# Poincare Patch covers half of the hyperboloid of AdS

+ 3 like - 0 dislike
50 views

We start with the general case of $AdS_{p+2}$ i.e AdS space in $p+2$ dimension. $$X_{0}^{2}+X_{p+2}^{2}-\sum_{i=1}^{p+1}X_{i}^{2} = R^2$$ This space has an isometry $SO(2,p+1)$ and is homogeneous and isotropic. The Poincare Patch is given by $$ds^2 = R^{2}\left(\frac{du^2}{u^2}+u^2(-dt^2 +d\mathbf{x}^{2})\right)$$ According to Equation (2.27) of the article http://arxiv.org/abs/hep-th/9905111, The second metric covers only half of the hyperboloid. Firstly, how do I show this. Secondly, when I go to the asymptotic limit (small radial distance), should the topology of the two spaces be different?

This post imported from StackExchange Physics at 2014-07-28 11:14 (UCT), posted by SE-user Debangshu
 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): Email me at this address if my answer is selected or commented on: 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$\hbar$ysic$\varnothing$OverflowThen 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.