What are the boundary asymptotics of harmonic symmetric transverse traceless rank-s tensors on $\mathbb{H}_n$ in the Poincare upper-half-space model?

Question

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?

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• 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 $z$)

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This post imported from StackExchange MathOverflow at 2014-09-09 10:51 (UCT), posted by SE-user user6818

Comments

you got a good answer at math.stackexchange.com/questions/499042/…

This post imported from StackExchange MathOverflow at 2014-09-09 10:51 (UCT), posted by SE-user Will Jagy

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@WillJagy As far as I can see that answer is not helping - in the Mobius transformation described there in the large $y$ limit at fixed $y_n$ the asymptotic relation is $y \sim \frac{1}{z}$ - this paper says that the large-y asymptotics are of the form $e^y$ and hence in $z$ coordinates they will look like $e^{\frac{1}{z}}$ - and this makes no sense in small $z$ - I would like to isolate all those harmonics which near $z=0$ scale as some say $z^a$ for a given number $a$ - but that can't be done in this form!

This post imported from StackExchange MathOverflow at 2014-09-09 10:51 (UCT), posted by SE-user user6818

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There are many, many isometries on the Poincare disk, or ball in more dimensions. You are free to apply those before the map to upper half space to get your desired conditions. what is your actual background?

This post imported from StackExchange MathOverflow at 2014-09-09 10:51 (UCT), posted by SE-user Will Jagy

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Can you tell as to what is the metric in the x, y and the z coordinates of the transformations given in that answer? Then I can be sure that it at least gets the start and the end metric as the two metrics which I have listed. [..can you give an example of the isometries of the Poincare disk that you mention?..is there a way to just write them down?...and hope that one of them does something in the exponent - I would have thought that there is some function $f$ such the $e^y = f(z)$...)

This post imported from StackExchange MathOverflow at 2014-09-09 10:51 (UCT), posted by SE-user user6818

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I actually don't see why $e^{1/z}$ "makes no sense". Why do you think there should be harmonics that scale like power law near the conformal boundary? After all, in hyperbolic models the area of the spheres near infinity grows exponentially. (Also, check your signs; isn't the large $y$ behaviour in the paper you cited $e^{-\rho y}$ where $\rho > 0$?)

This post imported from StackExchange MathOverflow at 2014-09-09 10:51 (UCT), posted by SE-user Willie Wong

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@WillieWong There are physics reasons to believe as to why a polynomial behaviour is the right answer. From other discussions that I had, I learnt that the way to convert the answer to my $z$ coordinates is to realize that the dependency on $y$ should be geometrically thought of as a dependency on the geodesic distance from the vertex of the hyperboloid. So one transforms that distance into the $z$ coordinates and then asks the question about the functions.

This post imported from StackExchange MathOverflow at 2014-09-09 10:51 (UCT), posted by SE-user user6818

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@WillieWong [....now that I think of it the basic issue is that since I want to transform the asymptotics of eigenfunctions of the Laplacian, it probably is not true that if I change coordinate systems then the new coordinate substituted into the old function will still keep them eigenfunctions of the Laplacian...]

This post imported from StackExchange MathOverflow at 2014-09-09 10:51 (UCT), posted by SE-user user6818

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@WillieWong I have given what I think is the right answer in this new question, mathoverflow.net/questions/144159/… [...it would be great to see you there...]

This post imported from StackExchange MathOverflow at 2014-09-09 10:51 (UCT), posted by SE-user user6818