FRW metric in higher dimensions (fast question?)

  • Thread starter ChrisVer
  • Start date
  • #1
ChrisVer
Gold Member
3,331
438

Main Question or Discussion Point

I was wondering if I can expand the FRW metric in d spatial dimensions, like:


[itex] g_{\mu \nu}^{frw} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & - \frac{a^2(t)}{1-kr^2} & 0 & 0 \\ 0 & 0 & - a^2(t) r^2 & 0 \\ 0 & 0 & 0 & -a^2 (t) r^2 \sin^2 \theta \end{pmatrix} \rightarrow g_{MN} = \begin{pmatrix} g_{\mu \nu}^{frw} & 0 \\ 0 & - 1_{(d-3) \times (d-3)} \end{pmatrix}[/itex]

Or should I insert a ##-a^2 (t) ## for the rest too?
 

Answers and Replies

  • #2
PeterDonis
Mentor
Insights Author
2019 Award
28,357
8,097
I haven't seen the FRW metric discussed for a different number of dimensions, but I would think the scale factor would have to be present in all the spatial terms.
 
  • #3
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,693
6,467
I haven't seen the FRW metric discussed for a different number of dimensions, but I would think the scale factor would have to be present in all the spatial terms.
Agreed. You would break isotropy otherwise unless I am not thinking straight... For isotropy to hold, the FRW metric in spherica coordinates should be pretty straight forward to generalise - just replace ##d\Omega^2## by the metric for the appropriate unit sphere.

This of course relies on the assumption tha you want to be isotropic also in the extra dimensions.
 
  • #4
ChrisVer
Gold Member
3,331
438
I think you are right. I found a paper
"Higher Dimensional FRW String Cosmological Models in a New Scalar-tensor Theory of Gravitation" where they added a fifth dimension in FRW with a term [itex] A^2(t) d \mu [/itex] instead of [itex]R^2(t)[/itex] to the original 3 spatial dimensions.
 

Related Threads on FRW metric in higher dimensions (fast question?)

  • Last Post
Replies
14
Views
1K
  • Last Post
Replies
6
Views
4K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
1K
Replies
3
Views
3K
Replies
9
Views
1K
  • Last Post
Replies
7
Views
3K
  • Last Post
Replies
5
Views
898
Top