## Updated roadmap

I have become more comfortable with the “T-dual” transformation and now I am confident on what to consider.

Exact adS can be described by the metric

After the T-transformation the metric returns to the same form but now ,

Both of these spaces can be described by embedding a hyper-surface in . They both have scale invariance.

There are two extra-dimensional spaces that one can obtain by embedding a hyper-surface in and reduce to adS in some limit.

**Case I:**

For this case anti-de Sitter space is obtained by setting the extra coordinate equal to . The metric in the original PoincarĂ© coordinates looks like:

After the T-transformation (leaving the coordinate intact) one obtains

**Case II:**

Now adS is reached when the extra coordinate vanishes. In PoincarĂ© coordinates,

While in the T-coordinates we obtain

Note that both of these cases have scale-invariance in the coordinates and . These two cases are atractive because they can be constructed straight from a higher-dimensional reduction. On the other hand one can just modify the adS metric and see what happens. Here are some of the modifications.

- Soft wall:
- Hard wall:
- Cutoff:

I might not want to keep the scale invariance so obvious for the coordinates.

Index Guysaid, on July 15, 2008 at 1:19 pmWhy insist so much in leaving the extra-coordinate unchanged? The metric actually diagonalized when one uses .

For case I one obtains

For case II we have

Notice that for case I the coordinate will not contain derivatives in the action…

Index Guysaid, on July 15, 2008 at 1:28 pmThe change of variables should be done in the T-dual space.