Gauge/diffeomorphism matter identity and field exchange — Examples

On ${\mathbb{R}}^4$ with $x^0=ct$, Minkowski metric $\eta =\mathrm{diag}(-1,1,1,1)$, and trivial bundle, set $A=0$, $V(u)=m_s^{2}u$ with $m_s>0$ in ${\mathrm{m}}^{-1}$, and

$\phi (t,\mathbf{x})=C\exp (-i\omega t+i\mathbf{k}\cdot \mathbf{x}),$

where $C\mathrm{\ne }0$ is dimensionless, $\mathbf{k}\in {\mathbb{R}}^3$ has units ${\mathrm{m}}^{-1}$, and $\omega :=c\sqrt{\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2+m_s^2}>0$ has units ${\mathrm{s}}^{-1}$. For the scalar action of Classical complex-scalar connection derivative and action, this is an exact solution of the local background scalar equation, equivalently stationary under compactly supported scalar variations. The infinite-volume action value need not be finite; only the local equation and compactly supported variations are used. If the coupling $\lambda \mathrm{\ne }0$, the same configuration has a nonzero derived current although $F=dA=0$, so it fails the sourced Maxwell equation of Sourced Maxwell sign and current conservation. It is therefore a background scalar solution, not a solution of the coupled scalar–Maxwell system. No relation to $\mathrm{\hslash }$ or empirical data is used.

• The M02/M03 suppliers are current draft carriers used only at their inspected statements. The seven typed foundational replacement carriers in the dependency closure are now authored on emg-mg02; no independent acceptance is asserted: Finite-real supremum and infimum domain, Real metric and pseudometric convention, Metric nonnegativity from the three axioms, Explicit rational and real power regimes, Sequence-limit criterion with explicit choice use, Boundaryless domain for geometric completeness, and Set-sized objects and definable-class schemas.

Work in boundaryless ambient $M={\mathbb{R}}^4$ and integrate the local divergence over the half-space subdomain $H=\{x^3\ge 0\}$. Give $M$ the flat metric $\eta =\mathrm{diag}(-1,1,1,1)$, coordinates $(x^0,x^1,x^2,x^3)$, and outward conormal $-d{x}^3$ at its boundary. All coordinates have length units and choose a length $\mathrm{\ell }>0$. Take dimensionless cutoffs $\chi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ and $\rho \in C_c^{\mathrm{\infty }}(\mathbb{R})$ with $\chi \ge 0$, $\chi =1$ near $0\in {\mathbb{R}}^3$, and $\rho =1$ near $0\in \mathbb{R}$. Define

$f(x):=\frac{x^3}{\mathrm{\ell }}\chi (x^0,x^1,x^2)\rho (x^3),h_{ab}:=f{\eta }_{ab}\mathrm{.}$

Then $h_{ab}=0$ on $\mathrm{\partial }H$, but its normal derivative is nonzero there. For the Einstein–Hilbert summand with coefficient $c^3\mathrm{/}(16\pi G_N)$, the local variation formula in Compact Einstein–Hilbert variation and SI coupling has a nonzero boundary flux for this variation. The diagram records only the chosen half-space and normal; the proof is the component calculation below.

• This is the local divergence flux on a subdomain of a boundaryless ambient manifold; it is not a boundary value problem for Einstein's equations.
• The seven typed foundational replacement carriers declared in deps are now authored on emg-mg02; no independent acceptance is asserted: Finite-real supremum and infimum domain, Real metric and pseudometric convention, Metric nonnegativity from the three axioms, Explicit rational and real power regimes, Sequence-limit criterion with explicit choice use, Boundaryless domain for geometric completeness, and Set-sized objects and definable-class schemas.