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Magnetic period obstructs a global potential

Example

Let $M = R_{x^{0}, x^{3}}^{2} \times T_{L}^{2}$ , where $T_{L}^{2} = (R / L Z)^{2}$ carries its standard global translation-invariant one-forms $d x^{1}, d x^{2}$ , and give $M$ the flat product Lorentz metric

$g = - (d x^{0})^{2} + (d x^{1})^{2} + (d x^{2})^{2} + (d x^{3})^{2} .$

Take future direction $\partial_{x^{0}}$ and orient $M$ by $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ . In the SI interpretation use the length-coordinate convention $x^{0} = c t$ of Electrovacuum tensor Cauchy system, with $c > 0$ and $x^{0}, x^{3}$ and local lifts of $x^{1}, x^{2}$ measured in metres; $L > 0$ is measured in metres. For a fixed $B_{0} \neq 0$ tesla, define the smooth two-form $F = B_{0} d x^{1} \land d x^{2}$ . Then $F$ and its Hodge dual have constant coefficients, so $d F = d (⋆_{g} F) = 0$ on the flat background. On the oriented spatial torus $S : = {x^{0} = x_{*}^{0}, x^{3} = x_{*}^{3}} ≅ T_{L}^{2}$ , oriented by $d x^{1} \land d x^{2}$ , its period is $\int_{S} F = B_{0} L^{2} \neq 0$ . The field has local potentials but no single-valued global one-form potential on $M$ . This example concerns the Maxwell field-strength sector on a prescribed background; it is not asserted to solve the coupled Einstein equation with that flat metric.

Facts & Assumptions

Given: The exterior-calculus and source-free Maxwell conventions of Electrovacuum tensor Cauchy system and the local potential conclusion of Local potential homotopy and Lorenz gauge reachability.

[F1]

Electrovacuum tensor Cauchy system: the source-free Maxwell equations are $d F = 0$ and $d (⋆_{g} F) = 0$ .

[F2]

Local potential homotopy and Lorenz gauge reachability: every smooth closed two-form has a potential on a star-shaped coordinate neighborhood.

Verification

Given: $L > 0$ , $B_{0} \neq 0$ , flat product coordinates, and $F = B_{0} d x^{1} \land d x^{2}$ .

1.1F1algebra

The global invariant one-forms $d x^{1}, d x^{2}$ on $T_{L}^{2}$ are closed and $B_{0}$ is constant, so $d F = 0$ . The flat Hodge dual has constant coefficients and is closed as well. On $S$ at fixed $x^{0}, x^{3}$ , direct integration over a fundamental square gives $\int_{S} F = \int_{[0, L]^{2}} B_{0} d x^{1} d x^{2} = B_{0} L^{2} \neq 0$ .

2.1step 1.1F2algebra∎

On each star-shaped coordinate neighborhood, [F2] supplies a local potential. If a global smooth one-form $A$ on $M$ satisfied $F = d A$ , its components $A_{1}, A_{2}$ are periodic in the torus variables. At fixed $x_{*}^{0}, x_{*}^{3}$ , direct integration over a fundamental square and periodicity give $\int_{S} F = \int_{[0, L]^{2}} (\partial_{1} A_{2} - \partial_{2} A_{1}) d x^{1} d x^{2} = \int_{0}^{L} (A_{2} (L, x^{2}) - A_{2} (0, x^{2})) d x^{2} - \int_{0}^{L} (A_{1} (x^{1}, L) - A_{1} (x^{1}, 0)) d x^{1} = 0$ . This contradicts the nonzero period in step 1.1, so no global potential exists although local potentials do.

Magnetic period obstructs a global potential

Example

Facts & Assumptions

Verification

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