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I am struggling to understand a general method to calculate the independent components of the Riemann Curvature Tensor (RCT).

Firstly, as far as I am aware the number of independent components of the RCT for a $d$-dimensional manifold is given by (1) which is equal to 1 for $d=2$.

$$\frac{d^2(d^2-1)}{12}\tag1$$

The question at hand deals with the simple 2-dimensional metric in (2).

$$ds^2=-dt^2+tdr^2\tag2$$

By calculating the Christoffel symbols I obtained $R^t_{rtr}=-\frac{1}{4t}$ and $R^r_{trt}=\frac{1}{4t^2}$ which is apparently correct. However here I have become stuck. I was under the impression that once I had identified one component of the RTC then I could easily work out the rest of them using the symmetry properties of the RCT (as there is only one independent component in 2-dimensions). The symmetry properties that I used are as follows:

$$R^\rho{}_{\sigma{}\mu{}\nu}=-R^\sigma{}_{\rho{}\mu{}\nu}\tag3$$

$$R^\rho{}_{\sigma{}\mu{}\nu}=-R^\rho{}_{\sigma{}\nu{}\mu}\tag4$$

$$R^\rho{}_{\sigma{}\mu{}\nu}=-R^\rho{}_{\mu{}\nu\sigma{}}\tag5$$

However, I can't see how the component's that I found follow these rules. Furthermore, once solutions were provided, the lecturer used a scaling factor $a^2(t)$ which I understand to be equal to $t$ (in this particular example) to calculate the components.

Would someone be able to correct my understanding of independent RCT components and also explain the use of a scaling factor to calculate the components?

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  • $\begingroup$ You can't just swap an up index with a down index and expect anything simple to happen. That operation involves two applications of metric tensor. (At time of writing the answer gives the symmetry properties correctly). $\endgroup$ Commented Feb 20 at 18:56

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Your symmetry properties do not make sense. The correct ones are $$ R_{\rho\sigma\mu\nu} = - R_{\sigma\rho\mu\nu} $$ $$ R^\rho{}_{\sigma\mu\nu} = - R^\rho{}_{\sigma\nu\mu} $$ $$ R^\rho{}_{\sigma\mu\nu} = - R^\rho{}_{\nu\sigma\mu} - R^\rho{}_{\mu\nu\sigma} $$

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