Kerr horizon mode and shifted horizon frequency

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M>0$, $0<\mathrm{\mid }a\mathrm{\mid }<M$, and use the length coordinates of the subextremal Kerr datum. Use the smooth ingoing extension with its proved compatible time orientation $\bar{N}={\mathrm{\partial }}_v-(2Mr\mathrm{/}\mathrm{\Sigma }){\mathrm{\partial }}_r$ from Kerr horizon angular velocity and the ergoregion. Set $H^{+}:=\{r=r_{+}\}$ and let $P\subset H^{+}$ be a smooth open patch in the hypersurface topology. Let $T$ and $\mathrm{\Phi }$ be the stationary and axial Killing fields restricted to $P$, and let $\chi :=T+{\mathrm{\Omega }}_H\mathrm{\Phi }$ with ${\mathrm{\Omega }}_H:=a\mathrm{/}(r_{+}^2+a^2)$ be the future null generator there; the supplier proves $g(\chi ,\bar{N})=-1$ on $H^{+}$. A smooth complex scalar $u\in C^{\mathrm{\infty }}(P,\mathbb{C})$ is an $(\omega ,m)$ horizon mode when $\omega \in \mathbb{R}$, $m\in \mathbb{Z}$, and

$Tu=-i\omega u,\mathrm{\Phi }u=imu\mathrm{.}$

Its shifted horizon frequency is $\tilde{\omega }:=\omega -m{\mathrm{\Omega }}_H$. By linearity of $\chi$, every such mode satisfies $\chi u=-i\tilde{\omega }u$. The parameter $\omega$ has units reciprocal to the Kerr time coordinate, $m$ is dimensionless, and ${\mathrm{\Omega }}_H$ has the same units as $\omega$. This is a local definition of mode data only; it does not assert that a nonzero mode extends to a global solution of the wave equation.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Subextremal Kerr exterior in length coordinates

Kerr horizon angular velocity and the ergoregion