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Single-center MP is extremal Reissner-Nordström exterior

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 $∣ q_{e} ∣ = M$ .

Statement

Let $m > 0$ have length units, let $σ \in {- 1, + 1}$ , put $ρ = ∣ x ∣$ on $D = R^{3} ∖ {0}$ , and use $H = 1 + m / ρ$ in Positive harmonic conformastatic electrovac data. Under the radial diffeomorphism $R = ρ + m$ from $ρ > 0$ to $R > m$ , the full metric and field data equal the dyonic Reissner-Nordström datum of Dyonic Reissner-Nordström metric and field domains with

$M = m, q_{e} = σ m, q_{m} = 0, f (R) = {(1 - \frac{m}{R})}^{2} .$

The parameters satisfy $∣ q_{e} ∣ = M$ ; this is the extremal parameter relation in the displayed RN convention. By Full harmonic Majumdar-Papapetrou Einstein-Maxwell verification and the geometric clause of Full dyonic Reissner-Nordström electrovac verification, these identical exterior data satisfy the source-free geometric electrovac equations. The identification is only between the open exteriors $ρ > 0$ and $R > m$ ; no horizon extension or global completeness follows.

Facts & Assumptions

Given: The single-center harmonic MP datum with $m > 0$ , the RN data convention, and the geometric electrovac equation.

[F1]

Positive harmonic conformastatic electrovac data gives $g = - H^{- 2} (d x^{0})^{2} + H^{2} (d ρ^{2} + ρ^{2} γ_{S^{2}})$ and $A_{σ} = σ (H^{- 1} - 1) d x^{0}$ .

[F2]

Dyonic Reissner-Nordström metric and field domains defines the RN metric and field by $g = - f (d x^{0})^{2} + f^{- 1} d R^{2} + R^{2} γ_{S^{2}}$ and $F = - q_{e} R^{- 2} d x^{0} \land d R + q_{m} ω_{S^{2}}$ .

[F3]

Full harmonic Majumdar-Papapetrou Einstein-Maxwell verification proves the MP data are electrovac; the first, geometric clause of Full dyonic Reissner-Nordström electrovac verification proves every RN datum on $f \neq 0$ is electrovac. Its separate SI tensor-normalization clause is not used.

[F4]

Geometric electrovac equations and flux normalization fixes the geometric Ricci equation $Ric = 2 Q (F)$ used to identify the common field data.

[F5]

The RN extremality convention above is $M^{2} = q_{e}^{2} + q_{m}^{2}$ for $M > 0$ ; when $q_{m} = 0$ , this is $∣ q_{e} ∣ = M$ .

[A1]

$m > 0$ , $ρ > 0$ , and $R = ρ + m$ ; hence $R > m$ and $f (R) \neq 0$ .

Proof

1.1givenA1F1algebraconstruct

On $ρ > 0$ , $H = 1 + m / ρ$ is smooth and positive. Direct differentiation gives $\partial_{i} (ρ^{- 1}) = - x^{i} ρ^{- 3}$ and $\partial_{i}^{2} (ρ^{- 1}) = - ρ^{- 3} + 3 (x^{i})^{2} ρ^{- 5}$ , so $Δ_{δ} H = m (- 3 ρ^{- 3} + 3 ρ^{2} ρ^{- 5}) = 0$ ; hence $H$ satisfies the hypotheses of [F1]. The map $ρ \mapsto R = ρ + m$ has smooth inverse $ρ = R - m$ on the stated open intervals and derivative $1$ , so it is a diffeomorphism. Since $H = R / (R - m)$ , we have $H^{- 1} = 1 - m / R$ and $d ρ = d R$ . Substitution into [F1] gives $g = - (1 - m / R)^{2} (d x^{0})^{2} + (1 - m / R)^{- 2} d R^{2} + R^{2} γ_{S^{2}}$ , because the sphere coefficient is $H^{2} ρ^{2} = R^{2}$ and the radial coefficient is $H^{2} = (1 - m / R)^{- 2}$ .

2.1givenA1F1F2step 1.1algebra

The potential transforms as $A_{σ} = σ ((R - m) / R - 1) d x^{0} = - σ m R^{- 1} d x^{0}$ . Differentiating gives $F_{σ} = - σ m R^{- 2} d x^{0} \land d R$ , which is exactly [F2] with $q_{e} = σ m$ and $q_{m} = 0$ . The coordinate map preserves the displayed orientation, and both future fields point along the positive $x^{0}$ direction on $R > m$ .

3.1givenF1F2F3F4F5A1step 1.1step 2.1∎

We have $M = m > 0$ , $∣ q_{e} ∣ = m = M$ , and $q_{m} = 0$ . By [F5], this is the extremal RN parameter relation, and $f (R) > 0$ on $R > m$ . The equality of the metrics in step 1.1 and two-forms in step 2.1 transfers the MP electrovac equations of [F3] to the identical RN exterior, with electrovac meaning the equations in [F4]. The conclusion does not extend the coordinate map to $ρ = 0$ or $R = m$ .

Single-center MP is extremal Reissner-Nordström exterior

RN extremality convention

Statement

Facts & Assumptions

Proof

Depends on

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