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A Deriving the Poincare patch from global coordinates in AdS##_{3}##

  1. Apr 25, 2017 #1
    I have been reading Thomas Hartman's lecture notes (http://www.hartmanhep.net/topics2015/) on Quantum Gravity and Black Holes.

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    In page 97, he derives (9.4), which is the metric of AdS##_{3}## in global coordinates:

    $$ds^{2} = \ell^{2}(-\cosh^{2}\rho\ dt^{2} + d\rho^{2} + \sinh^{2}\rho\ d\phi^{2}).$$

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    In page 100, he states that, expanding the metric at large ##r## under the coordinate change

    $$t^{\pm} = t \pm \phi, \qquad \rho = \log(2r),$$

    we can show that, to leading order, the induced metric on the hyperboloid AdS##_{3}## becomes

    ##ds^{2} = \ell^{2} \left(\frac{dr^{2}}{r^{2}}-r^{2}dt^{+}dt^{-}\right).##

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    I find, under the coordinate change, that

    ##ds^{2} = \ell^{2} \left(\frac{dr^{2}}{r^{2}} - \frac{1}{4}dt^{+2} - \left(r^{2} + \frac{1}{16r^{2}} \right) dt^{+}dt^{-} - \frac{1}{4}dt^{-2} \right).##

    Of course, the term in ##1/r^{2}## drops off at large ##r## but I am not able to get rid of the components in ##dt^{+2}## and ##dt^{-2}##. Am I missing something here?

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    He then goes on to mention that these are Poincare coordinates, but does not make contact with the usual way in which the metric of AdS##_{3}## is written in Poincare coordinates:

    $$ds^{2} = \frac{\ell^{2}}{z^{2}}(dz^{2}-dt^{2}+d\vec{x}^{2}).$$

    What am I missing here?
     
    Last edited: Apr 25, 2017
  2. jcsd
  3. May 1, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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