Teukolsky Equation

The resulting master equation is commonly known as the Teukolsky equation, which was derived in the Boyer–Lindquist coordinates, and can be written in the following form:

${\displaystyle \begin{array}{c}\left[\frac{{\left(r^2+a^2\right)}^2}{\mathrm{\Delta }}-a^2{\sin }^2\theta \right]\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{t}^2}+\frac{4Mar}{\mathrm{\Delta }}\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }t\mathrm{\partial }\varphi }+\left[\frac{a^2}{\mathrm{\Delta }}-\frac{1}{{\sin }^2\theta }\right]\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\varphi }^2}\\ -{\mathrm{\Delta }}^{-s}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left({\mathrm{\Delta }}^{s+1}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }r}\right)-\frac{1}{\sin \theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(\sin \theta \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\theta }\right)\\ -2s\left[\frac{a\left(r-M\right)}{\mathrm{\Delta }}+\frac{i\cos \theta }{{\sin }^2\theta }\right]\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\varphi }\\ -2s\left[\frac{M\left(r^2-a^2\right)}{\mathrm{\Delta }}-r-ia\cos \theta \right]\frac{\mathrm{\partial }\psi }{\mathrm{\partial }t}\\ +\left(s^2{\cot }^2\theta -s\right)\psi =4\pi \mathrm{\Sigma }T.\end{array}}$

Here, ${\displaystyle M}$ is the mass of the black hole; ${\displaystyle a}$ is the angular momentum per unit mass, which corresponds to rotation; ${\displaystyle \mathrm{\Delta }=r^2-2Mr+a^2}$; ${\displaystyle \mathrm{\Sigma }=r^2+a^2{\cos }^2\theta }$; ${\displaystyle \psi }$ the perturbing field for the Kerr black hole; ${\displaystyle T}$ is the stress–energy tensor (source term); and ${\displaystyle s}$ the different types of perturbations, such that ${\displaystyle s=0,\pm \frac{1}{2},\pm 1,\pm 2}$, which depict scalar, neutrino, electromagnetic, and gravitational perturbations respectively.

This equation enables investigation of small disturbances around a Kerr black hole. This was foundational because the Kerr geometry is less symmetric than the Schwarzschild geometry, making it less obvious that the perturbation equations could be decoupled and separated nicely into radial and angular ordinary differential equations. Teukolsky’s review paper on the Kerr metric reviews all the current research for Kerr such as black hole stability, quasinormal modes, and others, in which some of these areas of current research for Kerr revolve around the Teukolsky equation.