The components $R^i{}_{jk\ell}$ of the curvature 2-forms form a tensor, known as the Riemann curvature tensor. Tensor components can be computed in any basis, then converted to any other basis using the appropriate change-of-basis transformations. Put differently, tensors are linear maps on vectors, and can therefore easily be evaluated on any vectors, not just on basis vectors.

As an example, we recompute the tidal forces on neighboring falling objects in the Schwarzschild geometry, but this time work in Schwarzschild coordinates rather than rain coordinates. The discussion in the previous sections leads to \begin{equation} \ddot{\uu} = -R^s{}_{\tau s\tau}\,\uu \end{equation} and it remains to compute $-R^s{}_{\tau s\tau}$. Assuming that the $s$ direction is spatial, so that the up and down indices don't matter, we can think of this expression as the invariant object “$-R(\hat{s},\hat{\tau},\hat{s},\hat{\tau})$”, where $\hat{s}$ and $\hat{\tau}$ denote the unit vectors in the $s$ and $\tau$ directions, and where $R$ is linear in each argument. But the $\tau$ direction corresponds to \begin{equation} \TT = \frac{1}{\sqrt{1-2m/r}}\left(\That-\sqrt{\frac{2m}{r}}\>\rhat\right) \end{equation} so that by linearity we must have \begin{equation} R^s{}_{\tau s\tau} = \frac{1}{1-2m/r} R^s{}_{tst} + \frac{2m/r}{1-2m/r} R^s{}_{rsr} \end{equation} If the $s$ direction corresponds to $\phat$, we obtain \begin{align} R^\phi{}_{\tau\phi\tau} &= \frac{1}{1-2m/r} R^\phi{}_{t\phi t} + \frac{2m/r}{1-2m/r} R^\phi{}_{r\phi r} \nonumber\\ &= \frac{1}{1-2m/r} \frac{m}{r^3} - \frac{2m/r}{1-2m/r} \frac{m}{r^3} = \frac{m}{r^3} \end{align} and if the $s$ direction corresponds to \begin{equation} \RHAT = \frac{1}{\sqrt{1-2m/r}} \left(\rhat-\sqrt{\frac{2m}{r}}\>\That\right) \end{equation} then \begin{align} R^R{}_{\tau R\tau} &= \frac{1}{1-2m/r} R^r{}_{trt} + \frac{2m/r}{1-2m/r} R^t{}_{rtr} \nonumber\\ &= -\frac{1}{1-2m/r} \frac{2m}{r^3} + \frac{2m/r}{1-2m/r} \frac{2m}{r^3} = -\frac{2m}{r^3} \end{align} (since terms involving $R^t{}_{ttt}$ or $R^r{}_{rrr}$ vanish).

These expressions agree with those previously computed in rain coordinates.