Ostensible Contradiction b/w Continuity & Cartan's Magic Formula

[mma]

Continuity equation is

dj+\partial_t\rho_t=0​

where j and \rho are a time-dependent 2-form and a time-dependent 3-form on the 3-dimensional space M 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 \mathbb R\times M instead of time-dependent forms on M, than the continuity equation tells that the integral of the J:=\rho+dt\wedge j 3-form on the boundary of any 4-dimensional cube is 0, hence dJ=0.

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

L_vJ=\iota_vdJ+d(\iota_vJ)=d(\iota_vJ)=dj​

On the other hand, L_vJ=\frac{\partial}{\partial t}\tilde\rho_t

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

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

dj-\partial_t\rho_t=0​

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

 

[quasar987]

I think the problem is the definition of J! If I compute dJ, I get

dJ=d_{sp}\rho+dt\wedge\partial_t\rho - dt\wedge d_{sp}j

where d_{sp} means the exterior differential wrt spatial coordinates only. Now, d_{sp}\rho=0 since rho is a 3-form on a 3-manifold, and so we see that dJ=0 iff \partial_t\rho - d_{sp}j = 0 which is not the conservation equation. On the other hand, with J:= -\rho +dt\wedge j we do get dJ=0, and your little playing around with Cartan's formula gives dj+\partial_t\rho=0 at the end.

 

[mma]

Oh, yes, this solves the problem. I think that I got lost beause the four current vector in Physics is defined with +rho, but now I recognised that it means a - sign when I turn it to differential form because of the - sign in the Minkowski metric.

Thank you very much, Quasar!