Charged photon trapping and extremal scaling

Let $m>0$ and $q_e,q_m\in \mathbb{R}$ be length parameters, put $Q^2=q_e^2+q_m^2$, and assume $0\le Q^2<m^2$. Use the asymptotically flat dyonic Reissner–Nordström metric with $M=m$ and $f(r)=1-2m\mathrm{/}r+Q^2\mathrm{/}{r}^2$ on the exterior $r>r_{+}:=m+\sqrt{m^2-Q^2}$. Here $r_{-}$ denotes the other algebraic horizon root $m-\sqrt{m^2-Q^2}$; when $Q^2>0$, $r_{\pm }$ are the local null Killing horizons of the selected RN model, and when $Q^2=0$, $r_{+}=2m$ is the larger root. An equatorial circular null geodesic means a smooth nonconstant affinely parametrized geodesic $\gamma (s)=(x^0(s),r_c,\pi \mathrm{/}2,\varphi (s))$ with $g(\dot{\gamma },\dot{\gamma })=0$ and nonzero angular momentum $\mathrm{\ell }=r_c^2\dot{\varphi }$. Its effective radial function is $h(r)=f(r)\mathrm{/}{r}^2$. The outer circular null radius is

$r_{\mathrm{p}\mathrm{h}+}=\frac{3m+\sqrt{9m^2-8Q^2}}{2}\mathrm{.}$

It lies strictly above $r_{+}$ and is a strict local maximum of $h$. Along an equatorial geodesic, define the conserved quantities $E:=f(r){\dot{x}}^0$ and $\mathrm{\ell }:=r^2\dot{\varphi }$. At fixed $E,\mathrm{\ell }$, the null radial geodesic equation is $\ddot{r}=-\frac{1}{2}{\mathrm{\ell }}^2{h}^{\mathrm{\prime }}(r)$. Its radial linearization at $r=r_{\mathrm{p}\mathrm{h}+}$, defined by writing $r=r_{\mathrm{p}\mathrm{h}+}+ϵ\rho (s)$ and retaining the terms of first order in $ϵ$, is $\ddot{\rho }-{\nu }^2\rho =0$, where ${\nu }^2:=-\frac{1}{2}{\mathrm{\ell }}^2{h}^{\mathrm{\prime }\mathrm{\prime }}(r_{\mathrm{p}\mathrm{h}+})>0$; it has the growing mode $\rho (s)=e^{\nu s}$. The other algebraic stationary radius is $r_{\mathrm{p}\mathrm{h}-}=(3m-\sqrt{9m^2-8Q^2})\mathrm{/}2$; this proposition assigns it no exterior-orbit interpretation. Coordinates $x^0,r,m,q_e,q_m$ have length units, angular coordinates are dimensionless, and the affine parameter has arbitrary normalization. This is a null-ray calculation for the exact RN metric; it proves no wave-decay or Morawetz estimate.

Dyonic Reissner-Nordström metric and field domains

RN roots, Killing horizons and surface gravity

Geodesic of an affine connection

The smaller algebraic stationary radius is not used to infer an additional exterior trapping orbit. Radial null geodesics have $\mathrm{\ell }=0$ and are a separate class. The argument establishes only the exact metric's circular-ray trapping and instability; no wave estimate follows from it.

The diagram is a schematic ordering of areal radii, not a distance scale. Panel (a) depicts $0<Q^2<m^2$; the proven exterior is $r>r_{+}$, and the proof above—not the drawing—establishes $r_{\mathrm{p}\mathrm{h}+}>r_{+}$. Panel (b) is the excluded extremal endpoint: the algebraic horizon roots coalesce at $m$, while the continuous limit of the outer stationary-radius formula is $2m$.

\begin{tikzpicture}[x=1cm,y=1cm]
  \draw[->] (0,0) -- (8,0) node[right] {$r$};
  \draw[densely dashed] (1.35,-0.18) -- (1.35,0.18);
  \draw[densely dashed] (3.15,-0.18) -- (3.15,0.18);
  \draw[very thick] (3.15,0) -- (8,0);
  \draw[fill=black] (5.55,0) circle (1.6pt);
  \node[below] at (1.35,-0.2) {$r_-$};
  \node[below] at (3.15,-0.2) {$r_+$};
  \node[above] at (5.55,0.12) {$r_{\mathrm{ph}+}$};
  \node[above] at (5.3,0.65) {subextremal exterior};
  \node[below] at (0,-0.48) {0};
  \node[below] at (3.15,-0.72) {$r>r_+$};
  \draw[->] (0,-1.55) -- (8,-1.55) node[right] {$r$};
  \draw[densely dashed] (3.15,-1.73) -- (3.15,-1.37);
  \draw[fill=black] (5.55,-1.55) circle (1.6pt);
  \node[below] at (3.15,-1.77) {$r_-=r_+=m$};
  \node[above] at (5.55,-1.42) {$r_{\mathrm{ph}+}=2m$};
  \node[above] at (5.2,-0.95) {extremal endpoint};
\end{tikzpicture}

Let $m>0$ and $q_e,q_m\in \mathbb{R}$ be length parameters with $q_e^2+q_m^2=m^2$, set the RN mass parameter $M=m$, and use the exact dyonic RN metric and two-form on $r>m$. For each dimensionless $\lambda >0$, define ${\mathrm{\Phi }}_{\lambda }(T,R,\omega )=(x^0=T\mathrm{/}\lambda ,r=m+\lambda R,\omega )$ on $D={\mathbb{R}}_T\times (0,\mathrm{\infty }{)}_R\times S^2$. Then

$g_{\lambda }:={\mathrm{\Phi }}_{\lambda }^{\ast }g=-\frac{R^2}{(m+\lambda R{)}^2}d{T}^2+\frac{(m+\lambda R{)}^2}{R^2}d{R}^2+(m+\lambda R{)}^2{\gamma }_{S^2},{\mathcal{F}}_{\lambda }:={\mathrm{\Phi }}_{\lambda }^{\ast }\mathcal{F}=-\frac{q_e}{(m+\lambda R{)}^2}dT\wedge dR+q_m{\omega }_{S^2}\mathrm{.}$

As $\lambda \downarrow 0$, the pair converges with every coordinate derivative uniformly on each compact subset of $D$ to

$g_0=-\frac{R^2}{m^2}d{T}^2+\frac{m^2}{R^2}d{R}^2+m^2{\gamma }_{S^2},{\mathcal{F}}_0=-\frac{q_e}{m^2}dT\wedge dR+q_m{\omega }_{S^2}\mathrm{.}$

Here $x^0,r,T,R,m,q_e,q_m$ have length units, $\lambda$ is dimensionless, and ${\gamma }_{S^2},{\omega }_{S^2}$ are the unit round metric and outward area form. “Convergence with every coordinate derivative uniformly on compact subsets” means that, in a finite smooth atlas, every coordinate derivative of each tensor coefficient converges uniformly on each compact subset. The convergence is only on $R>0$: it asserts neither extension through $R=0$, uniform control over the whole original exterior, nor a decoupled theory. In the positive-electric subcase $q_e=m$, $q_m=0$, the displayed pullbacks agree exactly with the inherited near-horizon theorem's field and metric formulas.

Dyonic Reissner-Nordström metric and field domains

Extremal RN to Bertotti-Robinson in C-infinity local scaling

The two-form is pulled back along the same coordinate map as the metric. The limiting field is not set to a constant by analogy: its electric coefficient is $-q_e\mathrm{/}{m}^2$ and its magnetic component remains $q_m{\omega }_{S^2}$. The uniformity statement is local in the scaled coordinate $R$ and deliberately excludes the boundary $R=0$.