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Local SI model limit from extremal RN to Bertotti–Robinson

Physical scope

Exact local smooth scaling limit of the selected electric extremal RN exterior to the Poincaré product chart of the source-free Einstein–Maxwell model. Convergence is only on compact subsets of R>0; no extension through the limiting boundary, global limit, or empirical claim is included.

Statement

Fix $L > 0$ . Select $m_{*} = \frac{c^{2} L}{G_{N}}, Q_{e} = + \sqrt{4 π ε_{0} G_{N}} m_{*}, Φ_{B} = 0.$ The map of SI and geometric solution parameter identifications gives $M = q_{e} = L$ , $q_{m} = 0$ , so the electric RN metric function is $f (r) = (1 - L / r)^{2}$ on $r > L$ . For every dimensionless $λ > 0$ , on $R_{T} \times (0, \infty)_{R} \times S^{2}$ define $Φ_{λ} (T, R, ω) = (x^{0} = T / λ, r = L + λ R, ω) .$ Let $g_{λ} = Φ_{λ}^{*} g$ and $F_{λ} = Φ_{λ}^{*} F$ for the SI extremal RN solution. Then $g_{λ} = - \frac{R^{2}}{(L + λ R)^{2}} d T^{2} + \frac{(L + λ R)^{2}}{R^{2}} d R^{2} + (L + λ R)^{2} γ_{S^{2}},$ and $F_{λ} = α^{- 1} F_{λ}$ where $F_{λ} = - \frac{L}{(L + λ R)^{2}} d T \land d R .$ As $λ ↓ 0$ , $(g_{λ}, F_{λ})$ converges in $C_{l o c}^{\infty}$ on $R_{T} \times (0, \infty)_{R} \times S^{2}$ to $g_{0} = - \frac{R^{2}}{L^{2}} d T^{2} + \frac{L^{2}}{R^{2}} d R^{2} + L^{2} γ_{S^{2}}, F_{0} = - \frac{1}{α L} d T \land d R .$ Under $T = L τ$ and $R = L z$ , the limit is exactly the electric Bertotti–Robinson data of Constant-radius Bertotti-Robinson product data with $f (z) = z^{2}$ , $z > 0$ . By Constant-radius charged product model, it is a solution of the same adopted SI model. The convergence and solution statements are limited to the stated open domains and do not assert an extension at $R = 0$ .

Facts & Assumptions

Given: $L > 0$ , $λ > 0$ , the extremal SI parameter values, the exact RN and product definitions, and the adopted zero-source equation model.

[F1]

Extremal RN to Bertotti-Robinson in C-infinity local scaling: for $M = q_{e} = L$ , $q_{m} = 0$ , the exact pullbacks and geometric fields converge smoothly on compact subsets of $R > 0$ to the $f = z^{2}$ Bertotti–Robinson product after $T = L τ$ , $R = L z$ . This mathematical theorem is draft with a complete current proof and Step 1 authoring-ready record; it is used conditionally.

[F2]

Dyonic RN as adopted Einstein–Maxwell model: the dyonic RN data with the parameter map of SI and geometric solution parameter identifications solve the adopted SI equations on the open domain where $r > 0$ and $f \neq 0$ . This physical theorem is draft with a complete current proof and Step 1 authoring-ready record; its solution claim is used conditionally.

[F3]

Constant-radius charged product model: the $f = z^{2}$ product with $L > 0$ solves the same source-free SI model on $z > 0$ . This physical theorem is draft with a complete proof authored in this batch and is used conditionally, without independent approval.

[F4]

SI and geometric solution parameter identifications: $q_{e} = \sqrt{G_{N} / (4 π ε_{0} c^{4})} Q_{e}$ , $M = G_{N} m_{*} / c^{2}$ , $Φ_{B} = 0$ gives $q_{m} = 0$ , and $α$ is a positive constant.

[F5]

Equation-based Einstein–Maxwell model: the same adopted postulate is used with $Λ = 0$ , $j = 0$ , and $T^{m} = 0$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the inherited geometric convention is at most ${A C}_{ω}$ ; the scaling map and compact convergence do not make a choice.

Proof

Given: The hypotheses and exact current statements in [F1]–[F5] and [A1].

1.1givenF2F4algebra

Substituting $m_{*}$ and $Q_{e}$ from the Statement into [F4] gives $M = L$ , $q_{e} = L$ , and $q_{m} = 0$ , with $L > 0$ . Hence $f (r) = (1 - L / r)^{2} > 0$ for $r > L$ , so the physical RN solution in [F2] applies throughout the image of each $Φ_{λ}$ .

2.1givenF1F4step 1.1algebra

The exact geometric pullbacks in [F1] yield the displayed $g_{λ}$ and $F_{λ}$ . Since $F_{λ} = α^{- 1} F_{λ}$ with constant positive $α$ , the same coordinate derivatives converge on each compact subset of $R > 0$ after multiplication by $α^{- 1}$ . This is the stated $C_{l o c}^{\infty}$ convergence for the SI fields as well as the metrics.

3.1givenF3F5A1step 2.1algebra∎

The coordinate substitution $T = L τ$ , $R = L z$ transforms the limiting pair into $g_{L, z^{2}}$ and $F_{L} = α^{- 1} F_{L}$ from [F3], on $z > 0$ . Thus the limit is itself a source-free SI model solution. The coordinate scaling uses no choice beyond [A1] and makes no claim about $R = 0$ .

Local SI model limit from extremal RN to Bertotti–Robinson

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources