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RN circular null radius and exterior instability

Statement

Let $m > 0$ and $q_{e}, q_{m} \in R$ be length parameters, put $Q^{2} = q_{e}^{2} + q_{m}^{2}$ , and assume $0 \leq Q^{2} < m^{2}$ . Use the asymptotically flat dyonic Reissner–Nordström metric with $M = m$ and $f (r) = 1 - 2 m / r + Q^{2} / r^{2}$ 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 $Q^{2} = 0$ , $r_{+} = 2 m$ is the larger root. An equatorial circular null geodesic means a smooth nonconstant affinely parametrized geodesic $γ (s) = (x^{0} (s), r_{c}, π / 2, ϕ (s))$ with $g (\dot{γ}, \dot{γ}) = 0$ and nonzero angular momentum $ℓ = r_{c}^{2} \dot{ϕ}$ . Its effective radial function is $h (r) = f (r) / r^{2}$ . The outer circular null radius is

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

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 $ℓ : = r^{2} \dot{ϕ}$ . At fixed $E, ℓ$ , the null radial geodesic equation is $\ddot{r} = - \frac{1}{2} ℓ^{2} h^{'} (r)$ . Its radial linearization at $r = r_{p h +}$ , defined by writing $r = r_{p h +} + ϵ ρ (s)$ and retaining the terms of first order in $ϵ$ , is $\ddot{ρ} - ν^{2} ρ = 0$ , where $ν^{2} : = - \frac{1}{2} ℓ^{2} h^{''} (r_{p h +}) > 0$ ; it has the growing mode $ρ (s) = e^{ν s}$ . The other algebraic stationary radius is $r_{p h -} = (3 m - \sqrt{9 m^{2} - 8 Q^{2}}) / 2$ ; 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.

Statement references

Dyonic Reissner-Nordström metric and field domains

RN roots, Killing horizons and surface gravity

Geodesic of an affine connection

Facts & Assumptions

Given: The dyonic RN metric with $m > 0$ , $0 \leq Q^{2} < m^{2}$ , and a smooth equatorial affinely parametrized null geodesic in the static exterior with nonzero angular momentum.

[F1]

Dyonic Reissner-Nordström metric and field domains: for $Q^{2} = q_{e}^{2} + q_{m}^{2}$ and $f (r) = 1 - 2 m / r + Q^{2} / r^{2}$ , the static metric is $g = - f (d x^{0})^{2} + f^{- 1} d r^{2} + r^{2} γ_{S^{2}}$ on each open set where $f \neq 0$ .

[F2]

The round metric on the sphere as an induced metric: in spherical coordinates the unit round metric is $γ_{S^{2}} = d θ^{2} + \sin^{2} θ d ϕ^{2}$ .

[F3]

RN roots, Killing horizons and surface gravity: when $Q^{2} > 0$ and $m^{2} \geq Q^{2}$ , the positive roots are $r_{\pm} = m \pm \sqrt{m^{2} - Q^{2}}$ and are local null constant-radius Killing horizons. Its theorem's $Q^{2} > 0$ hypothesis is used only for this horizon interpretation; the $Q^{2} = 0$ root is handled directly below.

[F4]

Geodesic of an affine connection: an affinely parametrized geodesic satisfies $D_{s} \dot{γ} = 0$ on its interval.

[F5]

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields: assuming ${A C}_{ω}$ , the metric-selected connection of a smooth nondegenerate metric has coordinate coefficients $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ .

[F6]

Explicit rational and real power regimes: for $a > 0$ , $\sqrt{a}$ is the unique positive real $s$ with $s^{2} = a$ .

[F7]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ asserts that every sequence of nonempty sets has a choice function. This is the assumption explicitly used by [F5].

[A1]

$m > 0$ , $q_{e}, q_{m} \in R$ , and $0 \leq Q^{2} = q_{e}^{2} + q_{m}^{2} < m^{2}$ .

[A2]

Adopt ${A C}_{ω}$ as required by [F5]. No further choice principle is used.

Proof

1.1givenF1F2F4F5F6F7A1A2algebra

Let $x = \sqrt{m^{2} - Q^{2}}$ . By [F6] and [A1], $0 < x \leq m$ , so $r_{-} = m - x \geq 0$ and $r_{+} = m + x > 0$ . Factoring gives $f (r) = (r - r_{+}) (r - r_{-}) / r^{2} > 0$ on $r > r_{+}$ . From [F1]–[F2] the equatorial metric components are $g_{00} = - f$ , $g_{r r} = f^{- 1}$ , $g_{θ θ} = r^{2}$ , and $g_{ϕ ϕ} = r^{2} \sin^{2} θ$ . Substitution in [F5] gives $Γ^{0}_{0 r} = f^{'} / (2 f)$ , $Γ^{ϕ}_{r ϕ} = 1 / r$ , $Γ^{r}_{00} = f f^{'} / 2$ , $Γ^{r}_{r r} = - f^{'} / (2 f)$ , $Γ^{r}_{ϕ ϕ} = - f r \sin^{2} θ$ , $Γ^{r}_{θ θ} = - f r$ , $Γ^{θ}_{r θ} = 1 / r$ , and $Γ^{θ}_{ϕ ϕ} = - \sin θ \cos θ$ . The $x^{0}$ and $ϕ$ geodesic equations of [F4] are therefore ${\ddot{x}}^{0} + (f^{'} / f) \dot{r} {\dot{x}}^{0} = 0$ and $\ddot{ϕ} + 2 (\dot{r} / r) \dot{ϕ} = 0$ . At $θ = π / 2$ , $\dot{θ} = 0$ , the $r$ equation is $\ddot{r} + \frac{1}{2} f f^{'} ({\dot{x}}^{0})^{2} - \frac{f^{'}}{2 f} {\dot{r}}^{2} - f r {\dot{ϕ}}^{2} = 0$ . The $θ$ equation is $\ddot{θ} + 2 (\dot{r} / r) \dot{θ} - \sin θ \cos θ {\dot{ϕ}}^{2} = 0$ , so $θ = π / 2$ , $\dot{θ} = 0$ is invariant. Differentiating the $x^{0}$ and $ϕ$ equations shows that $E : = f {\dot{x}}^{0}$ and $ℓ : = r^{2} \dot{ϕ}$ are constant along the geodesic.

2.1givenF1A1step 1.1algebra

The null condition is $- f ({\dot{x}}^{0})^{2} + f^{- 1} {\dot{r}}^{2} + r^{2} {\dot{ϕ}}^{2} = 0$ . Multiplying by $f > 0$ and using the constants from 1.1 gives ${\dot{r}}^{2} + ℓ^{2} h (r) = E^{2}$ , where $h = f / r^{2}$ .

2.2givenF1F4F5A1step 1.1algebra

For a circular orbit $r = r_{c}$ , the radial geodesic equation from 1.1 has $\dot{r} = \ddot{r} = 0$ and reads $\frac{1}{2} f f^{'} ({\dot{x}}^{0})^{2} - f r_{c} (\dot{ϕ})^{2} = 0$ . The null equation and $ℓ \neq 0$ imply $r_{c} f^{'} (r_{c}) - 2 f (r_{c}) = 0$ , equivalently $h^{'} (r_{c}) = 0$ . Direct differentiation gives $h^{'} (r) = - 2 (r^{2} - 3 m r + 2 Q^{2}) / r^{5}$ , so its algebraic roots are $r_{p h \pm} = (3 m \pm \sqrt{9 m^{2} - 8 Q^{2}}) / 2$ . They are real because $9 m^{2} - 8 Q^{2} = m^{2} + 8 (m^{2} - Q^{2}) > 0$ .

3.1givenF1F3F6A1step 2.2algebra

Put $x = \sqrt{m^{2} - Q^{2}}$ , so $0 < x \leq m$ . Then $\sqrt{9 m^{2} - 8 Q^{2}} = \sqrt{m^{2} + 8 x^{2}} > x$ , and $r_{p h +} - r_{+} = \frac{m + \sqrt{m^{2} + 8 x^{2}} - 2 x}{2} = \frac{(m - x) + (\sqrt{m^{2} + 8 x^{2}} - x)}{2} > 0.$ For $Q^{2} > 0$ , [F3] identifies $r_{+}$ as the outer local horizon. For $Q^{2} = 0$ , [F1] gives $f = 1 - 2 m / r$ , whose larger root is $r_{+} = 2 m$ ; the same inequality applies. Thus $f (r_{p h +}) > 0$ and the outer root lies in the static exterior.

3.2givenA1step 2.2algebra

At the outer root, $r^{2} - 3 m r + 2 Q^{2} = 0$ and $2 r_{p h +} - 3 m = \sqrt{9 m^{2} - 8 Q^{2}}$ , so $h^{''} (r_{p h +}) = - 2 \sqrt{9 m^{2} - 8 Q^{2}} / r_{p h +}^{5} < 0$ . Hence $h$ has a strict local maximum there.

4.1givenF1F4F5F6A1step 1.1step 2.1step 2.2step 3.1step 3.2algebraconstruct∎

Since $f (r_{p h +}) > 0$ , [F6] defines $\sqrt{h (r_{p h +})}$ . Choose $ℓ > 0$ and $E = ℓ \sqrt{h (r_{p h +})}$ . The explicit curve $x^{0} (s) = E s / f (r_{p h +})$ , $r (s) = r_{p h +}$ , $θ (s) = π / 2$ , and $ϕ (s) = ℓ s / r_{p h +}^{2}$ is null and satisfies the $x^{0}, ϕ, θ$ geodesic equations from 1.1. Its radial geodesic equation is exactly the circular condition in 2.2, so it is an affinely parametrized circular null geodesic. For a general equatorial null geodesic, the radial geodesic equation from 1.1 and the constants in 1.1 give $\ddot{r} = - \frac{E^{2} f^{'} (r)}{2 f (r)} + \frac{f^{'} (r)}{2 f (r)} {\dot{r}}^{2} + \frac{f (r) ℓ^{2}}{r^{3}} .$ Using the null first integral 2.1, $E^{2} = {\dot{r}}^{2} + ℓ^{2} f (r) / r^{2}$ , cancels the two ${\dot{r}}^{2}$ terms and yields $\ddot{r} = - \frac{ℓ^{2}}{2} h^{'} (r)$ . Thus the first radial variation $ρ$ at the outer circular orbit, with $E, ℓ$ fixed, satisfies $\ddot{ρ} + \frac{1}{2} ℓ^{2} h^{''} (r_{p h +}) ρ = 0$ , equivalently $\ddot{ρ} - ν^{2} ρ = 0$ , where $ν^{2} = - \frac{1}{2} ℓ^{2} h^{''} (r_{p h +}) > 0$ by 3.2. This equation has the growing solution $ρ (s) = e^{ν s}$ , as asserted.

Remarks

The smaller algebraic stationary radius is not used to infer an additional exterior trapping orbit. Radial null geodesics have $ℓ = 0$ 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 $r_{p h +} > r_{+}$ . 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 $2 m$ .

\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}

RN circular null radius and exterior instability

Statement

Statement references

Facts & Assumptions

Proof

Remarks

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