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.

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