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A divergence-free current with nonzero face-flux density: J=x^1∂_2 in flat space

Example

Let $M = R^{4}$ carry the identity chart, the flat metric $g = diag (- 1, 1, 1, 1)$ , the time orientation $N = \partial_{0}$ and the chart orientation, with metric-selected connection $\nabla$ and volume form ${v o l}_{g}$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Oriented smooth manifolds and oriented charts, The metric volume form, current flux through 3-faces and finite-chain charges). Let

$J : = x^{1} \partial_{2}$

be the smooth vector field of A smooth vector field is a smooth section of the tangent bundle. Then:

$\nabla_{a} J^{a} = \partial_{2} x^{1} = 0$ : the current is divergence-free.
The current $3$ -form is $ι_{J} {v o l}_{g} = x^{1} d x^{0} \land d x^{1} \land d x^{3}$ , and $d (ι_{J} {v o l}_{g}) = 0$ identically, consistently with clause (a) of Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current.
The pointwise flux density of $J$ through a non-null face with adapted unit normal $ν^{*}$ is $ε g (J, ν^{*}) \sqrt{∣ \det (g (v_{i}, v_{j})) ∣}$ ; for the coordinate directions this is $0 for ν^{*} = \pm \partial_{0}, \pm \partial_{1}, x^{1} for ν^{*} = \partial_{2}, - x^{1} for ν^{*} = - \partial_{2} .$ Thus the divergence-free current has nonzero flux density through the faces normal to $\partial_{2}$ : vanishing divergence does not mean vanishing flux through every face, and the flux balance $Φ_{J} (Σ_{2}) - Φ_{J} (Σ_{1}) = - Φ_{J} (B)$ of clause (b) of Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current controls the side boundary genuinely.

Facts & Assumptions

Given: The flat Lorentzian four-manifold $(R^{4}, g)$ with identity chart, $g = diag (- 1, 1, 1, 1)$ , metric-selected connection $\nabla$ , volume form ${v o l}_{g}$ and the vector field $J = x^{1} \partial_{2}$ .

[L1]

${v o l}_{g} = d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ , $w = 1$ , and $Γ \equiv 0$ on the identity chart (The metric volume form, current flux through 3-faces and finite-chain charges, The derivative of the metric density and the chart formula for the divergence of a vector field, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

[L2]

$ι_{\partial_{2}} {v o l}_{g} = (- 1)^{2} d x^{0} \land d x^{1} \land d x^{3}$ and $ι_{J} {v o l}_{g}$ is obtained by multiplying the $\partial_{2}$ -insertion by the component $J^{2} = x^{1}$ (Interior product of a form by a vector field, The wedge product of differential forms).

[L3]

$d (ι_{J} {v o l}_{g}) = (\nabla_{a} J^{a}) {v o l}_{g}$ (Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current clause (a)).

[L4]

On a non-null face with velocity basis $(v_{1}, v_{2}, v_{3})$ and adapted unit normal $ν^{*}$ , $ε = g (ν^{*}, ν^{*})$ and the pullback of $ι_{J} {v o l}_{g}$ on $(v_{1}, v_{2}, v_{3})$ is $ε g (J, ν^{*}) \sqrt{∣ \det (g (v_{i}, v_{j})) ∣}$ (Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current clause (c), Chart independence of the metric volume form, its value on orthonormal frames, and the unit normal and density of a non-null hyperplane).

Verification

technique · direct; compute the divergence, the current form and its derivative, then evaluate the flux densities on the coordinate normals

1.1L1algebra

(Divergence.) By [L1], $\nabla_{a} J^{a} = \partial_{a} J^{a} = \partial_{0} 0 + \partial_{1} 0 + \partial_{2} x^{1} + \partial_{3} 0 = 0$ .

2.1step 1.1L2L3algebra

(Current form.) By [L2] and $J^{2} = x^{1}$ , $ι_{J} {v o l}_{g} = x^{1} d x^{0} \land d x^{1} \land d x^{3}$ . Its exterior derivative is $\partial_{1} x^{1} d x^{1} \land d x^{0} \land d x^{1} \land d x^{3} = 0$ , because $d x^{1}$ occurs twice; consistently, [L3] and step 1.1 give $d (ι_{J} {v o l}_{g}) = 0$ .

3.1step 2.1L4algebra∎

(Flux densities.) At a point of a face with adapted unit normal $ν^{*} = \partial_{2}$ (so $ε = g (\partial_{2}, \partial_{2}) = 1$ and $\sqrt{∣ \det (g (v_{i}, v_{j})) ∣} = 1$ for a coordinate face), [L4] gives the flux density $g (J, \partial_{2}) = g (x^{1} \partial_{2}, \partial_{2}) = x^{1}$ ; for $ν^{*} = - \partial_{2}$ it is $- x^{1}$ . For $ν^{*} = \pm \partial_{1}$ , $g (J, \pm \partial_{1}) = 0$ ; for $ν^{*} = \pm \partial_{0}$ , $g (J, \pm \partial_{0}) = 0$ . This is the displayed list of values. Since $x^{1} \neq 0$ on the open set ${x^{1} \neq 0}$ , the flux density through the faces normal to $\partial_{2}$ is nonzero there although the divergence of $J$ vanishes identically, which completes the verification.

Remarks

Reading. The example isolates the pointwise flux density: a divergence-free current can have nonzero density through faces normal to $\partial_{2}$ . The flux-balance theorem concerns integrals over oriented chains and states exactly how the slice fluxes differ by the side flux; whether a whole side-chain flux vanishes depends on that chain and its orientation.

A divergence-free current with nonzero face-flux density: J=x^1∂_2 in flat space

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources