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RS 1, massless scalar field, and sep of vars.

  1. Feb 29, 2008 #1
    Hey, this is rather involved but I hope someone can help me out.

    I am reading http://arxiv.org/abs/0704.3626 ( the casimir force in randall sundrum models) and am trying to get from equation 2.1 :

    [tex]g^{\mu\nu}\partial_\mu\partial_\nu\Phi+e^{2ky}\partial_y(e^{-4ky}\partial_y\Phi)=0[/tex]

    to equation 2.2 ...'The general solution for the non-zero modes can be eppressed in terms of Bessel functions of the 1st and 2nd kind as:

    [tex]\chi(y)=e^{2ky}(a_1 J_2(\frac{m_Ne^{ky}}{k}) +a_2Y_2(\frac{m_Ne^{ky}}{k}) )[/tex]

    I'll show you my attempts and if anyone has the patience can maybe help me.

    First express field [tex]\Phi[/tex] as:
    [tex]\Phi(x_\mu,y)=\phi(x_\mu)\chi(y)[/tex]

    Plug this into the original differential equation

    [tex](g^{\mu\nu}\partial_\mu\partial_\nu\phi)\chi+e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)\phi=0[/tex]

    Divide by [tex]\phi\chi[/tex] to obtain


    [tex] \frac{ (g^{\mu\nu}\partial_\mu\partial_\nu\phi) }{ \phi}+


    \frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=0[/tex]

    As the first term only depends on [tex]x_\mu[/tex] and the second on y, each term in the equation must be a constant, so for the [tex]\chi(y)[/tex] term we can write:

    [tex]\frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=n^2[/tex]

    where [tex]n^2[/tex] is some constant.

    Tidying this up a bit we get:

    [tex] e^{-2ky}\partial^2_y\chi} -4ke^{-2ky}\partial_y\chi-n^2\chi=0[/tex]

    Make this easier to compare to Bessels equation by writing

    [tex]e^{2ky} \rightarrow x^2, \chi\rightarrow y[/tex]

    My final equation now looks like:

    [tex]x^2y''-4kx^2y'-n^2y=0[/tex]

    Whereas Bessels equation is

    [tex]x^2y''+xy'+(x^2-n^2)y=0[/tex]

    So this is my problem, I obtain an expression that is not quite bessels equation. Can anyone see the error of my logic??

    Thank you so much.
     
    Last edited: Feb 29, 2008
  2. jcsd
  3. Feb 29, 2008 #2

    kdv

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    But I don't see how you can make that change of variable. In Bessel equation, the parameter "x" is the variable differentiated with respect to! So you must be more careful. I will look into it when I have some time tonight.
     
  4. Mar 1, 2008 #3
    Cool Kdv, any help appreciated, and I see what you are saying about the change of variable so thank you on that point.
     
  5. Mar 2, 2008 #4
    Hmmm I seem to remember working this out before. I think the problem is that you have to add boundary terms to make it work right, that is, to get a good definition of von Neuman or Dirichlet B.C.'s.

    Check out the Lecture notes by Gherghetta : http://arxiv.org/abs/hep-ph/0601213. I think he works it out in detail, but I could be wrong.

    Also you have to be careful. The separation of variables that you should use is

    [tex]\Phi(x_{\mu},y) = \sum_n \phi_n(x_{\mu})\chi_n(y)[/tex]

    up to some constant that gives you the correct mass dimension. This gives you the KK tower, etc.
     
  6. Mar 2, 2008 #5

    kdv

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    I can't get it to work out. But I suspect there is something more to the story. They mention treating separately the N=0 mode from the others and this does not show up at all in the steps you followed. See the post by BenTheMan too.
     
  7. Mar 2, 2008 #6

    George Jones

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    I couldn't get it to work out either, so I glanced at the reference given in the original post's paper. This reference is the the link given in BenTheMan's post. As BenTheMan said, there's another term.
     
  8. Mar 2, 2008 #7
    Ok guys, I did have a look at the Les Houches Lectures last week but it didn't help much. I'll take another look this evening and see if I can make progress. Thanks all for having a look at this.

    Cheers for the correction in my separation of vars Ben. Hope all is good with you!!

    Rich
     
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