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 Hn=EAdSn 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 Hn in the Poincare patch!
How does one convert between the two? Is there a known transformation?
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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)
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Or is there a reference where these results have already been found in terms of the Poincare patch coordinates?
- In the hyperbolic model of Hn the space is thought of a zero-set in Rn+1 of the equation, x20−∑ni=1x2i=a2 and then one uses the coordinates y∈[0,∞) and and →n∈Sn−1 to write, x0=acoshy and →x=a→nsinhy and then the metric is, ds2=a2[dy2+sinh2ydΩ2n−1]
Here dΩ2n−1 is the standard metric on Sn−1.
(..and this is the metric in equation 2.15 in the linked paper..)
- In the Poincare patch model of Hn it is thought of as the half-space xn>0 in Rn with the metric, ds2=a2z2(dz2+∑n−1i=1dx2i)
(relabeling xn as z)
This post imported from StackExchange MathOverflow at 2014-09-09 10:51 (UCT), posted by SE-user user6818