Deriving AdS Poincare Coordinates from Global Coordinates

[formodular]

Is there a straight-forward, motivated, derivation of AdS Poincare coordinates, e.g. as given here:

https://en.wikipedia.org/wiki/Anti-de_Sitter_space#Poincar.C3.A9_coordinates
starting from global coordinates, as given here:

https://en.wikipedia.org/wiki/Anti-de_Sitter_space#Global_coordinates
The coordinate transformations starting from global line element (in the wikipedia notation) ##ds^2 = \alpha^2(-\cosh^2 \rho d \tau^2 + d \rho^2 + \sinh^2 \rho d \Omega_{n-2}^2)## to ##ds^2 = \frac{\alpha^2}{z^2}(dz^2 + dx_{\mu} dx^{\mu})## seem to be pulled out of thin air, even in Zee's Gravity book - is there a simple straight-forward way to motivate the coordinate transformations?

 

[PAllen]

My guess would be that the transform was derived from 'guessing' a target metric. Generally, the gauge freedom in GR means that any of several broad families for metric forms (e.g. Gaussian, Riemann Normal, harmonic, etc.) can be specified for a patch of any manifold whatsoever. Then, a question becomes how large patch can they cover. So, without knowing the history, I would guess that, guided by these general types of metrics, someone guessed that the Poincare form was sufficiently general to capture quasilocal curvature, then derived explicit coordinate transformations to achieve this. If your guessed metric works (doesn't constrain the metric beyond what can be achieved by gauge freedom), you end up with differential equations for the transform that are, in principle solvable. Actually solving them will vary greatly in difficulty, depending on the particular case. Inspired guesswork typically plays a role.

 

[formodular]

Zee (sort of) shows Poincare coordinates as coming from solving
$$(T^2 - X^2) + (W^2 - Y^2) = 1$$
for
$$W^- W^+ = (W - Y)(W + Y) = 1 + (X^2 - T^2)$$
and then setting ##X = x/w##, ##T = t/w## to find
$$W^- W^+ = 1 + \tfrac{x^2}{w^2} - \tfrac{t^2}{w^2} = \tfrac{1}{w}[w + \tfrac{1}{w}(x^2 - t^2)]$$
so that we can define
$$W^- = \tfrac{1}{w},$$
$$W^+ = w + \tfrac{1}{w}(x^2 - t^2).$$
Now from ##Y = W - 1/w## we get
$$W = \tfrac{1}{2}[\tfrac{1}{w} + w + \tfrac{1}{w}(x^2 - t^2)]$$
and from ##W = Y + 1/w## we get
$$Y = \tfrac{1}{2}[- \tfrac{1}{w} + w + \tfrac{1}{w}(x^2 - t^2)].$$
He calls these Poincare coordinates.

It would be great to have a really deep feeling for why one would even think to do any of this, and especially why one would want to set ##X = x/w##, ##T = t/w##, and ##W^- = \tfrac{1}{w}##.