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FRW metric in higher dimensions (fast question?)

  1. Feb 3, 2015 #1


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    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?
  2. jcsd
  3. Feb 3, 2015 #2


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    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.
  4. Feb 3, 2015 #3


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    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.
  5. Feb 3, 2015 #4


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    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.
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