FRW from spaces of constant cuvature

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Discussion Overview

The discussion revolves around the derivation of the Friedmann-Robertson-Walker (FRW) metric from spaces of constant curvature, specifically in the context of the cosmological principle which posits that space is homogeneous and isotropic. Participants seek clarification on the mathematical foundations and derivations related to the curvature metrics associated with different values of k (-1, 0, +1).

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant expresses difficulty in finding a derivation of the FRW metrics from constant curvature spaces and requests assistance.
  • Another participant suggests consulting a specific chapter from a set of lecture notes, indicating it may contain relevant information.
  • A follow-up inquiry is made about specific equations and figures in the suggested lecture notes, indicating a need for further clarification on those points.
  • One participant reiterates their struggle to locate the necessary derivations and references a mathematical statement regarding the curvature tensor of N-dimensional Riemannian spaces, seeking help with the derivation of that calculation.
  • Access issues to the suggested lecture notes are mentioned, with one participant noting that the link was initially accessible but later became restricted.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the derivation of the FRW metrics or the specific content of the suggested lecture notes. Multiple requests for clarification and assistance indicate ongoing uncertainty and lack of resolution.

Contextual Notes

Limitations include potential access issues to external resources, as well as the need for specific mathematical derivations that remain unresolved within the discussion.

Mike2
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FRW from spaces of constant curvature

I'm reading how the Friedmann Robertson Walker metric (with its k=-1,0,+1) is derived by considering spaces of constant curvature in order to satisfy the cosmological principle that space is the same everywhere. But I'm not able to find any derivation of these metrics of constant curvature. And that's where we get k=-1,0,+1. Any help out there? Thanks.
 
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Try chapter 2 of this lectures:
http://astro.uwaterloo.ca/~mjhudson/teaching/phys787/
 
Last edited by a moderator:
hellfire said:
Try chapter 2 of this lectures:
http://astro.uwaterloo.ca/~mjhudson/teaching/phys787/

Thanks for the tip. I'm trying to go through it. Where did he come up with equation 2.11? What is S() in Fig 2.1? Thanks.
 
Last edited by a moderator:
I don't seem to be able to find it.
 
Mike2 said:
I'm reading how the Friedmann Robertson Walker metric (with its k=-1,0,+1) is derived by considering spaces of constant curvature in order to satisfy the cosmological principle that space is the same everywhere. But I'm not able to find any derivation of these metrics of constant curvature. And that's where we get k=-1,0,+1. Any help out there? Thanks.
The best I've been able to find is statements like the following:
"An N dimensional Riemannian space is of constant curvature if its curvature tensor obeys

[tex]\[R_{\mu \nu \gamma \lambda } = \frac{R}{{N(N - 1)}}(g_{\mu \gamma } g_{\nu \lambda } - g_{\mu \lambda } g_{\nu \gamma } ),\,\,\,\,\,\,\,R = const.,\][/tex]

where R/N(N-1) = [tex]\varepsilon\[/tex]K-2 is called the Gaussian curvature."

Can someone show me how that calculation was derived? Or would it more appropriate to post this question to the Tensor Analysis & Differential Geometry forum? Thanks.
 
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hellfire said:
Try chapter 2 of this lectures:
http://astro.uwaterloo.ca/~mjhudson/teaching/phys787/

Unfortunately they are blocking access to this.
 
Last edited by a moderator:
I am sorry, when I posted the link the access was free to all chapters of the lectures.
 

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