# Index Concordia

## Yet another candidate

Posted in Relativity by Index Guy on July 9, 2008

Today I found yet another space that looks promising. This one looks even more nicer than the previous cases. To obtain the metric one can start with the product of 3-d Minkowski and d-dimensional Minkowski space. The coordinates can be taken as

$C^{M} = (Z^A, z, Y^m),$

with $A$ a two-dimensional Minkowski index and $m$ a d-dimensional Minkowski index. Let us write the metric as:

$ds^2 = -2dZ_{+}dZ_{-} + dz + \eta_{mn}dY^{m}dY^{n}.$

Now one imposes the condition:

$C^2 = 0,$

and introduces “Poincaré coordinates” that satisfy this constrain. These can be defined as:

$Y^m = \displaystyle\frac{R}{r}y^m,$      $\sqrt{2}Z_{+} = \displaystyle\frac{R^2}{r},$      $\sqrt{2}Z_{-} = \displaystyle \frac{r z^2}{R^2} + \displaystyle\frac{\eta_{mn}y^m y^n}{r}.$

Then writing the metric in terms of these coordinates one obtains:

$ds^2 = \displaystyle\frac{R^2}{r^2}dy^2 + \frac{z^2}{r^2}dr^2 + \frac{2z}{r}drdz + dz^2.$

One can see that when we set $z = R$ we obtain exact d-dimensional anti-de Sitter space. This metric is very atractive. It contains more symmetry than the other cases that I have consider previously. Besides the first term being invariant under Poincaré transformations, one also has an invariance under the rescalling of the $(y^m , r)$ coordinates. I shall explore this space further.