[MUSIC] Okay, we have started our consideration of this hyperboloid, hypersurface. Which is actually hyperboloid as we will see in a moment. Where A and B run from 0 to D. So it's a hyperboloid embedded into D + 1 dimensional space, Minkowskian metric. So we made observation that it is a space of constant positive curvature for two reasons, that under the change of X0, out of this X0 to iX D plus 1, it is mapped to. To a sphere, D Dimensional sphere embedded into D plus 1 dimensional Euclidean space. That's the first way of observing. The second observation was made that this is a homogeneous space. Which is SO(D,1), isometer of the space SO(D,1), which is Laurentian group of D + 1 dimensional. D + 1 dimensional Minkowskian space time. Over SO(D-1,1), which is a stabilizer, a sub group of this group, which doesn't move arbitrary point of this space. And this tells us that every point in this space is equivalent under the action of this group to every other and every direction is equivalent to every other. Up to the difference between space like and time like directions. So there are several other observations that can be made. On the basis of this is the following thing. How do we define using this weak rotation as convenient to further the moves, how do we define distance on this sphere? On the sphere we define the distance as follows. So if we have a two-dimensional sphere, it's easy to see. So the distance between arbitrary two points on the sphere of radius r is nothing but the r times this angle. This angle between two. So we have our original, two radius vectors. Directing to the two points, one and two. And the distance, Geodesic distance is just this segment, which is which segment where I will explain in a moment. So the distance is just this angle measured in radiance times r. So the geodesic distance is just r theta where r is the radius. But how do we define this distance? We take two points, so we have two vectors. X. 1, it is a vector defining this point. So it, because it's laying on the sphere, it obeys this. We have X2, second vector. X2 which is laying on the sphere. And to define this, we just take the scalar product of X1 to X2. So X1. On the sphere, we have this. X1 times X2. Just scalar product gives us R squared times cosine of this guy, which is nothing but the Geodesic distance between this point divided by R. Cosine of this angle. This theta is just equal to this, because this is L12. Great. So in fact, weak rotation, helps us to see, that the same procedure can be applied in this case. Even without drawing the pictures. Well this is in dimensions this procedure and this sphere is similar. So using weak rotation we can define so called hyperbolic distance. If we have 2XA1 being the relation. Both X1 and X2 are being this condition. So laying on this hyperboloid. This is a hyperboloid, we will see in a moment that this is a hyperboloid. And so if they obey this, we can define their hyperbolic distance between them. Taking the scalar product of X1 to X2, we can define cosine of H times L12 where L12, is the Geodesic distance between points on the hyperboloid, divided by H squared. So this formula can be obtain from this one under the vic and change of r to 1 over H. So, the thing is obvious in this case. Now to see that the fact that this is a hyperboloid, let me just draw if for the case when D equals 2, too. So, when D equals to 2 too, when D equals to 2, we have the following situation basically. We have X0 squared with a minus sign, plus X1 squared, plus X2 squared. X2 squared equal to age to the second power. Which then tells us that for every X0, we have a circle. X1 squared plus X2 squared is equal to H squared plus X0 squared. So on this space of, When we have X0, X2, X1. So this guy is nothing but the one sheeted hyperboloid as follows. So it has minimal radius of the circle at X0 equals to 0, so this is just 1 over H. But then as X0 is increasing the radius is increasing to infinity and increasing to infinity towards backward in zero direction. So, under Wick rotation, this hyperboloid is mapped to the sphere. And this, we can relate geodesic distance to the hyperbolic distance using this formula on, well, this is a spherical distance probably. It's better to call this spherical distance. Hyperbolic distance is this quantity. It's called z12 and like this one which is Geodesic distance. And one last thing that I have to say is the following. How do we obtain geodesics on the sphere? We take a plane which passes through there, suppose we want to find a geodesic between point 1 and 2, how do we do that. And we take a plane which goes through these three points. This one and these two and this one. There's a unique plane. It cuts out from the sphere, an equator and the segment of this equator which joins these two points is exactly the curve of the minimal lengths joining these two points. So here, it can be seen exactly also using this weak rotation that to obtain geodesics, we just have to cut this hyperboloid by the planes passing through the origin. So, by cutting this hyperboloid with the planes which are passing through the origin, we can encounter several situations. First situation is when the curve which is cut out by this plane on the hyperboloid is ellipse. This corresponds to the segments of this ellipses are space-like Geodesic. Then, if we cut like this, we cut out hyperbolas. Segments of these hyperbolas are time-like Geodesic. And if we cut by a plane which cuts this hyperboloid under the angle of 45 degrees we obtain two straight lines which are general traits of this hyperboloid. Everyone knows that you obtain a hyperboloid you have to take just a straight line and rotate it around some, orbit it around some along some circle, then you have hyperboloid. So this lines I call [INAUDIBLE]. So if you take a plane which cuts this hyperboloid under the angle of 45 degrees, you obtain two light-like lines in ambient space, and life-like lines on the hyperboloid and this light-like Geodesic in two dimensional decision space. So this classifies all possible types of Geodesics in the sitter space and the situation is very similar to the sphere which can be seen from the Wick rotation. [MUSIC]