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I was trying to find components of the Riemann tensor and it occurred to me that there could be an issue with my notation. For example, if one particular component of the tensor is

$$ R^{\theta}{}_{t\theta t}, $$

and I wanted to find the component $R^t{}_{\theta\theta t}$, then I could

$$ R^t{}_{\theta\theta t}=g^{tt}(-g_{\theta\theta} R^{\theta}{}_{t\theta t}); $$ i.e. lower the $\theta$ index with the $g_{\theta\theta}$ component of the metric, then use the fact that $R_{t\theta\theta t}=-R_{\theta t\theta t}$ and finally, raise the $t$ index with the $g^{tt}$ component of the inverse metric.

I know that there should only be two indices the same on any one term, but, in this case these indices are neither free nor being summed over. So, is this notation allowed and is it correct to use the metric in this way?

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The $\theta$'s and $t$'s here are not indices (i.e., dummy indices), they're specific values of the indices. For example, if you have coordinates $(t,r,\theta,\phi)$, then you could also notate $R^\theta{}_{t\theta t}$ as $R^2{}_{020}$.

The manipulation you did with lowering the index, applying antisymmetry, and raising it again is a valid manipulation, and there is no need to explicitly notate the raising and lowering using the metric. If you do want to explicitly notate it, then you need indices on the metric that are dummy indices, i.e., variables that take on more than one value. These days most people use Greek letters for concrete indices, so you'd have stuff like $g^{\mu\nu}$.

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