Charged test-scalar scattering and superradiant flux

Let $M>0$ and $Q\in \mathbb{R}$ satisfy $0<\mathrm{\mid }Q\mathrm{\mid }<M$. Put $r_{\pm }:=M\pm \sqrt{M^2-Q^2},f(r):=1-\frac{2M}{r}+\frac{Q^2}{r^2},r>r_{+}\mathrm{.}$ Use the static exterior ${\mathcal{M}}_{\mathrm{e}\mathrm{x}\mathrm{t}}={\mathbb{R}}_t\times (r_{+},\mathrm{\infty }{)}_r\times S^2$ with the electric specialization $q_e=Q$, $q_m=0$, and $t=x^0$ of the Reissner--Nordström metric and potential fixed in the preceding definition. Thus $g=-fd{t}^2+f^{-1}d{r}^2+r^2{\gamma }_{S^2},A=-\frac{Q}{r}dt,A_t⟶0(r⟶\mathrm{\infty })\mathrm{.}$ Here $r_{+}$ is the positive outer root in the selected local model, not a claim about an event horizon in a maximal spacetime. Set ${\mathrm{\Phi }}_H:={\lim }_{r\to \mathrm{\infty }}{A}_t(r)-{\lim }_{r\downarrow r_{+}}{A}_t(r)=\frac{Q}{r_{+}}\mathrm{.}$ These are endpoint limits: $r=r_{+}$ and $r=\mathrm{\infty }$ are not points of ${\mathcal{M}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$.

For $e\in \mathbb{R}$, use the trivial complex line bundle ${\mathcal{M}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\times \mathbb{C}$ and regard $\psi$ as a smooth map ${\mathcal{M}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\to \mathbb{C}$. In a product chart put $D_a\psi :=({\mathrm{\partial }}_a-ie{A}_a)\psi$. Define the massless charged wave operator in the static chart by ${\mathrm{\square }}_{g,A}\psi :=-f^{-1}{D}_t^2\psi +\frac{1}{r^2}{\mathrm{\partial }}_r\text{\hspace{0.17em}⁣}\left(r^{2}f{\mathrm{\partial }}_r\psi \right)+\frac{1}{r^2}{\mathrm{\Delta }}_{{\gamma }_{S^2}}\psi ,D_t={\mathrm{\partial }}_t+\frac{ieQ}{r}\mathrm{.}$ The equation used here is ${\mathrm{\square }}_{g,A}\psi =0$; it is a fixed-background, minimally coupled test-field equation and does not evolve or source $g$ or $A$.

Let $\omega >0$ and $\mathrm{\ell }\in {\mathbb{N}}_0$. Supply a smooth real function $Y:S^2\to \mathbb{R}$ satisfying ${\int }_{S^2}{Y}^{2}d{\omega }_{S^2}=1,{\mathrm{\Delta }}_{{\gamma }_{S^2}}Y=-\mathrm{\ell }(\mathrm{\ell }+1)Y\mathrm{.}$ No existence assertion about $Y$ is made. Use the tortoise coordinate $r_{\ast }(r):={\int }_{2r_{+}}^{r}f(\rho {)}^{-1}d\rho ,$ which is strictly increasing because $f>0$ on the exterior. Under the subextremal hypotheses the supplied RN scalar-coordinate result proves that $r_{\ast }$ is a smooth coordinate from $(r_{+},\mathrm{\infty })$ onto $\mathbb{R}$; hence $r_{\ast }\to -\mathrm{\infty }$ as $r\downarrow r_{+}$ and $r_{\ast }\to +\mathrm{\infty }$ as $r\to \mathrm{\infty }$. A separated mode has the form $\psi (t,r,\mathrm{\Omega })=A_{\mathrm{i}\mathrm{n}}{e}^{-i\omega t}Y(\mathrm{\Omega })\frac{u(r_{\ast })}{r},$ where $A_{\mathrm{i}\mathrm{n}}\mathrm{\ne }0$ is a fixed incident amplitude with units $[\psi ]\mathrm{m}$, $u\in C^{\mathrm{\infty }}(\mathbb{R};\mathbb{C})$ is the normalized radial function, and its reflection and transmission coefficients are dimensionless.

Write ${\omega }_H:=\omega -e{\mathrm{\Phi }}_H=\omega -eQ\mathrm{/}{r}_{+}$ and, for any reference length $L>0$, define the Coulomb phase $\mathrm{\Theta }(r_{\ast }):=\omega r_{\ast }-eQ\log (r\mathrm{/}L),{\mathrm{\Theta }}^{\mathrm{\prime }}(r_{\ast }):=\omega -\frac{eQf(r)}{r}\mathrm{.}$ A normalized scattering solution is a smooth radial solution of ${\mathrm{\square }}_{g,A}\psi =0$ for which, for some $R,T\in \mathbb{C}$, $u=e^{-i\mathrm{\Theta }}+R{e}^{i\mathrm{\Theta }}+o(1),{\mathrm{\partial }}_{r_{\ast }}u=-i{\mathrm{\Theta }}^{\mathrm{\prime }}{e}^{-i\mathrm{\Theta }}+i{\mathrm{\Theta }}^{\mathrm{\prime }}R{e}^{i\mathrm{\Theta }}+o(1)(r⟶\mathrm{\infty }),$ and $u=T{e}^{-i{\omega }_H{r}_{\ast }}+o(1),{\mathrm{\partial }}_{r_{\ast }}u=-i{\omega }_{H}T{e}^{-i{\omega }_H{r}_{\ast }}+o(1)(r\downarrow r_{+})\mathrm{.}$ The error terms tend to zero in the indicated limits. These boundary conditions prescribe unit incoming amplitude at infinity. When ${\omega }_H\mathrm{\ne }0$, the condition as $r_{\ast }\to -\mathrm{\infty }$ selects the $e^{-i{\omega }_H{r}_{\ast }}$ branch at the outer-root end. At ${\omega }_H=0$, the two exponential phases coincide, so the condition does not distinguish ingoing from outgoing directions. They define admissible scattering data; they assert neither existence nor uniqueness of such a solution.

The logarithm in $\mathrm{\Theta }$ is required by the long-range term $-2\omega eQ\mathrm{/}r$ in the asymptotic radial coefficient. Changing $L$ changes constant phases, not flux magnitudes. All scattering data here use the gauge $A_t(\mathrm{\infty })=0$, so the mode label $\omega$ is the asymptotic gauge-covariant frequency. Under a constant shift $A_t^{\mathrm{\prime }}=A_t+C$, the field transformation ${\psi }^{\mathrm{\prime }}=e^{ieCt}\psi$ changes the coordinate mode label to ${\omega }^{\mathrm{\prime }}=\omega -eC$. The asymptotic kinetic frequency ${\omega }_{\mathrm{\infty }}:=\omega +e{\lim }_{r\to \mathrm{\infty }}{A}_t(r)$ is invariant because ${\omega }^{\mathrm{\prime }}+e{\lim }_{r\to \mathrm{\infty }}{A}_t^{\mathrm{\prime }}={\omega }_{\mathrm{\infty }},$ and ${\mathrm{\Phi }}_H$ is invariant because the constant cancels from its difference. Thus the invariant horizon frequency is ${\omega }_{\mathrm{\infty }}-e{\mathrm{\Phi }}_H$, which equals ${\omega }_H=\omega -e{\mathrm{\Phi }}_H$ in the chosen gauge.

Use geometric units $G=c=1$ and length coordinates, with $[t]=[r]=[r_{\ast }]=[M]=[Q]=[L]=\mathrm{m}$, $[e]=[\omega ]={\mathrm{m}}^{-1}$, $f$ dimensionless, and $A_t$ dimensionless; then $\mathrm{\Theta }$ is dimensionless and each term in ${\mathrm{\square }}_{g,A}$ has units $[\psi ]{\mathrm{m}}^{-2}$. The coupling $e$ is the charge coupling in the chosen field normalization. This item uses geometric units and asserts no SI value or conversion for $e$. This definition excludes scalar mass, magnetic charge, mirrors, anti-de Sitter boundaries, bound states, and complex-frequency quasinormal boundary conditions. A scalar mass parameter $\mu \ge 0$, with units ${\mathrm{m}}^{-1}$, would change the asymptotic propagating wave number (when $\omega >\mu$) to $\sqrt{{\omega }^2-{\mu }^2}$ and require a different scattering normalization. The real-frequency Wronskian identity makes no claim about complex-frequency mode problems with other boundary conditions.

Dyonic Reissner-Nordström metric and field domains

RN roots, Killing horizons and surface gravity

Subextremal RN scalar mode potential and conditional energy

The source conventions are not silently substituted for the local convention: the background one-form is exactly $A_t=-Q\mathrm{/}r$ and $D_t={\mathrm{\partial }}_t-ie{A}_t$. The long-range phase above is used instead of an unmodified plane-wave phase at infinity. No quantum pair-production claim, nonlinear instability, or coupled Einstein--Maxwell perturbation result is part of this definition.

Let the metric, gauge, parameters, tortoise coordinate, harmonic $Y$, and operator be exactly those of the charged radial scattering definition. For $\varpi (r):=\omega -eQ\mathrm{/}r$, a smooth separated solution $\psi =A_{\mathrm{i}\mathrm{n}}{e}^{-i\omega t}Y(\mathrm{\Omega })u(r_{\ast })\mathrm{/}r$ of ${\mathrm{\square }}_{g,A}\psi =0$ has radial function satisfying $\frac{d^{2}u}{d{r}_{\ast }^2}+[{(\omega -\frac{eQ}{r})}^2-f(r)(\frac{\mathrm{\ell }(\mathrm{\ell }+1)}{r^2}+\frac{f^{\mathrm{\prime }}(r)}{r})]u=0.$ Here $f^{\mathrm{\prime }}$ denotes the derivative with respect to $r$, and derivatives of $u$ are with respect to $r_{\ast }$.

Massless charged-scalar radial scattering data

This is a conditional separation identity for the explicitly supplied harmonic and mode. It proves neither existence of a spherical harmonic nor existence of a scattering solution.

Let $u$ be any scattering solution as defined on the fixed subextremal electric RN exterior, with real $\omega >0$, real $e,Q$, unit incoming amplitude, and reflection and transmission coefficients $R,T$. Then its radial Wronskian flux $j:=\mathrm{Im}(\bar{u}{\mathrm{\partial }}_{r_{\ast }}u)$ is constant and $\mathrm{\mid }R{\mathrm{\mid }}^2=1-\frac{\omega -eQ\mathrm{/}{r}_{+}}{\omega }\mathrm{\mid }T{\mathrm{\mid }}^2\mathrm{.}$ Consequently, if $0<\omega <eQ\mathrm{/}{r}_{+}$ and $T\mathrm{\ne }0$, the reflected radial flux magnitude exceeds the incident radial flux magnitude, since their ratio is $\mathrm{\mid }R{\mathrm{\mid }}^2>1$. This is conditional on an admitted scattering solution and does not assert its existence, a complex-frequency mode, or a nonlinear instability.

Massless charged-scalar radial scattering data

Charged radial equation on subextremal electric RN

The flux here is the conserved radial Wronskian flux of the separated classical test-field equation. The identity is a linear fixed-background scattering calculation. It gives no conclusion about field backreaction, quantum pair production, trapping, exponential growth, or nonlinear stability.