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Harmonic static multicenter electrovac model — Examples

1 · Prerequisites

2 · Summary

Scope

These two thought experiments specialize exact conditional solutions already proved on the A page. The first has a nonzero null electromagnetic field and nonzero spacetime curvature while its four listed scalar contractions vanish. The second fixes two distinct harmonic centers and computes their regular-sphere SI fluxes. Both are hypothetical model calculations; neither reports measurements or infers that an observed pair of black holes exists.

Two-center geometry

The diagram is a schematic cross-section of Euclidean space at fixed $x^{0}$ . The marked points are omitted centers, the smaller circles are sections of coordinate spheres with $0 < r < d / 2$ , and the larger circle is a section of an outer sphere with $R > d$ . Arrows show outward orientation. The drawing records the setup only; harmonicity, field equations, and fluxes follow from the complete argument in Two-center ideal static charged configuration and its suppliers.

\begin{tikzpicture}[scale=0.9]
  \coordinate (a1) at (0,0);
  \coordinate (a2) at (4,0);
  \draw[dashed,thick] (a1) circle (5);
  \draw[thick] (a1) circle (0.8);
  \draw[thick] (a2) circle (0.8);
  \fill (a1) circle (2pt);
  \fill (a2) circle (2pt);
  \node[below=4pt] at (a1) {$a_1$};
  \node[below=4pt] at (a2) {$a_2$};
  \node at (2,0.32) {$d$};
  \node at (-0.95,0.65) {$S_1(r)$};
  \node at (4.95,0.65) {$S_2(r)$};
  \node at (-2.9,4.1) {$S_R$};
  \draw[->] (-0.57,0.56) -- (-0.34,0.73);
  \draw[->] (4.57,0.56) -- (4.8,0.73);
\end{tikzpicture}

The radii in the sketch are not to scale. Each circle represents the intersection of a two-sphere with the displayed spatial plane.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

Thought experimentStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Nonzero null-wave field and curvature with four specified scalar contractions vanishing

Physical scope

Hypothetical exact null-wave solution of the selected classical SI Einstein–Maxwell model, specialized to one nonzero constant transverse profile. It illustrates a pointwise tensor distinction only and makes no observation, finite-energy, asymptotic-flatness, or empirical-validity claim.

Statement

Choose a constant $ν > 0$ with units $m^{- 1}$ and the profiles $a (u) = ν$ , $b (u) = 0$ , $h (u, x, y) = 0$ in Smooth null electrovac wave candidate data. Then $P = - ν^{2} (x^{2} + y^{2}), g = - 2 d u d v + d x^{2} + d y^{2} - ν^{2} (x^{2} + y^{2}) d u^{2},$ $F = ν d u \land d x, F = α^{- 1} F .$ Use orientation $d u \land d v \land d x \land d y$ and future unit field $N = \partial_{u} + (P + 1) \partial_{v} / 2$ from the null-wave definition. This is a smooth source-free solution of the adopted SI model of Equation-based Einstein–Maxwell model by Exact coupled electromagnetic null-wave model. Its electromagnetic field is nonzero. Define the scalar curvature by $S : = g^{a b} {Ric}_{a b}$ , using the curvature convention of Geometric electrovac equations and flux normalization. Then $F_{a b} F^{a b} = 0, S = 0, {Ric}_{a b} {Ric}^{a b} = 0, R_{a b c d} R^{a b c d} = 0.$ At the same time $R_{u x u x} = ν^{2} \neq 0$ , so the spacetime is not flat. These zero scalar contractions therefore do not imply a vanishing electromagnetic field or flat spacetime. This is a hypothetical model calculation, not an observation.

Facts & Assumptions

Given: A positive inverse-length constant $ν$ , the exact null-wave definitions, the selected equation-based physical postulate, and the current mathematical invariant proposition.

[F1]

Smooth null electrovac wave candidate data: the smooth null-wave metric, potential and two-form are determined by $a (u), b (u), h (u, x, y)$ and the prescribed transverse harmonicity.

[F2]

Exact coupled electromagnetic null-wave model: every datum of that profile class is an exact source-free solution of the selected SI model after $F = α^{- 1} F$ . This current physical theorem is draft with a complete authored proof and Step 1 authoring-ready record, not independent approval.

[F3]

Null wave curvature and zero quadratic invariants: for the null-wave datum, $F_{a b} F^{a b} = S = {Ric}_{a b} {Ric}^{a b} = R_{a b c d} R^{a b c d} = 0$ , its exact coordinate curvature formula is $R_{u i u j} = - \frac{1}{2} P_{i j}$ for $i, j \in {x, y}$ , and if $a^{2} + b^{2} > 0$ then the Riemann tensor is nonzero. This mathematical proposition is draft with a complete current proof and Step 1 authoring-ready record; the result is used conditionally.

[F4]

SI and geometric solution parameter identifications: $α > 0$ is constant, so $F = α^{- 1} F$ and $F_{a b} F^{a b} = α^{- 2} F_{a b} F^{a b}$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): only the inherited ${A C}_{ω}$ context is used; the constant-profile substitution and tensor scaling require no choice.

Proof

Given: The hypotheses in [F1]–[F4] and [A1], with $ν > 0$ .

1.1givenF1A1algebra

With $a = ν$ , $b = h = 0$ , the potential in [F1] is $A = - ν x d u$ and direct exterior differentiation gives $F = ν d u \land d x$ . This is a smooth datum on $R^{4}$ , with nonzero $F$ because $ν > 0$ .

2.1givenF2F4step 1.1

Apply [F2] to the datum in step 1.1. The geometric two-form is scaled by the positive constant $α^{- 1}$ , so the SI field $F$ remains nonzero and the metric-field pair solves the adopted source-free equations.

3.1givenF1F3F4step 1.1algebra∎

Since $P = - ν^{2} (x^{2} + y^{2})$ , direct differentiation gives $P_{x x} = - 2 ν^{2}$ . The exact supplier formula in [F3] therefore gives $R_{u x u x} = - \frac{1}{2} P_{x x} = ν^{2} \neq 0$ . The same supplier gives $F_{a b} F^{a b} = 0$ , $S = 0$ , ${Ric}_{a b} {Ric}^{a b} = 0$ , and $R_{a b c d} R^{a b c d} = 0$ ; by [F4], $F_{a b} F^{a b} = α^{- 2} F_{a b} F^{a b} = 0$ . Thus this exact model datum has nonzero electromagnetic field and spacetime curvature although all four stated scalar contractions vanish.

Thought experimentStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

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.

Harmonic static multicenter electrovac model — Examples

1 · Prerequisites

2 · Summary

Scope

Two-center geometry

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Nonzero null-wave field and curvature with four specified scalar contractions vanishing

Statement

Facts & Assumptions

Proof

Two-center ideal static charged configuration

Statement

Facts & Assumptions

Proof

Sources