Anti-de Sitter spacetime metric and its geodesics

[MManuel Abad]

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.

 

[Ben Niehoff]

In the last metric, put

[tex]r = b \sinh \rho[/tex]
and see if you can figure it out from there. Hint: AdS_n can be considered as a hyperboloid of one sheet in [itex]R^{n-1, 2}[/itex].

This hint serves to tell you how to find the geodesics as well. The isometries of AdS are the same as the isometries of the ambient [itex]R^{n-1, 2}[/itex], and hence geodesics are intersections between the one-sheeted hyperboloid of AdS and planes through the origin of [itex]R^{n-1, 2}[/itex]. (Similar to how geodesics on the 2-sphere are the intersections between the sphere and planes through the origin of R^3.)

 

[MManuel Abad]

Wow, man, that was really useful! I think I understand now how to go from the hyperboloid description to the AdS metric as Witten writes it. Nevertheless I still do not figure out how to go from this (Witten's) metric to one of the former ones.

It took me a while, but now I think I get it:

The AdS space can be considered as an "sphere" on a real space with hyperbolic metric [itex]G_{\mu\nu}=diag(-1,-1,+1,...,+1)[/itex]:

[itex]\sum\limits_{i=1}^{n}x_{i}^{2}-y_{1}^{2}-y_{2}^{2}=-L^{2}[/itex]

This can be achieved by the parameterization:

[itex]Y_{1}=b\cosh\rho\sin\tau[/itex]
[itex]Y_{2}=b\cosh\rho\cos\tau[/itex]

[itex]X_{1}=b\sinh\rho\cos\phi_{1}[/itex]
[itex]X_{2}=b\sinh\rho\sin\phi_{1}\cos\phi_{2}[/itex]
...
[itex]X_{n-1}=b\sinh\rho\sin\phi_{1}\cos\phi_{2}...\sin \phi_{n-2} \cos\phi_{n-1}[/itex]
[itex]X_{n}=b\sinh\rho\sin\phi_{1}\cos\phi_{2}...\sin \phi_{n-2} \sin\phi_{n-1}[/itex]

where capital letter have been used for the representation of the coordinates of points on the hyperboloid (we just used hyper-spherical coordinates)

Then, if one is not lazy (I am, so I just did it for n=1 and 2) one can find that, using the formula for the induced metric:

[itex]g_{ab}=G_{\mu\nu}\frac{\partial X^{\mu}}{\partial\xi^{a}}\frac{\partial X^{\nu}}{\partial\xi^{b}}[/itex]

where [itex]a,b \in \{ 0,1,...n \}[/itex] and [itex]\xi^{0}=\tau,\ \xi^{1}=\rho,\ \xi^{2}=\phi_{1}...\ \xi^{n}=\phi_{n-1}[/itex], that:

[itex]ds^{2}_{AdSn+1}=-b^{2}\cosh^{2}\rho d\tau^{2}+b^{2}d\rho^{2}+b^{2}\sinh^{2}\rho d\Omega_{n-1}^{2}[/itex]

which, under the transformations [itex]t=b\tau[/itex] and [itex]r=b\sinh\rho[/itex], yield the metric as Witten's wrote it.

Nevertheless, as I said before, I still don't know how to get from this (Witten's) metric to those I wrote at the beginning of my post.

Also, I know how to find the geodesics, but I hope you can help me finding some references where it they are obtained. I need these references because I need to cite the (preferably original) work where this geodesics appear.

 

[atyy]

http://www.aei.mpg.de/~gielen/ads.pdf

 

[MManuel Abad]

Wow, thanks, atyy. It appears I did it in another way: in the pdf file you gave me they first obtain the metric in the last form I wrote, and after that in the hyperbolic-functions form. I did it the other way-around :P I can see how they obtain the other forms of the metric, that was really helpful!

Anything about the references with the geodesics?

 

[atyy]

There seems to be something about geodesics in Chapter 5 of http://books.google.com/books?id=SbZpFMOwhzMC&source=gbs_navlinks_s .

 

[Ben Niehoff]

I haven't read what Aty posted, but once you see how to get AdS metrics from the hyperboloid, just try to think of other ways to parametrize the hyperboloid. For example, what happens if you define

[tex]\begin{align}
t &= X^0/X^{n+1}, \\
x^1 &= X^1/X^{n+1}, \\
& \vdots \\
x^n &= X^n/X^{n+1},
\end{align}[/tex]
where [itex]X^0, X^{n+1}[/itex] are the two negative-signature coordinates in [itex]R^{n-1,2}[/itex]?

Edit: As for geodesics, if you know how to obtain them from the hyperboloid, then all you have to do is some algebra. I don't think this is something you have to cite...it's a simple fact that works in homogeneous spaces.

 

[MManuel Abad]

Yes, Ben Niehoff. Actually, in the atyy's first reference, they begin with that very same parametrization :D Thanks a lot.

Thanks for the book, atyy