We know that $SO(d,2)$ is the isometry group of $AdS_{d+1}$. Let's only consider $AdS_3$ in this question.

In Poincare coordinates ($r,t,x)$, these can be grouped as follows :

  1. Two translations $$r'=r\; ;\; t'=t+a \;;\; x'=x$$ $$r'=r\; ; \; t'=t \; ; \; x'=x+a$$

  2. One boost $$r'=r\; ; \; t'=tcosh(\alpha) + xsinh(\alpha) \; ; \; x'=tsinh(\alpha) + xcosh(\alpha)$$

  3. One dilatation $$r'=\lambda r \; ; \; t'={t \over \lambda} \; ; \; x'={x \over \lambda}$$

  4. Two special conformal transformations which transform $(r,t,x)$ in a complicated manner (hence, not writing them down).

The metric of $AdS_3$ in Poincare coordinates : $$ds^2 = ({{l^2}\over{r^2}})dr^2 + (l^2r^2)(-dt^2+dx^2)$$ There are 6 generators, as expected, corresponding to the $SO(2,2)$ isometry.

I'm trying to derive them from the $SO(2,2)$ rotation isometry of the Embedding Space $\mathbb{R}^{2,2}$.

The metric of the Embedding space in $(X_{-1},X_0,X_1,X_2)$ coordinates : $$ds^2 = -dX_{-1}^2 - dX_0^2 + dX_1^2 + dX_2^2$$

In the figure (please refer the link below), I have listed down the six $4\times4$ matrices, each of which rotate two of the 4 Embedding space coordinates. In the figure P stands for Poincare and G stands for Global. Ignore the G's.

For each matrix, I'm trying to match its corresponding isometry in Poincare coordinates, using the following coordinate transformation between Poincare and Embedding space.

$$r = {{(X_{-1}-X_2)}\over{l}} \; ; \; t={{X_0}\over{(X_{-1}-X_2)}} \; ; \; x={{X_1}\over{(X_{-1}-X_2)}} $$

As we can see from the figure, matrix 3 and 4 correspond to the dilatation and boost isometry in Poincare coordinates respectively. But, none of the other matrices (matrices 1,2,5,6) leave $r$ invariant $(r'=r)$, while we know that the translation isometry in Poincare coordinates (given above) leaves $r$ invariant.

Here's my question:

Q : How do I get the translation isometry of $AdS_3$ in Poincare coordinates from the $SO(2,2)$ isometry of the Embedding space, when none of the other matrices leave $r$ invariant?

I'm sorry but I'm facing an error when I try to upload my figure in the body of my message. Please check the link : https://imgur.com/a/Kxn9xKS

Any help would be much appreciated!

Thank you for taking the time to read this question.


To make clear how I'm able to derive the isometric coordinate transformation in Poincare coordinates from the isometric coordinate transformation of the Embedding space, please look at the following picture : https://imgur.com/a/zvEGJZ3

In this picture, I've derived one of the isometric coordinate transformation of 2-d Euclidean space in polar coordinates from the translation isometry of the same metric in Cartesian coordinates. If I did the same thing for the rotation isometry in Cartesian coordinates, I will get the following isometry in polar coordinates : $$ r \rightarrow r \; \; ; \; \; \theta \rightarrow \theta + \alpha$$

I'm using the same approach to determine the isometric coordinate transformations in Poincare coordinates of $AdS_3$ using the known isometric coordinate transformations in Embedding space.

I agree that the manifolds that I'm comparing are not the same, like in the 2-d Euclidean case I showed above. But $AdS_3$ is a submanifold (the hyperboloid: $X_{-1}^2 + X_0^2 - X_1^2 - X_2^2 = l^2$) of $\mathbb{R}^{2,2}$, and the metric of $AdS_3$ is the induced metric on the hyperboloid (in Poincare coordinates, it's given above in the main question). So, the procedure should work fine.

Please let me know if you have any questions regarding my question.


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