Calculate Radial Transformation

The areal coordinate $\hat{r}$ of each sphere is calculated by

$$\hat{r} = \hat{r}(r) = \left[ \frac{1}{4\pi} \int\sqrt{\g... ...heta \theta } \gamma_{\phi \phi }}d\theta d\phi \right]^{1/2}$$ (part3121)

from which

$$\frac{d\hat{r}}{d\eta} = \frac{1}{16\pi \hat{r}} \int\frac{... ...{\gamma_{\theta \theta }\gamma_{\phi \phi }}} \ d\theta d\phi$$ (part3122)

Note that this is not the only way to combine metric components to get the areal radius, but this one was used because it gave better values for extracting close to the event horizon for perturbations of black holes.