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A Correct coordinate transformation from Poincare-AdS##_3## to global AdS##_3##

  1. Oct 9, 2017 #1
    Consider the transformation from Poincare-AdS##_3## geometry to global AdS##_3## geometry:

    $$ds^{2} = \frac{dr^{2}}{r^{2}} + r^{2}g_{\alpha\beta}dx^{\alpha}dx^{\beta}, \qquad \text{Poincare-AdS$_3$}$$
    $$ds^{2} = \frac{dr^{2}}{r^{2}} + r^{2}\left(-dt^{2}+r^{2}d\phi^{2}\right), \qquad \text{Poincare-AdS$_3$}$$
    $$ds^{2} = - r^{2}dt^{2} + \frac{dr^{2}}{r^{2}} + r^{4}d\phi^{2}, \qquad \text{Poincare-AdS$_3$}$$
    $$ds^{2} = -\cosh^{2}\rho\ d\tau^{2} + d\rho^{2} + \sinh^{2}\rho\ d\varphi^{2}, \qquad \text{global AdS$_3$}$$

    where the transformation of coordinates is as follows:

    $$\rho = \ln r, \qquad \tau = \left(\frac{2e^{\rho}}{e^{\rho}+e^{-\rho}}\right)t, \qquad \varphi = \left(\frac{2e^{2\rho}}{e^{\rho}-e^{-\rho}}\right)\phi.$$

    ------------------------------------------------

    The transformation ##\rho = \ln r## simply rescales the radial distance ##r## by the logarithmic function.

    The transformation with ##\displaystyle{\tau = \left(\frac{2e^{\rho}}{e^{\rho}+e^{-\rho}}\right)t}## rescales the time ##t## by the factor ##\displaystyle{\frac{2e^{\rho}}{e^{\rho}+e^{-\rho}}}##. For example, at ##\rho = 0##, we have ##\tau = t##, and at ##\rho = \infty##, we have ##\tau = 2t##.

    The transformation with ##\displaystyle{\varphi = \left(\frac{2e^{2\rho}}{e^{\rho}-e^{-\rho}}\right)\phi}## rescales the angle ##\phi## by the factor ##\displaystyle{\frac{2e^{2\rho}}{e^{\rho}-e^{-\rho}}}##. For example, at ##\rho = 0##, we have ##\varphi = \infty##, and at ##\rho = \infty##, we have ##\varphi = \infty##.

    -------------------------

    Have I made a mistake in my interpretation of the transformation ##\displaystyle{\varphi = \left(\frac{2e^{2\rho}}{e^{\rho}-e^{-\rho}}\right)\phi}##?
     
  2. jcsd
  3. Oct 13, 2017 #2

    Ben Niehoff

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    I don't think any of your interpretations are correct. I wouldn't use "rescale" to describe any of these, especially not for ##\tau## and ##\varphi##. The new coordinates are a mixture of all of the old ones, not merely "rescaled".

    To give the best interpretion of these, I would draw some diagrams.
     
  4. Nov 24, 2017 #3
    You may have a look at this online worksheet. At the end of it, the transformation from Poincaré coordinates to global ones is considered. This is for AdS4, but I guess you can easily adapt to AdS3.
     
  5. Dec 11, 2017 at 3:32 AM #4
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