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We consider a massless charged scalar field $\Phi$ on the Reissner–Nördstrom black hole space:

$$ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\theta^2+r^2\sin^2\theta\,d\varphi^2, \tag{1}$$

with $f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}$ and known horizons $r_\pm=M\pm\sqrt{M^2-Q^2}$. The dynamics of the scalar field is determined by

$$g^{\mu\nu}D_\mu D_\nu\Phi=0,\quad\text{with}\quad D_\mu=\nabla_\mu-iq A_\mu,\quad\text{and }\quad A_\mu=\left(-\frac{Q}{r},0,0,0\right). \tag{2}$$

The scalar field modes can be written as

$$\phi_{\omega\ell m}=\frac{e^{-i\omega t}}{r}\mathcal{N}_\omega X_{\omega\ell}(r)Y_{\ell m}(\theta,\varphi), \tag{3}$$

where $\ell=0,1,\dots$, $m=-\ell,\dots,\ell$; $X_{\omega\ell}(r)$ are radial functions, $Y_{\ell m}(\theta,\varphi)$ are the usual spherical harmonics and $\mathcal{N}_{\omega}$ is a normalization factor, which is the point of my question.

The tortoise coordinate $r_{\ast}$ is defined as $dr_{\ast}=f(r)^{-1}dr$, which lead to the following radial equation:

$$\left(-\frac{d^2}{dr_{\ast}^2}+V(r)\right)X_{\omega\ell}=0, \tag{4}$$

with a potential that has the following asymptotic values:

$$V(r)=\begin{cases} -\tilde{\omega}=-\left(\omega-\frac{qQ}{r_+}\right), & \text{ if } r_\ast\rightarrow-\infty\;(r\rightarrow r_+) \\ -\omega, & \text{ if } r_\ast\rightarrow+\infty\;(r\rightarrow\infty) \end{cases}. \tag{5}$$

The asymptotic behavior of the radial part of the solution modes is the following:

$$X_{\omega\ell}^{(in)}(r)=\begin{cases} T^{(in)}_{\omega\ell}e^{-i\tilde{\omega}r_\ast}, & \text{ if } r_\ast\rightarrow-\infty\;(r\rightarrow r_+) \\ \boxed{I^{(in)}_{\omega\ell}e^{-i\omega r_\ast}}+R^{(in)}_{\omega\ell}e^{i\omega r_\ast}, & \text{ if } r_\ast\rightarrow+\infty\;(r\rightarrow\infty) \end{cases}, \tag{6}$$

which are called in modes represents to waves incoming from the past null infinity, are partly reflected back to the future null infinity and also partly transmitted to the future horizon. In the same way, the so-called up modes:

$$X_{\omega\ell}^{(up)}(r)=\begin{cases} \boxed{I^{(up)}_{\omega\ell}e^{i\tilde{\omega}r_\ast}}+R^{(up)}_{\omega\ell}e^{-i\tilde{\omega}r_\ast}, & \text{ if } r_\ast\rightarrow-\infty\;(r\rightarrow r_+) \\ T^{(up)}_{\omega\ell}e^{i\omega r_\ast}, & \text{ if } r_\ast\rightarrow+\infty\;(r\rightarrow\infty) \end{cases}, \tag{7}$$

which represent waves outgoing near the past event horizon, partly reflected back to the future horizon and party transmitted to the future null infinity.

Question:

To find the normalization constant $\mathcal{N}_{\omega}$, it is defined the Klein-Gordon inner product as

$$\langle\Phi_1\Phi_2\rangle=i\int_\Sigma\left(\left(D_\mu\Phi_1\right)^\ast\Phi_2-\Phi_1^\ast\left(D_\mu\Phi_2\right)\right)\sqrt{-g}\;d\Sigma^\mu, \tag{8}$$

where I know that $d\Sigma^\mu=d\Sigma\;n^\mu$. The statement says that by using a Cauchy surface close to the union of the past event horizon and past null infinity for the in and up modes, we find, as computed in this paper (eq. 16)

$$\mathcal{N}_{\omega\ell}=\frac{1}{\sqrt{4\pi|\omega|}}, \tag{9}\label{9}$$

which seems to be a quick exercise of computation as in the flat KG inner product computed here and here. I dont understand the configuration previous to the computation: a Cauchy surface close to the union of the past event horizon and past null infinity, and how this would lead to \eqref{9}.

EDIT: Use the Penrose diagram:

My reasoning is as follows. In the Penrose diagram for a RN black hole,

penrose_RN_bh

I must choose a Cauchy surface near the past null infinite and the past event horizon. So, it seems that I must consider only the $I_{\omega\ell}^{in/up}$ parts of the "waves", and then to compute the KG inner product. But it is weird to me to have a Cauchy surface near that regions. Also, I dont know if my arrowed waves (red/blue) depicting the process are correct.

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    $\begingroup$ Maybe it will help if you provided a link to the original paper/book where you got the above calculations. Someone may find the answer given the entire context of the particular article. $\endgroup$
    – joseph h
    Aug 24, 2021 at 4:20
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    $\begingroup$ @josephh, of course. Just edited my question. $\endgroup$ Aug 24, 2021 at 4:33
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    $\begingroup$ Is your green curve the Cauchy surface You have in mind ? I would have thought it should belong to region I, very close to the past horizon and the past null infinity, but "above" them $\endgroup$ Aug 25, 2021 at 12:40
  • $\begingroup$ @AlanGarbarz, yes; indeed, it is very close and above the intersection between the past horizon and the past null infinity. It was a drawing mistake. I will correct the picture. thanks $\endgroup$ Aug 25, 2021 at 13:01

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