FRW Metric in d Dimensions: Can I Expand?

[ChrisVer]

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


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

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

 

[PeterDonis]

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.

 

[Orodruin]

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.

 

[ChrisVer]

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 $A^2(t) d \mu$ instead of $R^2(t)$ to the original 3 spatial dimensions.