I am currently with a difficulty in deriving the space-space components of the Ricci tensor in the flat FLRW metric $$ds^2 = -c^2dt^2 + a^2(t)[dx^2 + dy^2 + dz^2],$$ to find: $$R_{ij} = \delta_{ij}[2\dot{a}^2 + \ddot{a}a].$$
How can I do that?
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3$\begingroup$ Do you understand the formulas for the Ricci tensor, the Riemann tensor, and the Christoffel symbols? If so, what difficulty are you having applying them in the case of the flat FLRW metric? $\endgroup$– GhosterCommented Aug 8, 2023 at 23:13
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1$\begingroup$ Please, show step-by-step. This kind of request will probably ensure that your question gets closed as “homework-like”. $\endgroup$– GhosterCommented Aug 8, 2023 at 23:14
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$\begingroup$ I do understand the formulas for the Ricci tensor and the Christoffel symbol, but i think i am struggling with the algebra in the final steps because i am finding a very similar result (in despite of the factor 2 in front of the a dot) $\endgroup$– gabrielCommented Aug 8, 2023 at 23:27
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$\begingroup$ If you have checked your algebra over and over, then maybe there is something about the formulas that you don’t understand. Can you get $R_{tt}=-3\ddot a/a$? $\endgroup$– GhosterCommented Aug 8, 2023 at 23:37
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$\begingroup$ Although this calculation doesn’t need a computer algebra system, a CAS would help you find your error. $\endgroup$– GhosterCommented Aug 8, 2023 at 23:40
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