We have so far related curvature to geodesic deviation by working in rain coordinates, a special coordinate system adapted to the radial geodesics we chose to study. But this relationship is geometric, and therefore independent of the coordinates we choose. To demonstrate this geometric invariance, we revisit the problem of geodesic deviation using our original Schwarzschild coordinates. First, we need the connection.

A straightforward but lengthy computation (see §Schwarzschild Curvature) shows that the (independent, nonzero) curvature 2-forms in these coordinates take the form \begin{align} \Omega^t{}_r &= \frac{2m}{r^3} \, \sigma^t \wedge \sigma^r \\ \Omega^t{}_\theta &= -\frac{m}{r^3} \, \sigma^t \wedge \sigma^\theta \\ \Omega^t{}_\phi &= -\frac{m}{r^3} \, \sigma^t \wedge \sigma^\phi \\ \Omega^r{}_\theta &= -\frac{m}{r^3} \, \sigma^r \wedge \sigma^\theta \\ \Omega^r{}_\phi &= -\frac{m}{r^3} \, \sigma^r \wedge \sigma^\phi \\ \Omega^\theta{}_\phi &= \frac{2m}{r^3} \, \sigma^\theta \wedge \sigma^\phi \end{align}

Remarkably, these expressions are formally the same as the corresponding expressions in rain coordinates.