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.