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

RN extremality convention

For this Reissner–Nordström parameterization with $M > 0$ , “extremal” means $M^{2} = q_{e}^{2} + q_{m}^{2}$ . In the purely electric case $q_{m} = 0$ , this is $∣ q_{e} ∣ = M$ .

Convergence convention

$C_{l o c}^{\infty}$ convergence on $R_{T} \times (0, \infty)_{R} \times S^{2}$ means uniform convergence of every coordinate derivative on each compact subset in a finite smooth atlas.

Statement

Fix $L > 0$ and take the extremal RN datum with $M = q_{e} = L$ , $q_{m} = 0$ , on $r > L$ in Dyonic Reissner-Nordström metric and field domains. For each dimensionless $λ > 0$ , define

$Φ_{λ} (T, R, ω) = (x^{0} = T / λ, r = L + λ R, ω), (T, R, ω) \in R \times (0, \infty) \times S^{2} .$

Writing $g_{λ} = Φ_{λ}^{*} g$ and $F_{λ} = Φ_{λ}^{*} F$ , one has

$g_{λ} = - \frac{R^{2}}{(L + λ R)^{2}} d T^{2} + \frac{(L + λ R)^{2}}{R^{2}} d R^{2} + (L + λ R)^{2} γ_{S^{2}}, F_{λ} = - \frac{L}{(L + λ R)^{2}} d T \land d R .$

As $λ ↓ 0$ , these tensors converge in $C_{l o c}^{\infty}$ on $R_{T} \times (0, \infty)_{R} \times S^{2}$ to

$g_{0} = - \frac{R^{2}}{L^{2}} d T^{2} + \frac{L^{2}}{R^{2}} d R^{2} + L^{2} γ_{S^{2}}, F_{0} = - \frac{1}{L} d T \land d R .$

Under $T = L τ$ , $R = L z$ , the limiting datum is exactly the $f (z) = z^{2}$ , $z > 0$ product in Constant-radius Bertotti-Robinson product data, and it is a geometric electrovac datum by Product electrovacuum equations from full factor Ricci. Here $C_{l o c}^{\infty}$ means uniform convergence of every coordinate derivative on every compact subset in a finite smooth atlas. The convergence is only on $R > 0$ ; no extension or convergence across $R = 0$ is asserted.

Facts & Assumptions

Given: The RN and Bertotti-Robinson definitions, the extremal parameters $M = q_{e} = L > 0$ , and $λ > 0$ .

[F1]

Dyonic Reissner-Nordström metric and field domains gives $f (r) = 1 - 2 M / r + (q_{e}^{2} + q_{m}^{2}) / r^{2}$ and $F = - q_{e} r^{- 2} d x^{0} \land d r$ when $q_{m} = 0$ .

[F2]

Full dyonic Reissner-Nordström electrovac verification proves the dyonic RN datum is source-free electrovac on every open domain with $r > 0$ and $f \neq 0$ . Only its geometric clause is used.

[F3]

Constant-radius Bertotti-Robinson product data defines $g_{L, z^{2}}$ and $F_{L}$ on $z > 0$ .

[F4]

Product electrovacuum equations from full factor Ricci proves this limiting product is electrovac.

[F5]

Geometric electrovac equations and flux normalization defines the geometric equations satisfied by both source and limiting data.

[A1]

$L > 0$ , $λ > 0$ , and $R > 0$ , so $r = L + λ R > L$ and every denominator displayed in the Statement is nonzero.

Proof

1.1givenA1F1F2algebraconstruct

The map $Φ_{λ}$ is a diffeomorphism from $R_{T} \times (0, \infty)_{R} \times S^{2}$ onto the RN exterior $r > L$ , since $x^{0} = T / λ$ and $r = L + λ R$ have smooth inverses. Its pullback satisfies $d x^{0} = d T / λ$ and $d r = λ d R$ , while $f (r) = (1 - L / r)^{2} = λ^{2} R^{2} / (L + λ R)^{2}$ . Substitution into the RN metric in [F1] gives the displayed $g_{λ}$ . The starting RN datum is a geometric electrovac by the first clause of [F2]; the claim here is the exact coordinate pullback and its limit.

2.1givenA1F1step 1.1algebra

The RN two-form in [F1] pulls back to $- \frac{L}{(L + λ R)^{2}} \frac{d T}{λ} \land (λ d R) = - \frac{L}{(L + λ R)^{2}} d T \land d R$ , which is the displayed $F_{λ}$ .

3.1givenA1step 1.1step 2.1algebra

At $λ = 0$ the coefficient formulas in the Statement are smooth for $R > 0$ . Fix any compact subset $K$ of $R_{T} \times (0, \infty)_{R} \times S^{2}$ and choose a finite smooth atlas on its sphere projection. In each chart, $R$ is bounded below by a positive number and $L + λ R \geq L$ for $0 \leq λ \leq 1$ . Every coordinate derivative of every displayed coefficient is therefore a continuous function of $(λ, T, R, θ, ϕ)$ on a compact set containing ${0} \times K$ ; uniform continuity implies that each such derivative converges uniformly on $K$ to its value at $λ = 0$ . Since this holds for every derivative order and the atlas is finite, the convergence is $C_{l o c}^{\infty}$ .

4.1givenF3F4F5step 3.1algebra∎

Substituting $T = L τ$ and $R = L z$ into $g_{0}, F_{0}$ gives $L^{2} (- z^{2} d τ^{2} + z^{- 2} d z^{2} + γ_{S^{2}})$ and $- L d τ \land d z$ , exactly [F3] with $f = z^{2}$ . By [F4] this limiting product satisfies the equations defined in [F5]. This identification is on $z > 0$ only and does not add a horizon chart.

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

RN extremality convention

Convergence convention

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources