# Index Concordia

## In search for anti-de Sitter space (II)

Posted in Relativity by Index Guy on June 30, 2008

I want to continue the discussion started here about a metric that may yield anti-de Sitter space as some limit.

We can write the “original” six coordinates in terms of the following five:

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

We now define the following symbols:

$\alpha = r^2 + y^2$   and   $d\beta = \displaystyle\frac{rdr + ydy}{r}.$

These symbols allow us to write the differentials in a more compact form:

$dZ^{0} = d\beta - \displaystyle\frac{R^2 + \alpha}{2r^2}dr,$

$dZ^{1} = -d\beta - \displaystyle\frac{R^2 - \alpha}{2r^2}dr$   and

$dY^{m} = \displaystyle\frac{rRdy^m - Ry^m dr}{r^2}.$

Recall the form of the metric:

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

The flat part, i.e. the two terms with the etas, will change to the usual five-dimensional anti-de Sitter metric in Poincare coordinates. The task then is to find an expression for the metric in the new coordinates introduced above. The answer is:

$ds^2 = \displaystyle\frac{R^2}{r^2} \left(dy^2 - dr^2 \right) + \displaystyle\frac{2y\cdot dy}{r^3}\left(r^2 + y^2 - R^2 - \displaystyle\frac{2 y^2 R^2}{r^2}\right)dr - Wdr^2,$

With

$W = \displaystyle\frac{R^2}{r^2} + \frac{r^2}{R^2} 2y^2\left(\frac{R^2}{r^4} + \frac{2}{R^2}\right) - y^2 y^2 \left( \frac{R^2}{r^6} - \frac{6}{R^2 r^2}\right) + \frac{4 y^2 y^2 y^2}{R^2 r^4} + \frac{y^2 y^2 y^2 y^2}{R^2 r^6}.$

Observe that $W$ is a polynomial in $\xi = y^2$. We are looking for a metric that will reduce to 5-d adS. So at some point we must have the coefficient functions of the two extra terms tend to zero or vanish. The vanishing of the second term in the metric implies:

$\xi = y^2 = \displaystyle\frac{r^2 - R^2}{\displaystyle\frac{2 R^2}{r^2} - 1}.$

Meanwhile the third term gives a quartic polynomial in $\xi$. I plugged it in Mathematica, and the four roots were not pretty. None of the roots seem to coincide with the condition from the second term. This leads me to believe that there is no way that the metric for anti-de Sitter space is reached. Hence this metric might not be what I am looking for…

Another question is how to interpret the off-diagonal terms in the metric. The next step is to try modifying the definition of traditional 5-d adS with the inclusion of some arbitrary function.

Update: According to Mathematica I made a bid doodoo. The full metric after the change of coordinates goes as:

$ds^2 = \displaystyle\frac{R^2}{r^2}\left(dy^2 + dr^2\right) + \displaystyle\frac{R^2}{r^6}\left( r y \cdot dy - y^2 dr\right)^2.$

This is certainly interesting. First, when we have “light-like” objects (i.e. $y^2 = y\cdot dy = 0$), the metric reduces to anti-de Sitter space. Second, it is kind of compact and looks like it can reduce to adS in some other cases. I am still not sure about the off-diagonal terms…

Second and last update: I am a complete moron. First of all, since I am considering constant $Z_{2}$ the differential will vanish! No matter in what coordinates. This was verified by Mathematica early in the morning. So the end result: I just found another way of obtaining 5-d anti-de Sitter space. This post contains highly incorrect contents.