[MUSIC] To conclude our discussion of the anti-de Sitter spacetime, let me just introduce Poincare patch and Poincare coordinates of the anti-de Sitter spacetime. Well to remind you, anti-de Sitter spacetime is given by this equation. -X 0 squared + sum over j from 1 to D- 1 X j squared- XD squared = -1 over H squared. And it is embedded into the space which has signature +, -, -, -, and +. Now one can solve this equation, this equation, as follows by giving X0 = z over 2 [1 + 1 over z squared (1 over H squared- x mu times x mu)]. And z here is greater than or equal to 0. X1 = z over 2 [1- 1 over z squared (1 over H squared + x mu times x mu)]. And X mu, big X, X mu = x mu over H, where mu = 2, ..., D. And one last point is that x mu times x mu = eta mu nu times x mu times x nu. But eta mu nu has a signature which is a bit different from the one that we are using. It has pluses, all pluses and one Remember that before we are using Minkowskian metric where one plus and the rest are minuses. Here it's a bit different. So for z greater than 0, for z greater than 0, one can see from this expressions that X0- X1 = 1 over (H z) squared. And it is actually greater than 0. So these coordinates, unlike the previous ones which cover entire into de Sitter space, these coordinates cover only half of it, given by this condition. And in fact, if one uses these coordinates, the induced metric on the anti-de Sitter space is just ds squared = 1 over (Hz) squared [dz squared + dx mu dx mu] which is very similar, very similar to the metric in expanding and contracting patches. And remember that we had like this (H eta plus minus squared) [d eta plus minus squared- dx squared]. But here, the important crucial difference with respect of this de Sitter metric here at this time. But here that is space-like. Time in anti-de Sitter space is among these coordinates. It's like dx vector squared- dt squared so time is this. So there the anti-de Sitter spacetime metric is stationary, it's actually even static while the de Sitter spacetime metric is time dependent. That is a crucial difference between these two spaces. Actually this coordinates according to this condition one can easily see what part of the entire, what half of the entire. In case of two dimensions, one can see easily what part of the entire anti-de Sitter space it covers. It's just like this half of the anti-de Sitter space in higher dimensions. In two it is a bit harder to see. But it still covers half of the entire anti-de Sitter spacetime. Well, this ends up our discussion of the anti-de Sitter space and spaces of constant positive and negative curvature and actually ends up our introduction to general theory of relativity. And so that's the end of the story. Good luck. Good luck at the exams. Good luck at the study of this course. I hope you will enjoy learning this subject and use it in your further studies of basics of theoretical physics, quantum field theory, gravity, etc., and etc. [SOUND]