[MUSIC] Let us describe the properties of the obtained metric. So the metric that we have obtained has the following form, where r is implicit function of u and v. To remind you, dU dV is equal to -r squared d omega squared and I remind you that u and v are related to a regional cardinal axis as follows. 2rg exponent of -t- r divided by 2rg times r divided by rg- 1, one-half. And v = 2rg exponent of t + r divided by 2rg multiplied by r divided by rg-1 one-half. And now we're going to draw and describe this space-time. Before doing that, let me remind that if we multiply u by v, we obtain that r divided by rg-1 Exponent of r divided by rg is- UV divided by 2rg squared and if we divided V by U, V divided by U is exponent of t divided by r g. So now one can see that for constant r we have hyperbolas, for constant t we have straight lines which are passing through the origin. So straight this constant so we get a straight line. And the hyperbolas degenerate, if 1r goes to rg, the hyperbolas degenerate to the lines defined by this equation. So to the lines U = 0, V = 0. So we want to draw the lines. But to do that let me stress that U and V are light like coordinates, so the equation dU = 0 and dV = 0 defines light rays for light ray propagation for constant angles. So it is convenient to define this new T-R and V= T+R coordinates. And in these new coordinates, T and R, so let me draw them. So this is T, this R, now one can see that the line V is this one. So this is line V, this is line U, this is line U. And one can see that constant r, constant r corresponds to hyperbolas, constant r corresponds to hyperbolas like this. So the situation is very similar to the one which we have encountered in Rindler space, as you remember in the first lecture. So this is R constant and constant T are straight lines passing through the origin. So this is T constant, T constant. So, again, this line corresponds, as one can see from this formula, this corresponds to R = rg. This is U0, U0. And this also corresponds to r= rg and simultaneously this corresponds to t = +infinity, as can be seen from here. And this corresponds to t = -infinity, also as can be seen from here. So this is U = 0 this line, and this V = 0, this line. And Schwarzschild coordinates cover only this quadrant. While this U and V are similar to Menkosky coordinates covering the whole space. So Schwarzschild coordinates, our region of T and R coordinates, are similar to Rindlerian coordinates and tau. Remember that we have encountered in the first lecture. But what is important here that using this coordinates, one can continue to R less than rg. Perhaps using, well anyway, one can continue beyond this point. But this metric is singular, as we discussed during this lecture, along the hyperbola which corresponds to r = 0. This hyperbola has two images here and here. So this is Kruskal–Szekeres coordinates and their properties and we are now ready to draw Penrose-Carter diagram for this case. So we have obtained Kruskal–Szekeres coordinates. Which is T big, well it's more proper to use x letter than R, here, because x is ranging from minus infinity to plus infinity where I assume that r is ranging from 0 to infinity. So x. And this is then, this is, V coordinate, this is an U white like U coordinate, and this Kruskal–Szekeres space-time is not valid beyond this point. So, it's not valid here, so, this is r = 0 this hyperbola, so it's not valid beyond this point. So the situation is more or less up to this, the situation is more or less similar to Rindler, relation between Rindler and flat space-time. We can transform from U, which is T- x and V, which is T + X to the, we can do the same transformation, basically as we did for Minkowski space to the new coordinates, psi and xi. From this two coordinates to this, the same transformation, because as you may notice, so we are concerned about this part only, the relevant two dimensional part of this metric. The relevant two dimensional part is just this guy, rg dV dU, so we concentrate on this and making this space non compact space, this here. Making this space the same transformation as before. So then the diagram, Penrose-Carter diagram in this case, is basically the same square as before, with one important difference. So what we have is that, so we basically should obtain the square, The same square as before. Psi, xi but with important difference because we have to chop off here, the upper and lower triangular part. Chopping off this. So, this is r = 0. r = 0 with this map. So, but, it is important that under the conformal map, under the standard conformal map, these curves would map to something which is curvy here. But by adjusting conformal factor, one can make them straight. And now so this is a Penrose-Carter diagram for the Schwarzschild spacetime. Let me discuss it in a greater detail. [SOUND] [MUSIC]