I have been reading Thomas Hartman's lecture notes (http://www.hartmanhep.net/topics2015/) on Quantum Gravity and Black Holes.(adsbygoogle = window.adsbygoogle || []).push({});

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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 metricat 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?

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

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