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Extremal dyonic RN throat as a local scaled limit

Statement

Let $m > 0$ and $q_{e}, q_{m} \in R$ 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 $λ > 0$ , define $Φ_{λ} (T, R, ω) = (x^{0} = T / λ, r = m + λ R, ω)$ on $D = R_{T} \times (0, \infty)_{R} \times S^{2}$ . Then

$g_{λ} : = Φ_{λ}^{*} g = - \frac{R^{2}}{(m + λ R)^{2}} d T^{2} + \frac{(m + λ R)^{2}}{R^{2}} d R^{2} + (m + λ R)^{2} γ_{S^{2}}, F_{λ} : = Φ_{λ}^{*} F = - \frac{q_{e}}{(m + λ R)^{2}} d T \land d R + q_{m} ω_{S^{2}} .$

As $λ ↓ 0$ , the pair converges with every coordinate derivative uniformly on each compact subset of $D$ to

$g_{0} = - \frac{R^{2}}{m^{2}} d T^{2} + \frac{m^{2}}{R^{2}} d R^{2} + m^{2} γ_{S^{2}}, F_{0} = - \frac{q_{e}}{m^{2}} d T \land d R + q_{m} ω_{S^{2}} .$

Here $x^{0}, r, T, R, m, q_{e}, q_{m}$ have length units, $λ$ is dimensionless, and $γ_{S^{2}}, ω_{S^{2}}$ 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.

Statement references

Dyonic Reissner-Nordström metric and field domains

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

Facts & Assumptions

Given: The exact dyonic RN datum with $M = m > 0$ , $q_{e}^{2} + q_{m}^{2} = m^{2}$ , a dimensionless $λ > 0$ , and $R > 0$ .

[F1]

Dyonic Reissner-Nordström metric and field domains: on $f \neq 0$ , $g = - f (d x^{0})^{2} + f^{- 1} d r^{2} + r^{2} γ_{S^{2}}$ and $F = - q_{e} r^{- 2} d x^{0} \land d r + q_{m} ω_{S^{2}}$ , where $f = 1 - 2 M / r + (q_{e}^{2} + q_{m}^{2}) / r^{2}$ .

[F2]

Extremal RN to Bertotti-Robinson in C-infinity local scaling: only for $M = q_{e} = L > 0$ , $q_{m} = 0$ , the map $Φ_{λ} (T, R, ω) = (x^{0} = T / λ, r = L + λ R, ω)$ has the stated exact pullbacks and $C_{l o c}^{\infty}$ limit on $R > 0$ . The local argument below extends the explicit coordinate calculation to the dyonic charge pair; [F2] is used only in its exact positive-electric case.

[A1]

$m > 0$ , $λ > 0$ , and $R > 0$ , so $r = m + λ R > m$ and $m + λ R \geq m > 0$ . The extremality identity is $q_{e}^{2} + q_{m}^{2} = m^{2}$ .

Proof

1.1givenA1F1algebraconstruct

The extremality identity gives $f (r) = (1 - m / r)^{2} = λ^{2} R^{2} / (m + λ R)^{2}$ . The map $Φ_{λ}$ is a diffeomorphism from $D$ onto $R_{x^{0}} \times (m, \infty)_{r} \times S^{2}$ , with inverse $T = λ x^{0}$ , $R = (r - m) / λ$ . Its differentials satisfy $d x^{0} = d T / λ$ and $d r = λ d R$ . Substitution in [F1] gives the displayed $g_{λ}$ .

2.1givenA1F1step 1.1algebra

The same pullback differentials give $d x^{0} \land d r = (d T / λ) \land (λ d R) = d T \land d R$ ; substituting $r = m + λ R$ in the two-form of [F1] gives the displayed $F_{λ}$ .

3.1givenA1F1step 1.1step 2.1algebra

Fix a compact $K \subset D$ and choose $0 < δ \leq R \leq B < \infty$ on its radial projection. For $A_{λ} (R) = (m + λ R)^{- 2}$ and each integer $j \geq 0$ , $\partial_{R}^{j} A_{λ} (R) = (- 1)^{j} (j + 1)! \frac{λ^{j}}{(m + λ R)^{j + 2}} .$ The denominator is at least $m^{j + 2}$ ; for $j = 0$ , $A_{λ} \to m^{- 2}$ uniformly on $[δ, B]$ , and for $j \geq 1$ the displayed derivatives tend uniformly to zero. Thus every radial derivative of $R^{2} A_{λ}$ converges uniformly to that of $R^{2} / m^{2}$ . The other radial metric coefficient is $m^{2} R^{- 2} + 2 m λ R^{- 1} + λ^{2}$ , and the angular coefficient is $m^{2} + 2 m λ R + λ^{2} R^{2}$ ; all their derivatives converge uniformly on $[δ, B]$ . The electric field coefficient is $- q_{e} A_{λ}$ , and the magnetic term is independent of $λ$ . The fixed smooth sphere tensors have bounded derivatives on compact subsets of each chart in a finite atlas covering the sphere projection of $K$ . Therefore every coordinate derivative of every coefficient of $g_{λ}$ and $F_{λ}$ converges uniformly on $K$ to the displayed $g_{0}$ and $F_{0}$ .

4.1givenF2step 1.1step 2.1step 3.1algebra∎

If $q_{e} = m$ and $q_{m} = 0$ , formulas 1.1–3.1 specialize to $M = q_{e} = m > 0$ , $x^{0} = T / λ$ , $r = m + λ R$ and $F_{λ} = - m (m + λ R)^{- 2} d T \land d R$ . These are exactly the current hypotheses and field pullback in [F2]; the same metric formula and local all-derivative convergence agree with that supplier. No statement about unprovided charge sectors is inferred from [F2].

Remarks

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 $- q_{e} / m^{2}$ and its magnetic component remains $q_{m} ω_{S^{2}}$ . The uniformity statement is local in the scaled coordinate $R$ and deliberately excludes the boundary $R = 0$ .

Extremal dyonic RN throat as a local scaled limit

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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