I am trying to quantize a real scalar field in the interior of a rotating black hole (3+1 D, asymptotically flat). My question is regarding the modes of the radial part of the equation (obtained after separation of variables).
In tortoise coordinates, the radial equation near the outer horizon is of the form: $$ \partial^2_{r_\star}\psi_{\omega ml}(r_\star) = -\omega_+^2\psi_{\omega ml}(r_\star) $$ and near the inner horizon: $$ \partial^2_{r_\star}\psi_{\omega ml}(r_\star) = -\omega_-^2\psi_{\omega ml}(r_\star) $$ m,l are quantum numbers obtained from the angular part. Here $\omega_\pm = \omega - m\Omega_\pm$. ($\Omega_\pm$ are the angular velocities of the rotation of the horizons and depend on $m$ and $r_\pm$). I write two bases of solutions for this equation, the first one is $\{\psi_{\omega ml}^{R+}\psi_{\omega ml}^{L+}\}$ determined by the boundary conditions at the outer horizon (note that in the interior $r_\star$ is timelike): $$ \psi^{R+}_{\omega ml} \sim \frac{1}{\sqrt{2\omega_+}}e^{-i\omega_+ r_\star} \;\;\;\; \psi^{L+}_{\omega ml} \sim \frac{1}{\sqrt{2\omega_+}}e^{i\omega_+ r_\star} \;\;\;\;\;\; \text{as} \;\; r_\star \rightarrow -\infty \;\;(\text{i.e.}\;\; r\rightarrow r_+\;\;\; \text{from inside}) $$ Similarly i write another basis $\{\psi_{\omega ml}^{R-}\psi_{\omega ml}^{L-}\}$ which is determined by boundary conditions on the inner horizon: $$ \psi^{R-}_{\omega ml} \sim \frac{1}{\sqrt{2\omega_-}}e^{-i\omega_- r_\star} \;\;\;\; \psi^{L-}_{\omega ml} \sim \frac{1}{\sqrt{2\omega_-}}e^{i\omega_- r_\star} \;\;\;\;\;\; \text{as} \;\; r_\star \rightarrow \infty \;\;(\text{i.e.}\;\; r\rightarrow r_-\;\;\; \text{from outside}) $$
I can relate the two sets of solutions using bogoliubov coefficients: $$ \psi^{R+}_{\omega ml} = A_{\omega ml} \psi_{\omega ml}^{R-} + B_{\omega ml}\psi^{L-}_{\omega ml} \;\;\;\;\;\;\; \psi_{\omega ml}^{L+} = (\psi_{\omega ml}^{R+})^\star = B_{\omega ml}^\star \psi_{\omega ml}^{R-} + A_{\omega ml}^\star\psi^{L-}_{\omega ml} \;\;\;\;\;\;\;\;...(1)$$ I write the full field expansion as: $$ \hat\Phi = \sum_{l,m}\int_0^\infty \frac{d\omega_+}{\sqrt{2\pi(r^2 + a^2)}}\left(\hat c_{\omega ml} \psi^{R+}_{\omega ml} + \hat d^\dagger_{\omega ml} \psi^{L+}_{\omega ml}\right)Z_{\omega ml}(\theta,\varphi)e^{-i\omega t} \;\; + \text{h.c.} $$ Here $Z_{\omega ml}$ are solutions of the angular parts (which evidently are spheroidal harmonics). Alternatively i can also expand the field as: $$ \hat\Phi = \sum_{l,m}\int_0^\infty \frac{d\omega_-}{\sqrt{2\pi(r^2 + a^2)}}\left(\hat e_{\omega ml} \psi^{R-}_{\omega ml} + \hat f^\dagger_{\omega ml} \psi^{L-}_{\omega ml}\right)Z_{\omega ml}(\theta,\varphi)e^{-i\omega t} \;\; + \text{h.c.} $$ I'm trying to figure out how I can relate the mode operators $c$ and $d$ to the mode operators $e$ and $f$ in terms of the bogoliubov coefficients. The primary problem i'm facing is that the integral in the two expansions are over different omega (one has an integral over positive $\omega_+$ while the other over positive $\omega_-$.). So when i substitute (1) in the expansion, I do not obtain the other expansion. It'd be great if someone can point me in the right direction.
