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How can I work out in detail the explicit coordinate transformation formulas needed to go from the "canonical" coordinates to the "Poincare patch"? I'm reading about AdS but the text takes the validity of the Poincare patch for granted, which troubles me. It only has a drawing showing the Poincare patch covering half the whole space, which doesn't help.

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I can recommend the article Introduction to the Maldacena Conjecture on AdS/CFT, arXiv:hep-th/9902131, by Petersen. I suspect that the first 8 pages would be helpful.

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