# Notational(?) Question in Whiting's Paper “Mode Stability of the Kerr Black Hole”

I am a math grad student attempting to read Bernard Whiting's paper "Mode Stability of the Kerr Black Hole." If you are in a university network, the paper should be easily found by a google search. At the beginning of the paper Whiting refers to the "master equation of Teukolsky" from the paper "Rotating Black Holes: Separable Wave Equations for Gravitational and Electromagnetic Perturbations." However, the equation Whiting writes down appears (to me) to be different than Teukolsky's. I assume I am missing something silly and was hoping that someone else has read this paper and can easily point out my mistake.

One needs to look at the paper for all the relevant definitions, but for ease of reference:

Whiting writes down

$\{\partial_r\Delta\partial_r - (1/\Delta)\{(r^2+a^2)\partial_t + a\partial_{\phi} - (r-M)s\}^2 - 4s(r+ia\cos\theta)\partial_t +$ $\partial_{\cos\theta}\sin^2\theta\partial_{\cos\theta} + (1/\sin^2\theta)\{a\sin^2\theta\partial_t + \partial_{\phi} + i\cos\theta\cdot s\}^2\}\psi = 0$

where $\Delta = r^2 - 2Mr + a^2$

while Teukolsky writes down

$\{(\frac{(r^2+a^2)^2}{\Delta} - a^2\sin^2\theta)\partial^2_t + \frac{4Mar}{\Delta}\partial^2_{t,\phi} + (\frac{a^2}{\Delta} - \frac{1}{\sin^2\theta})\partial^2_{\phi} - \Delta^{-s}\partial_r(\Delta^{s+1}\partial_r) -$ $\frac{1}{\sin\theta} \partial_{\theta}(\sin\theta\partial_{\theta}) - 2s(\frac{a(r-M)}{\Delta} + \frac{i\cos\theta}{\sin^2\theta})\partial_{\phi} - 2s(\frac{M(r^2-a^2)}{\Delta}-r-ia\cos\theta)\partial_t$ $+ (s^2\cot^2\theta - s)\}\psi = 0$

When I expanded out Whiting's equation out I did not get Teukolsky's equation. For example, it is clear that the $\partial_r$ terms are different. I also got that the constant terms (no derivative terms) did not agree.

Probably the definition of $\psi$ are slightly different? Judging by the fact that you get a different $\partial_r$ term and a different constant term, I am guess that Whiting's equation is Teukolsky's but commuted against $\Delta^l$ for some power $l$. (In other words, in Whiting's equation, replace $\psi \to \Delta^l\psi$ and divide the entire equation by $\Delta^l$, you should be able to find a $l$ (which depends on $s$, I think) that makes the two equation appear the same.)
(Observe that $\Delta^l$ commutes through the entire equation except for terms which have derivatives in $r$.)
• And if it works, it is the same trick used in deriving Huygen's principle from spherical means ... (yes, I read your Generals transcript ;-)) – Willie Wong Jun 24 '11 at 2:35
• The factor is $\Delta^{s/2}$ – mmeent Jul 26 '18 at 7:56