Appendix: Mathematical Details

Rain Curvature

The basis 1-forms for the Schwarzschild geometry in rain coordinates are \begin{align} \sigma^T &= dT \\ \sigma^R &= \sqrt{\frac{2m}{r}}\>dR \\ \sigma^\theta &= r\,d\theta \\ \sigma^\phi &= r\,\sin\theta\,d\phi \end{align} where \begin{align} dT &= dt + \frac{\sqrt{\frac{2m}{r}}}{1-\frac{2m}{r}} \>dr \\ dR &= \sqrt{\frac{r}{2m}} \frac{dr}{1-\frac{2m}{r}} + dt \end{align} In these expressions, $r$ is an implicit function of $R$ and $T$, since $$dR - dT = \left(\sqrt{\frac{r}{2m}} - \sqrt{\frac{2m}{r}}\right) \frac{dr}{1-\frac{2m}{r}} = \sqrt{\frac{r}{2m}} \>dr$$ or equivalently $$dr = \sqrt{\frac{2m}{r}} \>(dR - dT) = \sigma^R - \sqrt{\frac{2m}{r}} \>\sigma^T$$

The structure equations therefore become \begin{align} d\sigma^T &= 0 = -\omega^T{}_m\wedge\sigma^m \\ d\sigma^R &= -\sqrt{\frac{m}{2r^3}} \>dr\wedge dR = \sqrt{\frac{m}{2r^3}} \>\sigma^T\wedge\sigma^R = -\omega^R{}_m\wedge\sigma^m \\ d\sigma^\theta &= dr\wedge d\theta = \left( \sigma^R - \sqrt{\frac{2m}{r}}\sigma^T \right) \wedge \frac{1}{r}\sigma^\theta = -\omega^\theta{}_m\wedge\sigma^m \\ d\sigma^\phi &= \sin\theta\,dr\wedge d\phi + r\,\cos\theta\,d\theta\wedge d\phi = -\omega^\phi{}_m\wedge\sigma^m \end{align} These equations suggest that \begin{align} \omega^R{}_T &= \sqrt{\frac{m}{2r^3}}\>\sigma^R = \frac{m}{r^2}\,dR \\ \omega^\theta{}_T &= -\sqrt{\frac{2m}{r^3}}\>\sigma^\theta = -\sqrt{\frac{2m}{r}}\>d\theta \\ \omega^\phi{}_T &= -\sqrt{\frac{2m}{r^3}}\>\sigma^\phi = -\sqrt{\frac{2m}{r}}\>\sin\theta\,d\phi \\ \omega^\theta{}_R &= \frac{1}{r}\,\sigma^\theta = d\theta \\ \omega^\phi{}_R &= \frac{1}{r}\,\sigma^\phi = \sin\theta\,d\phi \\ \omega^\phi{}_\theta &= \frac{1}{r}\cot\theta\,\sigma^\phi = \cos\theta\,d\phi \end{align} and it is easy to check that these educated guesses actually do satisfy the structure equations, remembering that \begin{align} \omega^T{}_R &= \omega^R{}_T, & \omega^T{}_\theta &= \omega^\theta{}_T, & \omega^T{}_\phi &= \omega^\phi{}_T, \nonumber\\ \omega^R{}_\theta &= -\omega^\theta{}_R, & \omega^R{}_\phi &= -\omega^\phi{}_R, & \omega^\theta{}_\phi &= -\omega^\phi{}_\theta \end{align} Since we are guaranteed a unique solution, we have found our connection 1-forms.

The curvature 2-forms are now given by \begin{align} \Omega^T{}_R = \Omega^R{}_T &= d\omega^R{}_T + \omega^R{}_m\wedge\omega^m{}_T \nonumber\\ &= -\frac{2m}{r^3} \>dr\wedge dR = \frac{2m}{r^3} \>\sigma^T\wedge\sigma^R \\ \Omega^T{}_\theta = \Omega^\theta{}_T &= d\omega^\theta{}_T + \omega^\theta{}_m\wedge\omega^m{}_T = d\omega^\theta{}_T + \omega^\theta{}_R\wedge\omega^R{}_T \nonumber\\ &= \frac{m}{r^2}\,(dR-dT)\wedge d\theta + d\theta\wedge \frac{m}{r^2}\,dR \nonumber\\ &= -\frac{m}{r^3}\,\sigma^T\wedge\sigma^\theta \\ \Omega^T{}_\phi = \Omega^\phi{}_T &= d\omega^\phi{}_T + \omega^\phi{}_m\wedge\omega^m{}_T = d\omega^\phi{}_T + \omega^\phi{}_R\wedge\omega^R{}_T \nonumber\\ &= \frac{m}{r^2}\,(dR-dT)\wedge \sin\theta\,d\phi + \sin\theta\,d\phi\wedge \frac{m}{r^2}\,dR \nonumber\\ &= -\frac{m}{r^3}\,\sigma^T\wedge\sigma^\phi \\ -\Omega^R{}_\theta = \Omega^\theta{}_R &= d\omega^\theta{}_R + \omega^\theta{}_m\wedge\omega^m{}_R = d\omega^\theta{}_R + \omega^\theta{}_T\wedge\omega^T{}_R \nonumber\\ &= 0 - \sqrt{\frac{2m}{r^3}}\>\sigma^\theta\wedge\sqrt{\frac{m}{2r^3}}\>\sigma^R = -\frac{m}{r^3}\,\sigma^\theta\wedge\sigma^R \\ -\Omega^R{}_\phi = \Omega^\phi{}_R &= d\omega^\phi{}_R + \omega^\phi{}_m\wedge\omega^m{}_R = d\omega^\phi{}_R + \omega^\phi{}_T\wedge\omega^T{}_R \nonumber\\ &= 0 - \sqrt{\frac{2m}{r^3}}\>\sigma^\phi\wedge\sqrt{\frac{m}{2r^3}}\>\sigma^R = -\frac{m}{r^3}\,\sigma^\phi\wedge\sigma^R \\ -\Omega^\theta{}_\phi = \Omega^\phi{}_\theta &= d\omega^\phi{}_\theta + \omega^\phi{}_m\wedge\omega^m{}_\theta = d\omega^\phi{}_\theta + \omega^\phi{}_R\wedge\omega^R{}_\theta + \omega^\phi{}_T\wedge\omega^T{}_\theta \nonumber\\ &= -\sin\theta\,d\theta\wedge d\phi - \sin\theta\,d\phi\wedge d\theta + \frac{2m}{r}\sin\theta\, d\phi\wedge\,d\theta \nonumber\\ &= \frac{2m}{r^3}\,\sigma^\phi\wedge\sigma^\theta \end{align} in agreement with the expressions given in §Rain Connection.