This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf

In this paper some of its most important results about the asymptotics of symmetric traceless transverse harmonic rank-s tensors on $\mathbb{H}_n = EAdS_n$ in equations 2.27, 2.28, 2.88, 2.89 are all given in hyperbolic coordinates.

But for reasons of physics one wants to write $\mathbb{H}_n$ in the Poincare patch!

How does one convert between the two? Is there a known transformation?

I would like to know if the large-y (variable defined below) behaviour as stated in the said equations of the linked paper can be converted to find the small-z behaviour (variable defined below)

Or is there a reference where these results have already been found in terms of the Poincare patch coordinates?

- In the hyperbolic model of $\mathbb{H}_n$ the space is thought of a zero-set in $\mathbb{R}^{n+1}$ of the equation, $x_0^2 - \sum_{i=1}^n x_i ^2 = a^2$ and then one uses the coordinates $y \in [0,\infty)$ and and $\vec{n} \in S^{n-1}$ to write, $x_0 = a cosh y$ and $\vec{x} = a \vec{n} sinh y$ and then the metric is, $ds^2 = a^2 [ dy^2 + sinh ^ 2 yd\Omega_{n-1}^2]$

Here $d\Omega_{n-1}^2$ is the standard metric on $S^{n-1}$.

(..and this is the metric in equation 2.15 in the linked paper..)

- In the Poincare patch model of $\mathbb{H}_n$ it is thought of as the half-space $x_n > 0$ in $\mathbb{R}^n$ with the metric, $ds^2 = \frac{a^2}{z^2}(dz^2 + \sum_{i=1}^{n-1}dx_i^2 )$

(relabeling $x_n$ as $z$)