# How to interpret this image of $AdS_5\times S^5$?

In the context of AdS/CFT an image like the following (coming from this article by David Mateos) is often shown:

but I'm not really sure if I interpret it correctly.

As the article says, we have a static spacelike stack of $N_c$ infinite 3-D branes (assumed to be at zero separation), whose energy density deforms the metric of Minkowski $9 + 1$-space to one that would obviously have to be constant on submanifolds that are the product of a 5-sphere transversal to the branes and $\Bbb R^3$, parallel.

Since the branes concentrate their energy in a lower-dimensional subspace, we get a horizon, which must also have the shape $S^5\times\Bbb R^3$. $R$ denotes the $S^5$ radius of the horizon.

As a Lorentzian manifold the resulting space is not equal to $AdS_5\times S^5$, but close to the branes the metric takes the form

$$ds^2 = \frac{r^2}{R^2}\eta^{\mu\nu}dx_\mu dx_\nu + \frac{R^2}{r^2}dr^2 + R^2d\Omega_5^2.$$

The last term is the standard metric on a Euclidean sphere of radius $R$, while the first two terms are the anti-de Sitter metric on a (locally) perpendicular factor.

Far from the branes the metric tends to the Minkowski metric.

Now for the image. Why is it drawn like this and why is it called a "throat geometry"? Is it just to say that the closer you get to the horizon, the larger distances become w.r.t. the coordinate distance, and an additional dimension is used so that Euclidian distance in this 11-dimensional space restricted to our 10-dimensional space (and depicted like a 2D surface in 3D) corresponds with the intrinsic distances in the actual space? If so, I guess the Schwarzschild solution also exhibits a "throat geometry".

I guess that the circles are just concentric $S^5$'s for different $r$, so we see the $S^5$ and from $AdS_5$ we only see the $r$-coordinate. If that is correct, we don't see any timelike directions, which is OK I guess since we are in a static situation, unless we'd like to see light-cones.

Can the $3 + 1$-dimensional Minkowski space on which the supersymmetric Yang-Mills theory dual to the string theory in $AdS_5\times S^5$ should live, and which is the boundary of our $AdS_5$ be appreciated? From what I understand, it is not there in the image at all (or just as a point).

Wouldn't it be more interesting to show some dimension parallel to the branes instead of one of $S^5$? Or would we be missing something then? What I have in mind is something like this:

In this image we see a cylinder, which is the horizon $S^5_R\times\Bbb R^3$, the stack of branes in red, and the same perpendicular coordinate $r$, now showing two dimensions of the $AdS$-factor. The Minkowski boundary would be any vertical line on the surface of the cylinder.

Is this indeed what it is? Or am I misunderstanding something?

• Have you tried to actually not just look at the picture but read 2.2 in the link you provided? Because if I wrote an answer it would essentially be that section. – OON Jun 25 '17 at 4:42
• @OON Fair point. I will make my question clearer later today and make it independent from what can be directly read in the article itself. For the moment I'll delete it. – doetoe Jun 25 '17 at 8:08
• @OON changed it, including the link by the way – doetoe Jun 25 '17 at 23:27