FRW from spaces of constant cuvature

[Mike2]

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.

 

[hellfire]

Try chapter 2 of this lectures:
http://astro.uwaterloo.ca/~mjhudson/teaching/phys787/

 

[Mike2]

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.

 

[quantum123]

I don't seem to be able to find it.

 

[Mike2]

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

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

where R/N(N-1) = \varepsilon\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.

 

[Hacky]

Try chapter 2 of this lectures:
http://astro.uwaterloo.ca/~mjhudson/teaching/phys787/

Unfortunately they are blocking access to this.

 

[hellfire]

I am sorry, when I posted the link the access was free to all chapters of the lectures.