Index Concordia

In search for anti-de Sitter space (I)

Posted in Relativity by Index Guy on June 29, 2008

Let us start by considering the direct product of a 3d Minkowski space with a 4d Minkowski space. The coordinates for this space then go as

X^{m} = \left(Z_{0}, Z_{1}, Z_{2}, Y_{0}, Y_{1}, Y_{2}, Y_{3}\right).

If we impose the condition X^2 = 0 we obtain the following relation:

-Z_{0}^{2} + Z_{1}^{2} -Y_{0}^{2} + Y_{1}^{2} + Y_{2}^{2} + Y_{3}^{2}= -Z_{2}^{2}

We can notice that if we fix the value of the Z_{2} coordinate we obtain one of the conditions for five-dimensional anti-de Sitter space. This will not produce anti-de Sitter space since the surface is not embeded in R^{2,4}. Nevertheless, I am not looking for anti-de Sitter space: I am looking for some space that will reduce to anti-de Sitter space in some limit. Defining \rho = -Z_{2}^{2} we can write

\rho = \eta_{ab}Z^{a}Z^{b} + \eta_{mn}Y^{m}Y^{n} \Longrightarrow \displaystyle\frac{d\rho}{2} = \eta_{ab}Z^{a}dZ^{b} + \eta_{mn}Y^{m}dY^{n}

with the understanding that the etas will represent flat metrics. Going back to the original, seven-dimensional metric we need

dZ_{2}^{2} = -\displaystyle\frac{d\rho^2}{4\rho} = \frac{-1}{\rho}\left( (ZdZ)^2 + 2(ZdZ)(YdY) + (YdY)^2 \right).

Now define the following tensors:

A_{ab} = \eta_{ca}\eta_{db}Z^{c}Z^{d}   ,   C_{mn} = \eta_{pm}\eta_{qn}Y^{p}Y^{q}   and   B_{am} = 2\eta_{ca}\eta_{pm}Z^{c}Y^{p}.

Then I claim the metric can be written as

ds^2 = \left( \eta_{ab} - \displaystyle\frac{A_{ab}}{\rho}\right)dZ^{a}dZ^{b} + \left(\eta_{mn} - \displaystyle\frac{C_{mn}}{\rho}\right)dY^{m}dY^{n} - \displaystyle\frac{B_{am}}{\rho}dZ^{a}dY^{m}.

Now we impose the condition that \rho = -R^2. With this condition we can use the five coordinates used in the Alday & Maldacena article as a starting point:

Y^{m} = \displaystyle\frac{R^2}{r}y^m   ,   Z_{0} + Z_{1} = \displaystyle\frac{R^2}{r}   and   Z_{0} - Z_{1} = \displaystyle\frac{r^2 + y^2}{r^2}.

The next task is to write the metric in terms of these coordinates and hope that we get something that will look like anti-de Sitter space with some extra terms.

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  1. […] under: AdS/CFT, Relativity — Index Guy @ 7:26 pm I want to continue the discussion started here about a metric that may yield anti-de Sitter space as some […]


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