Let us start with the following background metric:
where the functions and are functions of the extra coordinate . The case of anti-de Sitter space corresponds to
Under the “T-duality” transformation
the metric in the dual space takes the form
Since we can always bring the metric in this form, we will just consider the initial case and see what can we do with it. Note that it could be the case that the change of variables between could be of the general form
This case could be more complicated… For now we will just stick with the first change of variables introduced.