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

1. Oct 9, 2017

### highflyyer

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:

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

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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 $\varphi = \infty$.

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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. Oct 13, 2017

### Ben Niehoff

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.

3. Nov 24, 2017

### egourgoulhon

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.

4. Dec 11, 2017 at 3:32 AM

Thank you.