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

by mma
Tags: cartan's magic, continuity equation
 P: 230 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?
 Sci Advisor HW Helper PF Gold P: 4,768 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.
 P: 230 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!

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