I am attempting to prove that the FLRW metric given by $$ds^2 = -dt^2 + g_{ij}dx^idx^j = -dt^2 + a^2(t)\left(d\vec{x}^2+K\frac{(\vec{x}\cdot d\vec{x})^2}{1-K\vec{x}^2}\right)$$ has $$R_{ij} = \left[\frac{a^{\prime\prime}}{a} + 2 \left(\frac{a^\prime}{a}\right)^2+\frac{K}{a^2}\right]g_{ij}.$$ The other cases, namely $R_{00}$ and $R_{0i}$ are easy enough to derive, however I am struggling to see how the above result is obtained. I am yet to find a source online that explicitly derives the result for the Ricci tensor in the above case. I am self-taught, and rely entirely upon online resources, so if someone could show me how this was derived, I would very much appreciate it. I think my confusion lies in calculating the Christoffel symbols in this case. It seems like it would be quite messy in Cartesian coordinates. There must surely be an easier way?
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$\begingroup$ It'd be useful to see where your calculation with the Christoffel symbols has got to (and why it hasn't worked)? $\endgroup$– EletieApr 23, 2021 at 18:02
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$\begingroup$ I hasn't got far at all, other than a general statement of $\Gamma_{jk}^i$ in terms of the metric function (this is all that many texts seem to give). I can calculate them fine when one or more index is 0, but when all three are spatial I seem to have a mental block. Where do I go from here? I await your answer to this problem, given your interest in it. $\endgroup$– wrb98Apr 23, 2021 at 19:43
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$\begingroup$ Sorry, missed your laste sentence. Why don't you try spherical coordinates? As far as I remember, it works out quite nicely in spherical coordinates. $\endgroup$– PhotonApr 23, 2021 at 19:50
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$\begingroup$ Would that give the same answer? As far as I was aware, the $i$, $j$ etc. notation referred to the $x$, $y$ and $z$ components in Cartesian coordinates $\endgroup$– wrb98Apr 23, 2021 at 19:57
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$\begingroup$ @wrb98 latin indices just refer to the first three spatial components whatever they may be, Greek refer to all four values of the index which includes the temporal pieces. $\endgroup$– TriatticusApr 23, 2021 at 20:51
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