Fixed-background charged scalar regime

Adopt the following classical test-scalar model in the geometric length-coordinate convention of the charged radial-scattering definition. Let $M>0$ and $Q\in \mathbb{R}$ satisfy $0<\mathrm{\mid }Q\mathrm{\mid }\le M$, put

$r_{\pm }:=M\pm \sqrt{M^2-Q^2},f(r):=1-\frac{2M}{r}+\frac{Q^2}{r^2},U:={\mathbb{R}}_t\times (r_{+},\mathrm{\infty }{)}_r\times S^2\mathrm{.}$

On $U$ prescribe

$g=-f(r)d{t}^2+f(r{)}^{-1}d{r}^2+r^2{\gamma }_{S^2},A=-\frac{Q}{r}dt,F=dA=-\frac{Q}{r^2}dt\wedge dr\mathrm{.}$

Here ${\gamma }_{S^2}$ is the unit round metric, $A_t\to 0$ at infinity, and $r_{+}$ is the larger positive root; in the extremal case $Q^2=M^2$ both roots equal $M$. The factorization $f(r)=(r-r_{+})(r-r_{-})\mathrm{/}{r}^2$ makes $f>0$ on this open exterior. For horizon terminology only, use the selected ingoing RN metric $g_{\mathrm{i}\mathrm{n}}=-f(r)d{v}^2+2dvdr+r^2{\gamma }_{S^2}$ on ${\mathbb{R}}_v\times (0,\mathrm{\infty }{)}_r\times S^2$. The hypersurface $H_{+}:=\{r=r_{+}\}$ is a null Killing horizon for $\xi :={\mathrm{\partial }}_v$: $g_{\mathrm{i}\mathrm{n}}^{-1}(dr,dr)=f(r_{+})=0$, ${\mathcal{L}}_{\xi }{g}_{\mathrm{i}\mathrm{n}}=0$, and $\xi$ is everywhere nonzero and tangent to $H_{+}$; moreover ${\xi }^{\mathrm{♭}}=dr$ on $H_{+}$, so $\xi$ is null and normal there and its nonzero-generator part is all of $H_{+}$. This local statement asserts no global causal-boundary property of a maximal spacetime extension. Orient the product by $dt\wedge dr\wedge {\omega }_{S^2}$, where ${\omega }_{S^2}$ is the outward area form of the unit round sphere; ${\star }_g$ denotes the corresponding Lorentzian Hodge map. Adopt the source-free background Maxwell branch $dF=0$ and $d({\star }_{g}F)=0$ on $U$. The fields are prescribed on the open exterior only. No maximal spacetime extension, continuation of the scalar field or its equation across the horizon, coupled Einstein equation, or actual astrophysical object is asserted by this postulate.

In the chosen global electric gauge, let $\psi :U\to \mathbb{C}$ be a smooth complex scalar and let $e\in \mathbb{R}$ be its fixed charge-coupling parameter. Define $D_a\psi :={\mathrm{\partial }}_a\psi -ie{A}_a\psi$, so $D_t={\mathrm{\partial }}_t+ieQ\mathrm{/}r$. The scalar Laplace–Beltrami operator on the unit sphere is ${\mathrm{\Delta }}_{{\gamma }_{S^2}}=\frac{1}{\sin \theta }{\mathrm{\partial }}_{\theta }(\sin \theta {\mathrm{\partial }}_{\theta })+\frac{1}{{\sin }^2\theta }{\mathrm{\partial }}_{\varphi }^2$ in the standard polar chart, with smooth global extension. Adopt the massless, minimally coupled equation

${\mathrm{\square }}_{g,A}\psi :=-f^{-1}{D}_t^2\psi +\frac{1}{r^2}{\mathrm{\partial }}_r\text{\hspace{0.17em}⁣}\left(r^{2}f{\mathrm{\partial }}_r\psi \right)+\frac{1}{r^2}{\mathrm{\Delta }}_{{\gamma }_{S^2}}\psi =0.$

For its energy bookkeeping, adopt the symmetric tensor assignment ${\mathsf{T}}_{ab}[\psi ;g,A,e]$ defined by the exact mathematical stress-tensor formula linked below. The scalar's stress is omitted from the equations for the prescribed background, and no scalar source is added to its Maxwell equation. This is a test-field model: omission of the scalar response is a modeling choice, not an estimate that the response is small.

The coordinates $t,r$ and parameters $M,Q$ have unit metres; the polar angles $\theta ,\varphi$ are dimensionless. The quantities $f$ and $A_t$ are dimensionless; $e$ has unit ${\mathrm{m}}^{-1}$; $F_{tr}$ has unit ${\mathrm{m}}^{-1}$; and the scalar field has a specified but otherwise arbitrary unit $[\psi ]$. In the product chart $x^a=(t,r,\theta ,\varphi )$, the metric and charged derivative have units $[g_{ab}]={\mathrm{m}}^2\mathrm{/}([x^a][x^b])$ and $[D_a\psi ]=[\psi ]\mathrm{/}[x^a]$. Thus the tensor formula gives $[{\mathsf{T}}_{ab}]=[\psi {]}^2\mathrm{/}([x^a][x^b])$, with no summation in these unit formulas: components with two $t,r$ indices have unit $[\psi {]}^2{\mathrm{m}}^{-2}$, those with one $t,r$ index and one angular index have unit $[\psi {]}^2{\mathrm{m}}^{-1}$, and those with two angular indices have unit $[\psi {]}^2$. The common component unit $[\psi {]}^2{\mathrm{m}}^{-2}$ in the stress-tensor definition applies in charts with all coordinate units equal to metres; other coordinate components inherit their units through the tensor transformation law. For the subextremal branch, define the gauge-normalized outer-root potential difference by ${\mathrm{\Phi }}_H:={\lim }_{r\to \mathrm{\infty }}{A}_t(r)-{\lim }_{r\downarrow r_{+}}{A}_t(r)=Q\mathrm{/}{r}_{+}$. The separate mathematical scalar model on the Einstein–Maxwell page uses SI bookkeeping and its own coupling parameters; no identification with those parameters is made here.

Separately adopted fixed-background Maxwell and charged probes

Dyonic Reissner-Nordström metric and field domains

RN roots, Killing horizons and surface gravity

Massless charged-scalar radial scattering data

Complex scalar stress tensor with a supplied potential

This postulate adopts a fixed exterior model with no empirical premise. The subextremal scattering consequences use the exact tortoise coordinate, incoming and reflected Coulomb phase, and horizon normalization of the mathematical definition. The extremal branch supplies no scattering result in this pair. Neither branch supplies a scalar or Einstein–Maxwell existence theorem, a small-backreaction limit, a nonlinear stability claim, or a quantum pair-production conclusion. The SI scalar model elsewhere on the Einstein–Maxwell pages uses separate coupling and field normalizations; no conversion or physical identification is adopted here.