[MUSIC] Okay, we are still considering this hyperboloid X0 squared plus. x1 squared plus x d squared equal to h minus second. And we going to now we discuss the generic things about the geometry of this space etc. Now we're going to provide coordinates in the space, or for example. This hyper void is embedded into the space with signature d x 0 squared minus d x 1 squared minus d x. D squared. So one can solve this equation very easily. For example, by X zero equal to hyperbolic sign of HT divided by H. And XI equal to NI times hyperbole cosin of HT divided by H where I is running from one to D. So, actually I is running all of theese guys and if ni squared sum over i = 1. So this defined the units here ni. We obtained that these things solve this equation. Now if we flag this into here. From ni we obtain d- 1 dimensionals sphere. We'll discuss it in a moment. So the metric that is induced on this space from this metric is equal to dt squared- hyperbolic of sine squared of Ht over h squared d omega D- 1 squared. So this is a sphere, D- 1 dimensional unit sphere and it appears exactly from this guy. Exactly from this guy. And how to see that? Let me explain it. So for example, this equation which is the difference d- 1 dimensional sphere in d dimensional space can be solved as follows. We can define n1 which is cosine, cosine of theta 1. And theta 1 is ranging between- pi over 2 of two P over P then N two equal to sine tetra one times cosine tetra two and tetra two is ranging between SP over two P over 2 and so forth. Up to md minus 1 which is sine of theta 1, sine theta 2 dot dot dot, sine Theta D minus 2 times cos of theta D minus 1 was theta D minus 1. All these angles ranging in this range and at the same time, theta D minus 1 is ranging between Minus pi and pi. Remember that, well let me finish first and then I'll explain what what is going on. ND is equal to sin TAT one sin TAT to two dot dot dot sin TAT to D minus two sin. Theta d- 1. So the situation is very similar with the case of two-dimensional sphere. Remember that are two angles phi which is ranging exactly in this range. And theta which is ranging in this range. This how we cover two-dimensional sphere only once. And that is exactly the same situation that we encounter here. So all the angles except this one take this range and only one angle ranges here. In case if D = 2, D = 2 we obtain only that one angle here phi or theta 1 which is ranging in this range in fact. So that is a circle. So we going to draw this in a moment again, but let me just say the following thing. The d omega d -1 squared. If we use this which solves this and plug it here, we obtain that this guy is just nothing but. Sum over g from 1 to d minus 1 and product of over i from 1 to g minus 1 of sins. Square of theta i times d theta j squared. This is a metric on you need d minus 1 dimensional sphere. Actually by the way, big rotation can be seen already on this metric. If we change Ht to i theta D minus pi over 2, if we do such a change in this metric. Then we map this metric to 1 over h squared times d omega. So this metric is map to this, d omega d squared. So which is exactly the map between de sitter space. And d dimensional sphere, and now one can straightforwardly see the meaning of this picture that I drew. So when we have h zero, which is in fact simply changes geniously, rising from minus infinity to plus infinity. This is a x1, x2. We obtain this hyperboloid every time slice constant x0 corresponds to constant t. So, every time slice of this hyperbola, this is just a sphere. D minus one dimensional sphere. For this two dimensional case, it's just a circle because in this case, we have a one dimensional sphere which is a circle and actually using this. Using this one can see that hyperbolic distance is expressed as false. It's just a minus hyperbolic sign of Ht1 times hyperbolic sign of Ht2 + hyperbolic cosine of Ht1 times hyperbolic cosine of Ht2 times cosine of omega. Where cosine of omega is nothing but n1 scale up of the form where n1 and n2 are at least vectors defining first and second point. So, now the geometry of the space becomes apparent and we're going to consider with the consideration of Penrose diagram to discuss causal properties of decision space. We're going to continue the discussion of Penrose diagram for the space. [MUSIC]
