Alphabeta Math

1 · Prerequisites

2 · Summary

Scope

These examples calculate two limits of the specified smooth coupling. A classical scalar plane wave can solve its matter equation on a prescribed flat background while failing the coupled Einstein equation; with nonzero electromagnetic coupling, the same zero-potential background also fails the sourced Maxwell equation. A separate constant electric field solves the source-free Maxwell equations on Minkowski space but has nonzero Hilbert stress, so the same flat metric fails the coupled Einstein equation. These are exact conditional model calculations, not observations or claims about particle ontology.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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Background scalar wave versus coupled source equations

Physical scope

Exact smooth plane-wave calculations in the specified classical scalar model on a flat background. They distinguish a matter equation solution from a solution of the coupled equations and make no empirical or quantum claim.

Example

Take the trivial global-potential branch of the scalar model on $M = R^{4}$ with global coordinates $(x^{0}, x)$ , where $x^{0} = c t$ and $x \in R^{3}$ , and Minkowski metric $η = diag (- 1, 1, 1, 1)$ . Use the trivial principal bundle $P = M \times U (1)$ with its product connection, so the global physical potential is $A = 0$ , and take $φ \in C^{\infty} (R^{4}; C)$ . Set $V (u) = m_{s}^{2} u$ , where $m_{s} > 0$ is a classical inverse-length parameter $[m^{- 1}]$ , not a rest mass. Let $C \in C ∖ {0}$ be dimensionless and let $k \in R^{3}$ have components in $m^{- 1}$ . Define the angular frequency $ω : = c \sqrt{∣ k ∣^{2} + m_{s}^{2}}$ , with unit $s^{- 1}$ . The smooth field $φ = C \exp (- i ω t + i k \cdot x)$ solves the scalar equation for every allowed value of $λ$ , because $A = 0$ makes $D_{a} = \partial_{a}$ . Its current, however, depends on $λ$ . The two cases below show why a solution of the scalar equation on a prescribed background need not solve the coupled Einstein–Maxwell system.

Facts & Assumptions

Given: The smooth scalar model and its coupled equations from Field equations and balances of the selected scalar coupling, the flat metric and fields in the Example, $m_{s} > 0$ , $C \neq 0$ , real $k$ , and the indexed countable-choice convention [A3].

[F1]

Field equations and balances of the selected scalar coupling: with $E_{φ} = D_{a} P^{a} - V^{'} (∣ φ ∣^{2}) φ$ , scalar stationarity requires $E_{φ} = 0$ ; the matter current and stress are $j^{a} = K_{φ} λ Im (\overline{φ} P^{a})$ and $T_{a b}^{φ} = K_{φ} Re (\overline{P_{a}} P_{b}) - \frac{K_{φ}}{2} g_{a b} L_{φ}$ ; the coupled Maxwell and Einstein equations use those sources.

[A1]

$Λ = 0$ , there is no additional matter stress, and the background potential is $A = 0$ , so $F = 0$ .

[A2]

The scalar is a smooth global complex field on the trivial bundle; $m_{s} > 0$ , $C \neq 0$ , $K_{φ} > 0$ , $c > 0$ , and $λ$ is any real number on this branch. In particular, both $λ = 0$ and $λ \neq 0$ are admissible.

[A3]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every sequence of nonempty sets has a choice function. This is the indexed countable-choice convention used by the compact integration suppliers; no full Axiom of Choice is assumed.

Verification

Given: The hypotheses and exact field definitions above.

1.1A1A2A3F1givenalgebra

Since $A = 0$ in the trivial branch [A2], $P_{a} = \partial_{a} φ$ . With $□ = - c^{- 2} \partial_{t}^{2} + Δ$ , $E_{φ} = (□ - m_{s}^{2}) φ = (ω^{2} / c^{2} - ∣ k ∣^{2} - m_{s}^{2}) φ = 0$ by the definition of $ω$ . Thus the plane wave solves the selected scalar equation on the flat background.

2.1A1A2A3F1step 1.1algebra

For the admissible $λ = 0$ case of [A2], the current formula in [F1] gives $j^{a} = 0$ . On shell, $L_{φ} = (- ω^{2} / c^{2} + ∣ k ∣^{2} + m_{s}^{2}) ∣ C ∣^{2} = 0$ by step 1.1, and $P_{0} = - i (ω / c) φ$ , so $Re (\overline{P_{0}} P_{0}) = ω^{2} ∣ C ∣^{2} / c^{2}$ . The stress formula in [F1] therefore gives $T_{00}^{φ} = K_{φ} ω^{2} ∣ C ∣^{2} / c^{2} > 0$ . In the Cartesian coordinates of $η$ , the metric components are constant; the Levi-Civita formula $Γ_{μ ν}^{ρ} = \frac{1}{2} η^{ρ σ} (\partial_{μ} η_{σ ν} + \partial_{ν} η_{σ μ} - \partial_{σ} η_{μ ν})$ therefore gives $Γ_{μ ν}^{ρ} = 0$ , and the curvature formula then gives $G_{00} = 0$ . With $Λ = 0$ and $T^{m} = 0$ , the coupled Einstein equation would require $0 = κ_{E} T_{00}^{φ}$ , impossible because $κ_{E} = 8 π G_{N} / c^{4} > 0$ .

2.2A1A2A3F1step 1.1algebra

For the admissible $λ \neq 0$ case of [A2] and the same $A = 0$ wave, $P^{0} = i (ω / c) φ$ , so $j^{0} = K_{φ} λ (ω / c) ∣ C ∣^{2} \neq 0$ . The sourced Maxwell equation would require $0 = \partial_{μ} F^{μ 0} = - μ_{0} j^{0}$ , because $F = 0$ ; this is impossible since $μ_{0} > 0$ . Thus the background scalar wave is not a coupled solution in this branch either.

3.1step 2.1step 2.2given∎

The exact calculations establish a conditional model example only. $m_{s}$ is not identified with $m c / ℏ$ , and no particle, observation, or quantum field interpretation is asserted.

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A flat Maxwell field need not solve the coupled Einstein equations

Physical scope

A smooth exact counterexample showing that a source-free Maxwell solution on a prescribed flat background need not satisfy the coupled action-based Einstein equation. It uses no empirical data or singular field.

Statement refuted

The false claim is: on $M = R^{4}$ with global length coordinates $(x^{0}, x)$ , $x^{0} = c t$ , and Minkowski metric $η = diag (- 1, 1, 1, 1)$ , every smooth source-free Maxwell field on the trivial principal $U (1)$ bundle, paired with that same flat metric, is a solution of the coupled action-based Einstein–Maxwell equations with $Λ = 0$ and no additional matter.

Facts & Assumptions

Given: The action-based field equations and SI stress convention of Einstein–Maxwell equations from compact action stationarity, on a flat background with zero cosmological parameter and no additional matter, under the indexed countable-choice convention [A2].

[F1]

Einstein–Maxwell equations from compact action stationarity: the smooth fixed-sector field system requires $d F = 0$ , $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ , and $G_{a b} + Λ g_{a b} = κ_{E} (T_{a b}^{E M} + T_{a b}^{m})$ , where $T_{a b}^{E M} = μ_{0}^{- 1} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d})$ .

[A1]

Let $M = R^{4}$ with global length coordinates $(x^{0}, x^{1}, x^{2}, x^{3})$ , $x^{0} = c t$ , and Minkowski metric $η = diag (- 1, 1, 1, 1)$ . Use the trivial principal bundle $P = M \times U (1)$ with product projection $π$ and its global trivialization. For a nonzero constant electric-field component $E_{0} \in R$ with SI unit $V m^{- 1}$ , defined relative to the observer $n = \partial_{x^{0}}$ in the $x^{1}$ direction, define the global physical potential $A \in Ω^{1} (M)$ by its components $A_{0} = E_{0} x^{1} / c$ and $A_{i} = 0$ for $i = 1, 2, 3$ . The normalized local representative is $a = A / κ_{0}$ , and the principal connection in this trivialization is $ω = - i z^{- 1} d z + π^{*} a$ for $z \in U (1)$ . Its physical curvature is $F = κ_{0} d a = d A$ , so $F_{01} = - E_{0} / c$ and all other independent components vanish.

Choose the zero-matter model with configuration space $C_{m} = {*}$ , no matter-field variations, and differentiable functional $S_{m} [g, ω, *] \equiv 0$ . Its matter first variations vanish for every admissible metric and connection variation, hence its current and Hilbert stress are $j = 0$ and $T^{m} = 0$ . Set $Λ = 0$ and $g = η$ .

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every sequence of nonempty sets has a choice function. This is the indexed countable-choice convention used by the compact integration suppliers; no full Axiom of Choice is assumed.

Counterexample

Given: The fields and parameter values in [A1].

1.1A1A2F1givenalgebra

The two-form $F$ is constant, so $d F = 0$ and $\partial_{μ} F^{μ ν} = 0$ for every $ν$ . Hence it satisfies the sourced equation in [F1] with $j = 0$ , and is a smooth source-free Maxwell solution on the prescribed Minkowski background.

1.2A1A2F1algebra

Its invariant is $F_{c d} F^{c d} = - 2 E_{0}^{2} / c^{2}$ and $F_{0 c} F_{0}^{c} = E_{0}^{2} / c^{2}$ . Substitution in the stress formula [F1] gives $T_{00}^{E M} = E_{0}^{2} / (2 μ_{0} c^{2}) > 0$ .

2.1A1A2F1step 1.2algebra∎

In Cartesian coordinates the components of $η$ are constant, so the Levi-Civita formula $Γ_{μ ν}^{ρ} = \frac{1}{2} η^{ρ σ} (\partial_{μ} η_{σ ν} + \partial_{ν} η_{σ μ} - \partial_{σ} η_{μ ν})$ gives $Γ_{μ ν}^{ρ} = 0$ . The curvature formula then gives zero Riemann and Einstein tensors, in particular $G_{00} = 0$ . With $Λ = 0$ and $T^{m} = 0$ from [A1], the coupled Einstein equation in [F1] would require $0 = κ_{E} T_{00}^{E M}$ , contradicting $κ_{E} = 8 π G_{N} / c^{4} > 0$ and step 1.2. This source-free Maxwell solution on a fixed background is not a solution of the coupled equations, refuting the claim.

Independent equation-based Einstein–Maxwell coupling — Examples

1 · Prerequisites

2 · Summary

Scope

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Background scalar wave versus coupled source equations

Example

Facts & Assumptions

Verification

A flat Maxwell field need not solve the coupled Einstein equations

Statement refuted

Facts & Assumptions

Counterexample

Sources