Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,873 answers , 20,701 comments
1,470 users with positive rep
502 active unimported users
More ...

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

+ 3 like - 0 dislike
218 views

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 $z$)



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

asked Sep 21, 2013 in Theoretical Physics by user6818 (955 points) [ revision history ]
retagged Aug 28, 2016
Most voted comments show all 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
@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
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
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
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
@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
@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

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ys$\varnothing$csOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...