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

[robousy]

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 :

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

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:

$$\chi(y)=e^{2ky}(a_1 J_2(\frac{m_Ne^{ky}}{k}) +a_2Y_2(\frac{m_Ne^{ky}}{k}) )$$

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

First express field $$\Phi$$ as:
$$\Phi(x_\mu,y)=\phi(x_\mu)\chi(y)$$

Plug this into the original differential equation

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

Divide by $$\phi\chi$$ to obtain


$$\frac{ (g^{\mu\nu}\partial_\mu\partial_\nu\phi) }{ \phi}+ <br /> <br /> <br /> \frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=0$$

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

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

where $$n^2$$ is some constant.

Tidying this up a bit we get:

$$e^{-2ky}\partial^2_y\chi} -4ke^{-2ky}\partial_y\chi-n^2\chi=0$$

Make this easier to compare to Bessels equation by writing

$$e^{2ky} \rightarrow x^2, \chi\rightarrow y$$

My final equation now looks like:

$$x^2y''-4kx^2y'-n^2y=0$$

Whereas Bessels equation is

$$x^2y''+xy'+(x^2-n^2)y=0$$

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.

 

[kdv]

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 :

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

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:

$$\chi(y)=e^{2ky}(a_1 J_2(\frac{m_Ne^{ky}}{k}) +a_2Y_2(\frac{m_Ne^{ky}}{k}) )$$

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

First express field $$\Phi$$ as:
$$\Phi(x_\mu,y)=\phi(x_\mu)\chi(y)$$

Plug this into the original differential equation

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

Divide by $$\phi\chi$$ to obtain


$$\frac{ (g^{\mu\nu}\partial_\mu\partial_\nu\phi) }{ \phi}+ <br /> <br /> <br /> \frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=0$$

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

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

where $$n^2$$ is some constant.

Tidying this up a bit we get:

$$e^{-2ky}\partial^2_y\chi} -4ke^{-2ky}\partial_y\chi-n^2\chi=0$$

Make this easier to compare to Bessels equation by writing

$$e^2ky \rightarrow x^2, \chi\rightarrow y$$

My final equation now looks like:

$$x^2y''-4kx^2y'-n^2y=0$$

Whereas Bessels equation is

$$x^2y''+xy'+(x^2-n^2)y=0$$

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.

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.

 

[robousy]

Cool Kdv, any help appreciated, and I see what you are saying about the change of variable so thank you on that point.

 

[BenTheMan]

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

$$\Phi(x_{\mu},y) = \sum_n \phi_n(x_{\mu})\chi_n(y)$$

up to some constant that gives you the correct mass dimension. This gives you the KK tower, etc.

 

[kdv]

Cool Kdv, any help appreciated, and I see what you are saying about the change of variable so thank you on that point.

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.

 

[George Jones]

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.

 

[robousy]

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