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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.

Background scalar wave versus coupled source equations

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources