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Hello, everybody. I have some doubts I hope you can answer:

I have read that the n+1-dimensional Anti-de Sitter (from now on [itex]AdS_{n+1}[/itex]) line element is given, in some coordinates, by:

[itex]ds^{2}=\frac{r^{2}}{L^{2}}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}]+\frac{L^{2}}{r^{2}}dr^{2}[/itex]

This can be written, with the change of coordinates [itex]z=\frac{L^{2}}{r}[/itex] as:

[itex]ds^{2}=\frac{L^{2}}{z^{2}}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}+dz^{2}][/itex]

and, with [itex]z=Le^{-y/L}[/itex]:

[itex]ds^{2}=e^{2y/L}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}]+dy^{2}[/itex]

Nevertheless I have also seen (in page 7 of this paper by Witten: http://arxiv.org/pdf/hep-th/9803131v2.pdf and in http://www.physics.ntua.gr/cosmo09/Milos2009/Milos Talks 2009/1st day/Charmousis Paper.pdf) another formulation of this spacetime:

[itex]ds^{2}=-(1+\frac{r^{2}}{b^{2}})dt^{2}+\frac{dr^{2}}{1+r^{2}b^{-2}} +r^{2}d\Omega^{2}_{n-1}[/itex]

with [itex]d\Omega^{2}_{n-1}[/itex] the line element for the unit [itex]S^{n-1}[/itex] sphere.

I'd like to know: what transformation relates this last expression of [itex]AdS_{n+1}[/itex] with the former ones? under what circumstances is each one of these formulations used? In the last reference I gave there is some kind of explanation about slicings, but I did not understand. Could you explain it to me, please?

Also, is there any relation between [itex]b[/itex] and [itex]L[/itex]?

Now, I'd really appreciate if you could also give me a link to a reference in which the null and time-like geodesics of this [itex]AdS_{n+1}[/itex] spacetime are obtained (you know, the motion of massless and massive particles), preferrable in the last (Witten's) representation of the metric. I was looking for the reference but I could not find it.

I appreciate your help.

I have read that the n+1-dimensional Anti-de Sitter (from now on [itex]AdS_{n+1}[/itex]) line element is given, in some coordinates, by:

[itex]ds^{2}=\frac{r^{2}}{L^{2}}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}]+\frac{L^{2}}{r^{2}}dr^{2}[/itex]

This can be written, with the change of coordinates [itex]z=\frac{L^{2}}{r}[/itex] as:

[itex]ds^{2}=\frac{L^{2}}{z^{2}}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}+dz^{2}][/itex]

and, with [itex]z=Le^{-y/L}[/itex]:

[itex]ds^{2}=e^{2y/L}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}]+dy^{2}[/itex]

Nevertheless I have also seen (in page 7 of this paper by Witten: http://arxiv.org/pdf/hep-th/9803131v2.pdf and in http://www.physics.ntua.gr/cosmo09/Milos2009/Milos Talks 2009/1st day/Charmousis Paper.pdf) another formulation of this spacetime:

[itex]ds^{2}=-(1+\frac{r^{2}}{b^{2}})dt^{2}+\frac{dr^{2}}{1+r^{2}b^{-2}} +r^{2}d\Omega^{2}_{n-1}[/itex]

with [itex]d\Omega^{2}_{n-1}[/itex] the line element for the unit [itex]S^{n-1}[/itex] sphere.

I'd like to know: what transformation relates this last expression of [itex]AdS_{n+1}[/itex] with the former ones? under what circumstances is each one of these formulations used? In the last reference I gave there is some kind of explanation about slicings, but I did not understand. Could you explain it to me, please?

Also, is there any relation between [itex]b[/itex] and [itex]L[/itex]?

Now, I'd really appreciate if you could also give me a link to a reference in which the null and time-like geodesics of this [itex]AdS_{n+1}[/itex] spacetime are obtained (you know, the motion of massless and massive particles), preferrable in the last (Witten's) representation of the metric. I was looking for the reference but I could not find it.

I appreciate your help.

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