MManuel Abad
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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 AdS_{n+1}) line element is given, in some coordinates, by:
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}
This can be written, with the change of coordinates z=\frac{L^{2}}{r} as:
ds^{2}=\frac{L^{2}}{z^{2}}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}+dz^{2}]
and, with z=Le^{-y/L}:
ds^{2}=e^{2y/L}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}]+dy^{2}
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:
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}
with d\Omega^{2}_{n-1} the line element for the unit S^{n-1} sphere.
I'd like to know: what transformation relates this last expression of AdS_{n+1} 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 b and L?
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 AdS_{n+1} 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 AdS_{n+1}) line element is given, in some coordinates, by:
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}
This can be written, with the change of coordinates z=\frac{L^{2}}{r} as:
ds^{2}=\frac{L^{2}}{z^{2}}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}+dz^{2}]
and, with z=Le^{-y/L}:
ds^{2}=e^{2y/L}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}]+dy^{2}
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:
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}
with d\Omega^{2}_{n-1} the line element for the unit S^{n-1} sphere.
I'd like to know: what transformation relates this last expression of AdS_{n+1} 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 b and L?
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 AdS_{n+1} 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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