Regarding Quasi-normal modes of black holes

Question

I am a Ph.D. student working on quasinormal modes of black holes. I am following the paper https://ui.adsabs.harvard.edu/abs/1985ApJ...291L..33S/abstract which is perhaps the first paper on calculating quasinormal modes using WKB approzimation upto 1st order. The basic equation that describes perturbation of a schwarzschild black hole is

d$^2$ $\psi$/dr$_*$$^2$ + {$\sigma$ $^2$ - [1 - (2/r)][$\lambda$/r$^2$ + (2$\beta$/r$^3$)]} $\psi$ = 0 (Equation 7 in tha paper).

From my understanding , the term [1 - (2/r)][$\lambda$/r$^2$ + (2$\beta$/r$^3$)] represents potential. (Please correct me if I am wrong). My doubt is

a] from where this potential is coming from ? Is this regge wheeler potential ?

b] Why $\beta$ = 1, 0, -3 for scalar, electromagnetic and gravitational perturbations respectively ?

It would be very helpful if someone could clear my doubts or provide any helpful refernces. Thank you in advance!

Answer 1

The equation in question in is the (radial) Teukolsky equation, which describes black hole perturbations. It was derived by Saul Teukolsky in this 1973 paper. One of the remarkable aspects of the Teukolsky equation is that you can write a single equation with just one free parameter $s$ that describes the spin of the perturbing field. When $s=0$ the equation describes (massless) scalar perturbations. When $s=\pm 1$, the equation describes (massless) vector perturbations the most obvious example being the electromagnetic field, and when $s=\pm 2$ it describes (massless) tensor perturbations with the most obvious example the gravitational field. (Any half integer will work.)

In the equation of the paper you mention they use $\beta= 2s+1$ for a scalar (s=0), EM (s=-1) and gravitational (s=-2) perturbation.

The equation in question is specialized to Schwarzschild, but the Teukolsky equation can also be written for perturbations of rotating Kerr black holes.