Adopted charged-background and effective-model regimes

Let $(M,g,N,o)$ have the smooth, boundaryless, oriented and time-oriented Lorentzian four-manifold structure, signature $(-,+,+,+)$, metric-selected connection ${\mathrm{\nabla }}^g$, and length-coordinate convention of Oriented Lorentzian fields, observers and coordinate conventions. Let $A\in {\mathrm{\Omega }}^1(M)$ be a supplied smooth real one-form, let $e\in \mathbb{R}$ be a fixed real coupling, and let $\psi \in C^{\mathrm{\infty }}(M,\mathbb{C})$ be a smooth complex-valued function. In the length-coordinate convention, assign dimensionless components to $A_a$ and unit ${\mathrm{m}}^{-1}$ to $e$, so $e{A}_a$ has derivative unit ${\mathrm{m}}^{-1}$; assign $\psi$ an arbitrary fixed unit $[\psi ]$. Define the complex covector

$D_a\psi :={\mathrm{\nabla }}_a^g\psi -ie{A}_a\psi \mathrm{.}$

Since $\psi$ is a scalar, ${\mathrm{\nabla }}_a^g\psi ={\mathrm{\partial }}_a\psi$. Define the real symmetric covariant tensor

${\mathsf{T}}_{ab}[\psi ;g,A,e]:=\overline{D_a\psi }{D}_b\psi +\overline{D_b\psi }{D}_a\psi -g_{ab}{g}^{cd}\overline{D_c\psi }{D}_d\psi ,$

with indices raised using $g^{-1}$. Its covariant components have unit $[\psi {]}^2{\mathrm{m}}^{-2}$. This is a mathematical tensor assignment. Its interpretation as a stress-energy tensor, and any field equation or conservation law, require a separate physical model.

Oriented Lorentzian fields, observers and coordinate conventions

The definition uses a globally supplied real one-form $A$ and a chosen complex trivialization. It asserts no global-bundle trivialization for a general charged field.

Use a smooth, boundaryless, oriented and time-oriented Lorentzian four-manifold $(M,g,N,o)$ with signature $(-,+,+,+)$, the metric-selected connection and the independent orientation $o$, and a fixed bundle sector with smooth connection and smooth real curvature two-form $F$ satisfying $dF=0$, as in the nonlinear constitutive definition linked below. Define its invariants

$X:=-\frac{1}{4}{F}_{ab}{F}^{ab},Y:=-\frac{1}{4}{F}_{ab}({\star }_{g}F{)}^{ab},$

Choose fixed real parameters $(\alpha ,\beta )\mathrm{\ne }(0,0)$ and a dimensionless $\delta \in (0,1)$. Adopt, as a classical source-free constitutive model on field configurations whose invariant values lie in

${\mathcal{D}}_{\alpha ,\beta ,\delta }:=\{(X,Y)\in {\mathbb{R}}^2:\mathrm{\mid }\alpha X\mathrm{\mid }+\mathrm{\mid }\beta Y\mathrm{\mid }<\delta \},$

the Lagrangian density

$L_{\mathrm{e}\mathrm{f}\mathrm{f}}(X,Y):=X+\alpha X^2+\beta Y^2\mathrm{.}$

The corresponding constitutive tensor in the supplier's convention is

$H^{ab}=(1+2\alpha X)F^{ab}+2\beta Y({\star }_{g}F{)}^{ab}\mathrm{.}$

Here ${\star }_g$ is the Hodge map determined by the metric and independent orientation of the Lorentzian geometry. This postulate adopts the displayed constitutive assignment, but does not adopt or assert a solution of a nonlinear field equation.

The pair $(\alpha ,\beta )$ is fixed within a selected model but otherwise arbitrary; this item assigns no numerical values. If the components of $F$ have unit $\mathfrak{F}$, then $X,Y,L_{\mathrm{e}\mathrm{f}\mathrm{f}}$ have unit ${\mathfrak{F}}^2$ and $\alpha ,\beta$ have unit ${\mathfrak{F}}^{-2}$, making the domain condition dimensionless. This is the mathematical normalization of the supplier, not an SI action prefactor.

The domain ${\mathcal{D}}_{\alpha ,\beta ,\delta }$ is an adopted regime label, not a proved error bound. No quantum origin, QED matching, coefficient value, derivative-correction control, characteristic-cone condition, hyperbolicity, causality, or solution-existence claim is included. The formal nonlinear field equations, if separately derived under the supplier's compact-variation hypotheses, do not establish that this model describes nature.

Nonlinear electromagnetic constitutive model with a $C^2$ Lagrangian

This is a selected phenomenological classical model with no empirical premise. The arbitrary parameters and invariant-domain threshold are declared inputs. Any claim of low-energy quantum derivation, approximation error, semiclassical consistency, gravitational coupling, or observed validity requires separate hypotheses and evidence.

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.

Use the fixed subextremal electric Reissner–Nordström exterior, gauge, charged derivative, real normalized spherical harmonic $Y$, and mode conventions of Massless charged-scalar radial scattering data. Let $\psi =A_{\mathrm{i}\mathrm{n}}{e}^{-i\omega t}Y(\mathrm{\Omega })u(r_{\ast })\mathrm{/}r$ be any admitted real-frequency scattering mode with $\omega >0$ and $A_{\mathrm{i}\mathrm{n}}\mathrm{\ne }0$, and let $R$ be its reflection coefficient. Form ${\mathsf{T}}_{ab}[\psi ;g,A,e]$ by the scalar stress-tensor definition. With $\xi ={\mathrm{\partial }}_t$, define the outward radial flux through the sphere of radius $r$ by

$\mathcal{P}(r):=-{\int }_{S^2}{r}^2{\mathsf{T}}^r{}_t[\psi ;g,A,e]d{\omega }_{S^2}\mathrm{.}$

Let ${\psi }_{\mathrm{i}\mathrm{n}}:=A_{\mathrm{i}\mathrm{n}}{e}^{-i\omega t}Y{e}^{-i\mathrm{\Theta }}\mathrm{/}r$ and ${\psi }_{\mathrm{r}\mathrm{e}\mathrm{f}}:=A_{\mathrm{i}\mathrm{n}}{e}^{-i\omega t}YR{e}^{i\mathrm{\Theta }}\mathrm{/}r$ be the two leading asymptotic components. Define their nonnegative directional flux magnitudes by

${\mathcal{P}}_{\mathrm{i}\mathrm{n}}:={\lim }_{r\to \mathrm{\infty }}{\int }_{S^2}{r}^2{\mathsf{T}}^r{}_t[{\psi }_{\mathrm{i}\mathrm{n}};g,A,e]d{\omega }_{S^2},{\mathcal{P}}_{\mathrm{r}\mathrm{e}\mathrm{f}}:=-{\lim }_{r\to \mathrm{\infty }}{\int }_{S^2}{r}^2{\mathsf{T}}^r{}_t[{\psi }_{\mathrm{r}\mathrm{e}\mathrm{f}};g,A,e]d{\omega }_{S^2}\mathrm{.}$

Then

${\mathcal{P}}_{\mathrm{i}\mathrm{n}}=2{\omega }^2\mathrm{\mid }{A}_{\mathrm{i}\mathrm{n}}{\mathrm{\mid }}^2,{\mathcal{P}}_{\mathrm{r}\mathrm{e}\mathrm{f}}=2{\omega }^2\mathrm{\mid }{A}_{\mathrm{i}\mathrm{n}}{\mathrm{\mid }}^2\mathrm{\mid }R{\mathrm{\mid }}^2,\frac{{\mathcal{P}}_{\mathrm{r}\mathrm{e}\mathrm{f}}}{{\mathcal{P}}_{\mathrm{i}\mathrm{n}}}=\mathrm{\mid }R{\mathrm{\mid }}^2\mathrm{.}$

This is a conditional tensor calculation for the specified mode asymptotics; it asserts neither that such a mode exists nor that its flux has been observed.

Massless charged-scalar radial scattering data

Complex scalar stress tensor with a supplied potential

The Coulomb phase in the scattering definition is retained; only its derivative limit ${\mathrm{\Theta }}^{\mathrm{\prime }}\to \omega$ enters the asymptotic stress calculation. The result concerns the prescribed geometric normalization and supplies no scattering-solution existence claim.

Within the adopted fixed-background charged-scalar model, consider any admitted real-frequency scattering mode on a subextremal electric Reissner–Nordström exterior with incident amplitude $A_{\mathrm{i}\mathrm{n}}\mathrm{\ne }0$, reflection coefficient $R$, and transmission coefficient $T\mathrm{\ne }0$. Let ${\mathrm{\Phi }}_H=Q\mathrm{/}{r}_{+}$ be the gauge-normalized potential difference and let ${\mathcal{P}}_{\mathrm{i}\mathrm{n}},{\mathcal{P}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ be the positive incoming and reflected time-translation energy-flux magnitudes defined by the adopted stress tensor. If

$0<\omega <e{\mathrm{\Phi }}_H,$

then

$\frac{{\mathcal{P}}_{\mathrm{r}\mathrm{e}\mathrm{f}}}{{\mathcal{P}}_{\mathrm{i}\mathrm{n}}}=\mathrm{\mid }R{\mathrm{\mid }}^2=1+\frac{e{\mathrm{\Phi }}_H-\omega }{\omega }\mathrm{\mid }T{\mathrm{\mid }}^2>1.$

Thus the reflected asymptotic mode flux magnitude exceeds the incident magnitude in this frequency band, conditional on an admitted scattering solution with nonzero transmission. The statement asserts neither existence nor uniqueness of such a solution.

Fixed-background charged scalar regime

Massless charged-scalar radial scattering data

Exact charged scattering Wronskian flux identity

Asymptotic time-translation flux of a charged scalar mode