Holography in Fefferman-Graham coordinates

[B]

Hi All,

I'm having some trouble with setting up the framework for a 5d bulk metric and its 4d boundary metric. Ultimately I'd like to solve the Einstein equations in the Fefferman-Graham coordinates. 

With pure gravity, the Einstein equation is R_ab + 4G_ab =0 in 5 dim (G_ab is the bulk metric), and with the following FG convention: 

ds^2 = G_ab dx^a dx^b = dr^2/(4r^2) + 1/r g_munu dx^mu dx^nu (r usually is rho)

I (analytically) know that this becomes:

1/2 Tr(g^-1 g'')- 1/4 Tr(g^-1 g'g^-1 g') = 0 (where ' denotes the derivate wrt the radial coordinate r/rho)

Now I would like to do this calculation with xAct, and run into some problems.

I have tried to use the product of a 1d and 4d manifold to describe the bulk, and use the warp product to describe the bulk metric. However, my problem then is to assign the right values to the 1d and 4d metric according to FG-coordinates. A more specific question here is, is it possible to define a chart to a 1d metric? 

More generally I'm asking whether someone has any experience with this holography-type of calculations in xAct. i.g. setting up the framework for the boundary values of any tensors/metrics/fields. (Am I correct there are no such examples available so far?)

To better understand where I'm coming from, here is the paper I've been studying. It concerns section 3, the derivation of eq. 3.14-3.18 from 3.2-3.3 with the above-mentioned FG coordinates.

Hope someone can help,

B

[Zhanfeng Mai]

Dear B,

  I have a file about FG expansion. I found it last in the Internet but I forgot the website and the author of this file. Thus I could not give you a correct reference. 

  The file has been attached and hopefully, it can help you.

在 2020年5月16日星期六 UTC+8上午12:23:34，B写道：

[B]

Dear Zhanfeng,

Thanks for your help, much appreciated! This code is McNees' code he shared earlier in this group, which has helped me a lot for a different calculation. My question is somewhat different though. Let me try to explain.

My question doesn't involve the FG expansion, it only involves its form where the bulk metric is split into the radial coordinate "rho" (or its square) and the boundary metric. This boundary metric is then expanded according to the FG expansion showed in McNees' code. However, I'm trying to get the boundary metric from the bulk metric. As this is a general principle in holography, I still hope someone with more xAct experience than me would be able to help me with this. 

Best,

B

Op zaterdag 16 mei 2020 06:00:58 UTC+2 schreef Zhanfeng Mai:

[Pedram Karimi]

Dear B

One trick is to use a line element and then build the boundary metric by it. It would be something like this:

ds2=1/z[]^2 (-dt^2 + dx^2 + dy^2 + dz^2)

matrix1 = CoefficientArrays[ds2 z[]^2 , {dx, dy, dt}, "Symmetric" -> True][[3]] // Normal // Simplify;

Now you can define the boundary metric on its manifold. Of course, this method would work for any given bulk metric.

you can also make some rules to eliminate the radius component of the metric. I personally don't recommend such a method.

Best,

Pedram

[B]

Hi Pedram,

Thanks for your reply, I have implemented in the attached notebook. However, my problem here is that you I want to have a general boundary metric whereas you have defined a Minkowski space. 

This time I have attached a notebook, and hope someone can help me with what I need. As mentioned before, such an holography computation is relatively standard, and I'm hoping that someone has experience with such a xAct approach.

Best,

B

Op maandag 18 mei 2020 17:22:53 UTC+2 schreef Pedram Karimi: