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

1. Apr 25, 2017

### spaghetti3451

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. May 1, 2017

### PF_Help_Bot

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