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Trivial-potential classical charged scalar system — Examples

1 · Prerequisites

2 · Summary

The scalar-zero branch with $V (0) = 0$ has zero current and stress and reduces exactly to the source-free Einstein–Maxwell equations when the electromagnetic two-form has a global potential. The scope remark records why this example does not extend to nontrivial bundles, and why $λ = 0$ alone does not force the scalar stress to vanish.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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Zero scalar with V(0)=0 reduces to electrovacuum

Example

Use Trivial-potential classical charged-scalar Einstein–Maxwell system and suppose $0 \in U$ and $V (0) = 0$ . Let $φ \equiv 0$ and $π_{a} \equiv 0$ . Then the scalar equation, current, and scalar stress vanish, and the full coupled field equations reduce exactly to the source-free Einstein–Maxwell equations. Conversely, any smooth source-free Einstein–Maxwell solution on this region that admits the global potential $F = d A$ is a solution of the defined charged-scalar system with $φ = 0$ . The condition $V (0) = 0$ is essential for the zero scalar stress conclusion. In particular, source-free initial data in this exact-potential sector that satisfy the local-development theorem's hypotheses have a smooth local realization in this branch.

Facts & Assumptions

Given: The mathematical system of Trivial-potential classical charged-scalar Einstein–Maxwell system, a smooth solution or compatible smooth initial data, and the zero scalar field.

[F1]

Trivial-potential classical charged-scalar Einstein–Maxwell system defines $π = D φ$ , the scalar equation, and the full coupled field equations; for $φ = 0$ , these give $π = 0$ and $D_{a} π^{a} = 0$ .

[F2]

The current and scalar Hilbert tensor are $j^{a} = K_{φ} λ Im (\overline{φ} π^{a})$ and $T_{a b}^{φ} = K_{φ} Re (\overline{π_{a}} π_{b}) - \frac{1}{2} K_{φ} g_{a b} L_{φ}$ in Trivial-potential classical charged-scalar Einstein–Maxwell system.

[L1]

Local classical Einstein–Maxwell–charged-scalar development gives smooth local existence for its exact-potential data satisfying the sourced Maxwell and total-stress Einstein constraints. The example uses that local conclusion with its stated constraints and choice assumptions.

[A1]

$0 \in U$ , $V (0) = 0$ , and all fields are smooth. For the existence clause, take exact-potential electrovacuum initial data satisfying the Gauss and Einstein constraints in Local classical Einstein–Maxwell–charged-scalar development; the zero scalar and its zero normal derivative complete the data.

[A2]

Assume ${A C}_{ω}$ and Dependent Choice $D C$ as required by the local-development supplier [L1].

Proof

1.1F1F2A1algebra

Since $φ = 0$ and $π = 0$ , the scalar equation is $0 - V^{'} (0) 0 = 0$ and $j^{a} = 0$ . Also $L_{φ} = V (0) = 0$ , so $T_{a b}^{φ} = 0$ . The condition $V (0) = 0$ is what removes the remaining term $- \frac{1}{2} K_{φ} g_{a b} V (0)$ .

2.1F1F2step 1.1algebra

Substituting $j = 0$ and $T^{φ} = 0$ into the system of [F1] leaves exactly $G_{a b} + Λ g_{a b} = κ T_{a b}^{E M}$ , $d F = 0$ , and $\nabla_{a} F^{a b} = 0$ . Conversely, an exact-potential source-free solution with $φ = π = 0$ satisfies each charged-scalar equation by step 1.1 and each remaining equation by its source-free hypothesis.

3.1A1A2L1step 1.1step 2.1∎

For exact-potential electrovacuum initial data, setting $φ_{0} = 0$ , $π_{n} = 0$ , and hence $π_{i} = 0$ makes $j = 0$ and $T^{φ} = 0$ on the initial slice. The sourced Maxwell and total-stress Einstein constraints in [L1] then reduce to the source-free electrovacuum constraints. Applying [L1] under [A2] gives a smooth local solution. On that solution the scalar block for $(φ, π)$ is a homogeneous first-order wave/transport system with the metric, $A$ and $F$ as given smooth coefficients: the potential term is the smooth coefficient $V' (∣ φ ∣^{2})$ times $φ$ , and every other scalar source is linear in $φ, π$ . The symmetric scalar-block energy with zero initial scalar data therefore satisfies $E' \leq C E$ and is zero by the same local uniqueness argument in [L1]. Thus $φ = π = 0$ persists. Step 2.1 then identifies the resulting equations with the zero-scalar branch. No global existence claim follows.

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Classical scalar sector and bundle limitation

Remarks

Trivial-potential classical charged-scalar Einstein–Maxwell system uses a globally defined real one-form $A$ and $F = d A$ , so it describes only the trivial-potential sector. It does not cover a connection on an arbitrary nontrivial principal $U (1)$ bundle, a scalar represented only by local sections with nontrivial transition functions, or singular currents. In this global real-potential sector $λ$ may be any real number; charge quantization belongs to a separately specified bundle representation and is not inferred here. The field $φ$ is a classical complex field, not a quantum amplitude, and no value of $ℏ$ is used.