modes will be given below.) The label $n$ distinguishes between modes incoming from the past null infinity ${\cal J}^-$ (denoted with $n = \leftarrow$) and those coming out from the past horizon $H^{-}$ (denoted with $n = \rightarrow$).[^2] The solutions with $A_\mu^{(0n;\,\omega lm)} = 0$, $(\mu \neq t)$, and $$A_t^{(0n;\,\omega l m)} = R_{\omega l}^{(0n)}(r)Y_{lm}e^{-i\omega t}$$ will be called “non-physical modes” because they satisfy the field equations (\[eqt\])-(\[eqang\]) but not the gauge condition $G=0$. Here $Y_{lm}$ is a scalar spherical harmonic on the unit $p$-sphere with $\tilde{\nabla}^2 Y_{lm} = -l(l+p-1)Y_{lm}$, where $l=0,1,2,\ldots$ and $m$ denotes a set of $p-1$ integers $(m_1,\ldots,m_{p-1})$ satisfying $l\ge m_{p-1}\ge \ldots \ge m_{2} \ge |m_1|$. (See Ref. [@CM] for a concise description of spherical harmonics on the $p$-sphere.) They are normalized as $$\int d\Omega_p \overline{Y_{lm}}Y_{l'm'} = \delta_{ll'}\delta_{mm'}\,,$$ where $d\Omega_p$ is the volume element of the unit $p$-sphere. The function $R_{\omega l}^{(0n)}(r)$ satisfies $$\left[ \frac{\omega^2}{f} 
+ \sqrt{\frac{f}{h}}\frac{1}{r^p}\frac{d}{dr}
\left( \frac{r^p}{\sqrt{fh}} \frac{d}{dr}
\right) - \frac{l(l+p-1)}{r^2}\right]R_{\omega l}^{(0n)}= 0\,.  \label{eqrA_t}$$ We will determine the normalization of the functions $R_{\omega l}^{(0n)}(r)$ later. The pure-gauge modes are given as $$A_\mu^{(3n;\,\omega lm)}=\nabla_{\mu}\Lambda^{(n\omega lm)}\,,$$ where $$\Lambda^{(n\omega lm)}= \frac{i}{\omega}
R_{\omega l}^{(0n)}(r)Y_{lm}e^{-i\omega t}
\,.$$ (As usual, the pure-gauge modes satisfy the field equations (\[eqt\])-(\[eqang\]) and $G=0$.) The 