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:

${\begin{aligned}{\Bigg \{}&\left({\frac {(r^{2}+a^{2})^{2}}{\Delta }}-a^{2}(1-z^{2})\right){\frac {\partial ^{2}}{\partial t^{2}}}+{\frac {4ar}{\Delta }}{\frac {\partial ^{2}}{\partial t\partial \phi }}+\left({\frac {a^{2}}{\Delta }}-{\frac {1}{1-z^{2}}}\right){\frac {\partial ^{2}}{\partial \phi ^{2}}}-\Delta ^{-s}{\frac {\partial }{\partial r}}\left(\Delta ^{s+1}{\frac {\partial }{\partial r}}\right)-{\frac {\partial }{\partial z}}\left((1-z^{2}){\frac {\partial }{\partial z}}\right)\\&-2s\left({\frac {a(r-1)}{\Delta }}+{\frac {iz}{1-z^{2}}}\right){\frac {\partial }{\partial \phi }}-2s\left({\frac {r^{2}-a^{2}}{\Delta }}-r-iaz\right){\frac {\partial }{\partial t}}+s{\frac {(s+1)z^{2}-1}{1-z^{2}}}{\Bigg \}}\Psi _{s}=4\pi \Sigma T_{s}\end{aligned}}$

Radial equation

$\left\{\Delta ^{-s}{\frac {d}{dr}}\left(\Delta ^{s+1}{\frac {d}{dr}}\right)+{\frac {K^{2}-2is(r-1)K}{\Delta }}+4is\omega r-\lambda \right\}{_{s}R_{lm\omega }}(r)={_{s}T_{lm\omega }}(r)$