci tensors.

pleasehelpmeno
Messages
154
Reaction score
0
Hi does anyone know a formal definition for why off diagonal ricci tensors are equal to zero in a symmetric standard FRW metric?
 
Physics news on Phys.org
Is this in a particular set of coordinates?
 
yeah sorry (t,r,\theta,\phi)
 
I think it follows from symmetry. For instance, [itex]R_{\theta t}[/itex] flips signs under a change of coordinates [itex]\theta\rightarrow \pi-\theta[/itex], so if it was nonzero it would violate isotropy. Under the same transformation [itex]R_{\theta\theta}[/itex] stays the same.
 
  • Like
Likes   Reactions: yancey

Similar threads

Graduate Question about Weinberg's GR book
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad Calculate Ricci Scalar & Cosm. Const of AdS-Schwarzschild Metric in d-Dimensions
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Can a conformal flat metric be curved?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad How do non-diagonal indices of a metric allow for local flatness?
  • · Replies 22 ·
Replies
22
Views
2K
Graduate Ricci tensor of schwarzschild metric
  • · Replies 6 ·
Replies
6
Views
3K
Graduate Calculating Ricci tensor in AdS space
  • · Replies 8 ·
Replies
8
Views
3K
Does anyone know which are Ricci and Riemann Tensors of FRW metric?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Energy distribution on Alcubierre walls
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Diagonal Matrix of Stress-Energy Tensor: Why?
  • · Replies 17 ·
Replies
17
Views
3K
Undergrad Ricci tensor for Schwarzschild metric
  • · Replies 7 ·
Replies
7
Views
7K