Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Mar 1, 2012 #1

    mma

    User Avatar

    Continuity equation is

    [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?
     
  2. jcsd
  3. Mar 1, 2012 #2

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Re: Ostensible contradiction between the continuity equation and Cartan's magic formu

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

    [tex]dJ=d_{sp}\rho+dt\wedge\partial_t\rho - dt\wedge d_{sp}j[/tex]

    where d_{sp} means the exterior differential wrt spatial coordinates only. Now, [itex]d_{sp}\rho=0[/itex] since rho is a 3-form on a 3-manifold, and so we see that dJ=0 iff [itex]\partial_t\rho - d_{sp}j = 0[/itex] which is not the conservation equation. On the other hand, with [itex]J:= -\rho +dt\wedge j[/itex] we do get dJ=0, and your little playing around with Cartan's formula gives [tex]dj+\partial_t\rho=0[/tex] at the end.
     
  4. Mar 2, 2012 #3

    mma

    User Avatar

    Re: Ostensible contradiction between the continuity equation and Cartan's magic formu

    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!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Ostensible contradiction between the continuity equation and Cartan's magic formula
  1. Cartan's Identity (Replies: 3)

  2. Cartan tensor (Replies: 2)

Loading...