I'm looking at Tod and Hughston Introduction to GR and writing the metric in the two forms;(adsbygoogle = window.adsbygoogle || []).push({});

[1]##ds^{2}=dt^{2}-R^{2}(t)(\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2}))##

[2] ##ds^{2}=dt^{2}-R^{2}(t)g_{ij}dx^{i}dx^{j}##

where

##g_{ij}dx^{i}dx^{j}=d\chi^{2}+\chi^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2}) ## for ##k=0##

##=d\chi^{2}+sin^{2}\chi{2}(d\theta^{2}+sin^{2}\theta d\phi^{2}) ## for ##k=1##

##=d\chi^{2}+sinh^{2}\chi{2}(d\theta^{2}+sin^{2}\theta d\phi^{2}) ## for ##k=-1##

Now in solving for the form [2] Tod computes the Ricci scalar of ##ds^{2}=d\chi^{2}+f^{2}(\chi)(d\theta^{2}+sin^{2}\theta d\phi^{2})## and finds ##R=-(2\frac{f''}{f}-\frac{1}{f^{2}}+\frac{(f')^{2}}{f^{2}}## then integrates, uses ##R=3k## and solves for all 3 cases ##k=0,\pm 1##.

My question

##R=3k## doesn't seem right to me, since in 3-d space we can write ##R_{ab}=2kg_{ab}##. Of course you could just define a constant ##K=2k##, but it uses the constant ##k## in the FRW metric of the form [1] not ##k##, comparing to Introduction to GR lecture notes by sean M.Caroll,I thought that this should be ##R=6k##

...In Caroll's notes he uses ##R_{ab}=2kg_{ab}## in the derivation and gives the FRW metric the same as in form [1] with small ##k##. So it doesn't look as though Tod has used ##K=2k##.

Can anyone help explain how Tod uses ##R=3k##?

Thanks.

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# R=6k or R=3k confused, FRW metric derivation, Maximally symm

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