Continuity equation is(adsbygoogle = window.adsbygoogle || []).push({});

[itex]dj+\partial_t\rho_t=0[/itex]

where [itex]j[/itex] and [itex]\rho[/itex] are a time-dependent 2-form and a time-dependent 3-form on the 3-dimensional space [itex]M[/itex] respectively. (see e.g. A gentle introduction to

the foundations of classical electrodynamics (2.5))

If we use differential forms on the 4-dimensional space-time [itex]\mathbb R\times M[/itex] instead of time-dependent forms on [itex]M[/itex], than the continuity equation tells that the integral of the [itex]J:=\rho+dt\wedge j[/itex] 3-form on the boundary of any 4-dimensional cube is 0, hence [itex]dJ=0[/itex].

If we apply Cartan's magic formula to [itex]J[/itex] and the vector field [itex]v:=\partial_t[/itex] then we get:

[itex]L_vJ=\iota_vdJ+d(\iota_vJ)=d(\iota_vJ)=dj[/itex]

On the other hand, [itex]L_vJ=\frac{\partial}{\partial t}\tilde\rho_t[/itex]

where [itex]\tilde\rho_t=\varphi_t^*\rho[/itex], where [itex]\varphi[/itex] is the flow of [itex]v(=\partial_t)[/itex], i.e. [itex]\tilde\rho_t[/itex] is the same time-dependent 3-form [itex]\rho_t[/itex] on [itex]\{0\}\times M\simeq M[/itex] as appear in the starting continuity equation.

Consequenty, from Cartan's magic formula we get [itex]\partial_t\rho_t=dj[/itex], i.e.

[itex]dj-\partial_t\rho_t=0[/itex]

So, there is a sign difference between this equation an the continuity equation. Were is the error?

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# Ostensible contradiction between the continuity equation and Cartan's magic formula

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