Scattering from an AdS₃ bubble and an exact AdS₃

We investigate the close relationship between the potential and absorption cross section for test fields in an AdS$_3$ bubble(a 5D black hole) and an exact AdS$_3$. There are two solutions in type IIB string theory: an AdS$_3$ bubble corresponds to the dilatonic solution, while an exact AdS$_3$ is the non-dilatonic solution. In order to obtain the cross section for an AdS$_3$ bubble, we introduce the \{out\}-state scattering picture with the AdS$_3$-AFS matching procedure. For an exact AdS$_3$, one considers the \{in\}-state scattering picture with the AdS$_3$-AdS$_3$ matching. Here the non-normalizable modes are crucially taken into account for the matching procedure. It turns out that the cross sections for the test fields in an AdS$_3$ bubble take the same forms as those in an exact AdS$_3$. This suggests that in the dilute gas and the low energy limits, the S-matrix for an AdS$_3$ bubble can be derived from an exact AdS$_3$ space.