Chapter 6: Geodesic Deviation

### Rain Coordinates II

Recall from § Rain Coordinates that \begin{align} \sigma^T &= dt + \frac{\sqrt{\frac{2m}{r}}}{1-\frac{2m}{r}} \>dr \\ \sigma^R &= \frac{dr}{1-\frac{2m}{r}} + \sqrt{\frac{2m}{r}} \>dt \end{align} which allows us to introduce rain coordinates ($T$,$R$) defined by \begin{align} dT &= dt + \sqrt{\frac{2m}{r}}\frac{dr}{1-2m/r} \\ dR &= \sqrt{\frac{r}{2m}}\frac{dr}{1-2m/r} + dt \end{align} The Schwarzschild line element in rain coordinates then takes the form $$ds^2 = -dT^2 + \frac{2m}{r} dR^2 + r^2 \left( d\theta^2 + \sin^2\theta\,d\phi^2 \right)$$ and a straightforward but lengthy computation (see § Rain Curvature) shows that the (independent, nonzero) curvature 2-forms in these coordinates are \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}

But what is the physical meaning of these expressions?