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Two-center ideal static charged configuration

Physical scope

Hypothetical two-center exact source-free SI model on the punctured exterior, with two selected positive SI mass parameters and one common sign for both electric flux parameters. Excluded centers are not included as sources or regular points. No horizon extension, stability, observed configuration, or empirical-validity claim is made.

Statement

Let $d > 0$ be the separation of the centers, measured in metres, and let $m_{*, 1}, m_{*, 2} > 0$ be selected SI masses. Put $a_{1} = (0, 0, 0)$ , $a_{2} = (d, 0, 0)$ , choose one sign $σ \in {- 1, + 1}$ , and set $m_{A} = G_{N} m_{*, A} / c^{2}$ for $A = 1, 2$ . On $D = R^{3} ∖ {a_{1}, a_{2}}$ take $H (x) = 1 + \frac{m_{1}}{∣ x - a_{1} ∣} + \frac{m_{2}}{∣ x - a_{2} ∣},$ and define the spheres with outward Euclidean orientation by $S_{A} (r) : = {x \in R^{3} : ∣ x - a_{A} ∣ = r} (0 < r < d / 2) .$

$S_{R} : = {x \in R^{3} : ∣ x ∣ = R} (R > d) .$

$g_{H} = - H^{- 2} (d x^{0})^{2} + H^{2} δ_{i j} d x^{i} d x^{j}, A_{σ} = σ (H^{- 1} - 1) d x^{0}, F = α^{- 1} d A_{σ},$ with $x^{0} = c t$ and $α$ from SI and geometric solution parameter identifications. This hypothetical datum solves the selected source-free SI model on $R_{x^{0}} \times D$ . For every displayed sphere $S$ , use $i_{s, S} (x) = (s, x)$ into $R_{x^{0}} \times D$ at any fixed $s$ , with transported outward orientation, as in Finite positive multicenter MP exterior and fluxes and Harmonic static multicenter electrovac model. Each integral over $S$ means the integral of the spacetime form pulled back by $i_{s, S}$ and is independent of $s$ . For each $S_{A} (r)$ , its SI electric flux is $ε_{0} c \int_{S_{A} (r)} ⋆_{g} F = σ \sqrt{4 π ε_{0} G_{N}} m_{*, A},$ and its magnetic flux is zero. On $S_{R}$ , the electric flux is the sum of the two displayed values and the magnetic flux remains zero. The example is a model calculation; it does not insert a source at either omitted point or claim that two observed black holes are in equilibrium.

Facts & Assumptions

Given: A separation $d > 0$ measured in metres, two positive SI masses, distinct centers, one common sign, positive SI constants, the exact harmonic MP data and multicenter result, and the source-free branch of the adopted physical model.

[F1]

Positive harmonic conformastatic electrovac data: for a smooth positive harmonic $H$ , the exact metric, orientation, potential and geometric field are the conformastatic MP data.

[F2]

Finite positive multicenter MP exterior and fluxes: for finite distinct centers and either common sign, $H = 1 + \sum_{A} m_{A} / ∣ x - a_{A} ∣$ is positive harmonic on the punctured exterior, satisfies the geometric source-free equations, and has local and total sphere fluxes $q_{e} = σ m_{A}$ and $q_{m} = 0$ .

[F3]

Harmonic static multicenter electrovac model: the same finite harmonic family with selected SI mass inputs solves the adopted SI source-free equation model on the punctured exterior. This current physical theorem is draft with a complete proof and Step 1 authoring-ready record; it is used conditionally, not as independent proof approval.

[F4]

SI and geometric solution parameter identifications: $q_{e} = α (4 π ε_{0} c)^{- 1} Q_{e}$ , $q_{m} = α (4 π)^{- 1} Φ_{B}$ , and $m_{A} = G_{N} m_{*, A} / c^{2}$ .

[F5]

Equation-based Einstein–Maxwell model: the equation-based physical law is an adopted assumption, here restricted to zero cosmological and material sources.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): only the inherited ${A C}_{ω}$ geometric convention is used, and the two-center substitutions and finite flux sums require no further choice.

Proof

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

1.1givenF1F2F4A1algebra

Since $d$ is a positive separation measured in metres, $a_{1} = (0, 0, 0)$ and $a_{2} = (d, 0, 0)$ are distinct length-coordinate points. By the Statement, $S_{A} (r) = {x \in R^{3} : ∣ x - a_{A} ∣ = r}$ for $0 < r < d / 2$ , and $S_{R} = {x \in R^{3} : ∣ x ∣ = R}$ for $R > d$ ; these spheres isolate either center and strictly enclose both centers, respectively. The mass map in [F4] gives $m_{1}, m_{2} > 0$ , and the displayed $H$ is exactly the $N = 2$ function in [F2]. Thus it is smooth, positive and harmonic on $D$ .

2.1givenF2F3step 1.1

With the supplied time-slice embeddings and pullback convention of [F2] and [F3], the exact geometric fields on $D$ satisfy the source-free equations and have local fluxes $q_{e} (S_{A}) = σ m_{A}$ , total outer flux $σ (m_{1} + m_{2})$ , and zero magnetic flux. Apply [F3] to the same $N = 2$ data to obtain the selected SI source-free model equations; the omitted points remain outside the model manifold.

3.1givenF2F4step 2.1algebra

By [F4], the SI flux corresponding to a geometric $q_{e}$ is $Q_{e} = (4 π ε_{0} c / α) q_{e}$ . Substituting $q_{e} = σ m_{A}$ and $m_{A} = G_{N} m_{*, A} / c^{2}$ , using $α^{2} = 4 π G_{N} / (μ_{0} c^{4})$ and $ε_{0} μ_{0} c^{2} = 1$ , gives $Q_{e} = σ \sqrt{4 π ε_{0} G_{N}} m_{*, A}$ for each local sphere; linearity gives their sum on the outer sphere. The magnetic flux is zero since $q_{m} = 0$ and $α > 0$ .

4.1givenF3F5A1step 2.1∎

The model deduction in step 2.1 is conditional on the adopted postulate [F5]. Neither the field equations on the punctured manifold nor the flux integrals assert material sources or regular fields at the omitted centers, and this construction contains no observation or stability argument.

Two-center ideal static charged configuration

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources