[MUSIC] Now, once we have been very detailed during the discussion of de Sitter space, anti-de Sitter space, we will describe more concisely as already can be understood from the previous discussion of the generic geometry of this anti-de Sitter space. But now we continue with the consideration of the coordinates in anti-de Sitter space. So we're going to solve this equation, -x 0 squared + sum over j from 1 to D-1 X j squared minus X D squared, 1 over H squared. So the solution for this case can be given like this. 1 over H hyperbolic cosine of rho, cosine of tau, XD = 1 over H hyperbolic cosine of rho, sine of tau. So basically, this plus this give us 1 over H hyperbolic cosine of rho squared. And X j is 1 over H hyperbolic sine of rho times nj, where nj squared = 1 and j is ranging from 1 to D-1. So, and then if we use, well, this basically defines those coordinates on D-2 dimensional sphere. And as a result, if we use these coordinates and plug them into the metric of ambient space, we get the induced metric in anti-de Sitter space, which has the following form, 1 over H squared hyperbolic cosine squared of rho d tau squared minus d rho squared minus sine squared of rho d omega d minus 2 squared, where this is the coordinates of the sphere given by this n's standing here. This is product. So one can see immediately that, well, one comment which is important to say here is the following, that if we are dealing with D = 3, so we consider two-dimensional anti-de Sitter space, no, if D = 2, sorry, two-dimensional anti-de Sitter space, because then here we have D+1 of course coordinates. For D=2, for the case of two-dimensional anti-de Sitter space, the equation that we obtained here is very similar to the equation for two-dimensional de-Sitter space. In fact, it just coincides with it. And what is important here that this factor is absent, for the case of D = 2, we don't have this factor. So the metric is just this. And in fact, in this case, one can easily see the two-dimensional anti-de Sitter space, two-dimensional anti-de Sitter space is the same kind of hyperboloid, but rotated by the angle p over 2, so it lays here. So this is X0, X1, and X2. That what happens, so the same kind of hyperboloid, but embedded into, yeah, better to say that actually, this is x2, and x1, because, and X, in this coordinates, and in X0 and X2 we have a circle. And that actually already is seen here. This tau, which is time, is ranging from 0 to 2 pi. So we have a compact time. So we have closed timelike curves in anti-de Sitter space, which is considered as physically unfavorable. And that's the reason one usually considers universal color of this space. So the same metric like this, but with tau ranging from minus infinity to plus infinity. That is what usually called by anti-de Sitter space in physics literature, frequently called. So, universal, how, the same metric like this, but with time ranging not in this range, but here. And one more comment, that in two-dimensional anti-de Sitter space, rho here is ranging in the limits from minus infinity to plus infinity. And constant tau slice is just hyperbola, one of the hyperbolas lying on this hyperboloid. And two-dimensional anti-de Sitter space has two boundaries, here and here. While, if we consider D > 2, starting with 3 and so forth, rho, with tau, the range of values of tau is not dependent on the dimensionality. It's always appropriate to have it like this, but we consider universal cover. But what happens with rho, that's only for D=2 like this. For D>2, rho is ranging from 0 to infinity. And that's how we cover uniquely only once the anti-de Sitter space. And actually, time slice, this part of the metric, for this case has the following form. It's 1 over H squared d rho squared plus hyperbolic sine of rho squared d omega D-2 squared. This is actually D dimension, D-1 dimensional Lobachevsky space. D-1 dimensional Lobachevsky space. We have encountered it for the three-dimensional case. And one can obviously see that the metric we have been considering before is obtained from this one when H = 1 and rho is exchanged for the chi. We have been using different notation for this, calling it rho before. So, and in this case, de Sitter space in dimension greater than 2, anti-de Sitter space has only one boundary, not two boundaries. In fact, while rho equal to minus and plus infinity corresponds to these two boundaries of the two-dimensional de Sitter space, rho equals to 0 corresponds to the center of the, topologically, this guy's disk. rho equals to 0 corresponds to its center, while rho is infinity corresponds to its boundary. So it is also important that the spatial boundary of this space, so when rho goes to plus infinity, is conformally equivalent to the cylinder. So of this space, of this space, when we take rho to infinity, the space that we obtain is conformally equivalent to R times S D-2. It's actually obvious, apparent from here, if we take rho to infinity, this becomes exponent of 2 rho and this becomes exponent of 2 rho. So we drop off this at the boundary. Exponent factor l gives us a conformal factor which we can drop off, but the rest is tau, which we've taken to be ranging in this, so R, and sphere, which is this. So that's the geometry of the anti-de Sitter space. [MUSIC]