- #1
binbagsss
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- 11
I'm looking at Tod and Hughston Introduction to GR and writing the metric in the two forms;
[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.
[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.