User:TimothyRias/Teukolsky Equation

The Teukolsky Equation is a complex scalar partial differential equation describing linear perturbations of fields on Kerr black holes (or more generally spacetimes of Petrov Type D).

Teukolsky Variables
d

The equation
The Kerr metric in (modified) Boyer-Lindquist coordinates (t,r,z=cos(Θ), φ)
 * $$ds^2 =-(1 - \frac{2r}{\Sigma})dt^2 + \frac{\Sigma}{\Delta} dr^2 + \frac{\Sigma}{1-z^2} dz^2

+ \frac{1-z^2}{\Sigma} (2a^2 r (1-z^2)+(a^2+r^2)\Sigma)d\phi^2- \frac{4ar(1-z^2)}{\Sigma}dtd\phi $$ with
 * $$\Delta = r(r-2) + a^2$$
 * $$\Sigma= r^2 + a^2 z^2$$

Teukolsky master equation:

\begin{align} \Bigg\{ &\left(\frac{(r^2+a^2)^2}{\Delta}-a^2 (1-z^2)\right)\frac{\partial^2}{\partial t^2} +\frac{4ar}{\Delta}\frac{\partial^2}{\partial t\partial \phi} +\left(\frac{a^2}{\Delta}-\frac{1}{1-z^2}\right)\frac{\partial^2}{\partial \phi^2} -\Delta^{-s} \frac{\partial}{\partial r}\left(\Delta^{s+1}\frac{\partial}{\partial r}\right) -\frac{\partial}{\partial z}\left((1-z^2)\frac{\partial}{\partial z}\right)\\ &-2s\left(\frac{a(r-1)}{\Delta}+\frac{iz}{1-z^2}\right)\frac{\partial}{\partial \phi} -2s\left(\frac{r^2-a^2}{\Delta}-r-iaz\right)\frac{\partial}{\partial t} +s \frac{(s+1)z^2-1}{1-z^2} \Bigg\}\Psi_s = 4\pi\Sigma T_s \end{align} $$ Radial equation
 * $$\left\{\Delta^{-s}\frac{d}{dr}\left(\Delta^{s+1}\frac{d}{dr}\right) +\frac{K^2-2is(r-1)K}{\Delta}+4is\omega r -\lambda  \right\}{_{s}R_{lm\omega}}(r) = {_{s}T_{lm\omega}}(r)

$$