Index Concordia

Three roads to anti-de Sitter space

Posted in Relativity by Index Guy on July 8, 2008

I want to summarize the three different routes I am pursuing to reach anti-de Sitter space.

First is the direct route. This means to start with anti-de Sitter space in the first place. Alday and Maldacena followed this path. For future references the metric for adS space in Poincaré coordinates looks like

ds^2 = \displaystyle\frac{R^2}{r^2}\left(\eta_{mn}dx^m dx^n + dr^2\right).

Second is to deform the exact adS metric by introducing some arbitrary function A(r) of the fifth coordinate. Right now my main struggle is what form of the deformation should I use. Here are some favorites:

  • ds^2 = \displaystyle\frac{R^2 - A^2}{r^2}\left(\eta_{mn}dx^m dx^n + dr^2\right)
  • ds^2 = \left(\displaystyle\frac{R^2 - A^2}{r^2}\right)dr^2 + \displaystyle\frac{R^2}{r^2}\eta_{mn}dx^m dx^n
  • ds^2 = \displaystyle\frac{R^2}{r^2}\left(1 + A^2\right)dr^2 + \displaystyle\frac{R^2}{r^2}\eta_{mn}dx^m dx^n

Lastly we can try something completely different. Along this road it was suggested to start with an extra spatial dimensions and impose some constraints. One gets a six-dimensional space. In Poincaré coordinates this looks like:

ds^2 = \displaystyle\frac{R^2}{r^2}\left(\eta_{mn}dx^m dx^n + dr^2\right) + \left(dz + \displaystyle\frac{z}{r}dr\right)^2 .

After the so-called “T-dual” transformation, the metric obtains the following form:

ds^2 = \displaystyle\frac{R^2}{\rho^2}\left(\eta_{mn}dy^m dy^n + d\rho^2\right) + \left(dz - \displaystyle\frac{z}{\rho}d\rho\right)^2 .

This latter approach is the most attractive in my humble opinion. It contains adS in the limit when the coordinate z vanishes. We also have an invariance under a scale transformation on the coordinates (x^m , r) or (y^m, \rho). What am I suppose to do with this metric? I want to consider classical strings propagating in any of these particular backgrounds.

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