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.
For this case anti-de Sitter space is obtained by setting the extra coordinate equal to . The metric in the original Poincaré coordinates looks like:
After the T-transformation (leaving the coordinate intact) one obtains
Now adS is reached when the extra coordinate vanishes. In Poincaré 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:
I might not want to keep the scale invariance so obvious for the coordinates.