Independent Components of the Riemann Curvature Tensor

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?

• 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). Commented Feb 20 at 18:56

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