[MUSIC] So, now we have obtained diagram for the black hole. Let me draw it more carefully. Now I'm going to use it to show some quality properties of the black hole space time. And show you why it's convenient to study physics and geometry of space time. So, the diagram that we have obtained. Is as follows let me draw it more carefully, so this is a like like line, like like line so this is, I didn't draw it carefully on the previous board so let me draw it. More carefully. So, this is parallel to this, this is parallel to this, and this is a straight angle, 90 degree angle. This corresponds to r = 0, this corresponds to r = 0. This corresponds to r = rg, this corresponds to r = rg. This part is [INAUDIBLE] by Schwarzschild coordinates. Only this, part. And now as usual white raise in this space, light raise in this space, it can be represented by straight lines going like this. Light rays are like this so this is scry minus, this is scry plus. This is scry -, this is scry +. And this is actually I0, I0, this is I-, this is infinity infinity, so as time goes by, as this big T, time goes by, this light rays are directed like this. This lines are also white rays, white like. As you remember this corresponds to v equals to zero and u equals to zero which is light rate. Of course. And what is important that this space now, let us describe properties of this space. Now, one can see immediately the difference of the space of this Penrose diagram with respect to the flat space diagram. In the flat space diagram, every observer had an excess, to all cache surfaces as it approached future infinity. Here, the situation is quite different. We have a funny saying that if someone, for example, appeared at a point here. For some reason appeared at a point here, light cone, future light cone, starting from this point, go like this, and one will inevitably end up at R equals 2,0 because any massive particle or observer has a whirl line going within the future directed light cone. So, inevitably it will end up in the singularity. So, if you found yourself inside a Schwarzschild black hole, you inevitably end up here. To avoid ending up there, r=0, is the same as to avoid next Monday, moreover from here never gets here. So for example, if an observer which is a fixed above a surface of a black hole grows along hyperbola which R constant that hyperbola in crucial space time as to something like like this here, so this is if you are fixed above the black hole you go along this red line. Or for example if you circle around the black hole because you forget about the angle part of the space time circulating around the black hole is not sensitive. This diagram is not sensitive to the circulation around the black-hole. So if you orbit around the black-hole at a fixed radius, your whirl line will look more or less the same. So an observer from here never see a signal from anyone which is here. From anyone which is live in here. So this region, this region is called black hole. Here the situation is quiet opposite if one appeared here it will inevitably run out from this part, you never to blame on our web. This place is called white hole. Hole. White hole. But we have some kind of double link. So we have a black hole and on top of that we have extend our region to the black hole, we have interior region of the black hole and we have also interior region of a white hole and exterior region to the white hole. And we have some kind of doubling. And it is actually easy to expect if one recalls that general theory of relativity. Is time reversal invariant. Well let me explain it in a bit different maner. So the as a way to represent general theory of relativity in the Hamiltonian from. Which goes beyond our lectures, and this Hamiltonian form, is in variant under the time reversal so then it means that for every solution of general theory of relativity there is a solution which is time reversal of that solution. So if one considers a space time which is static so which is in variant under this transformation and time translation. It is inevitable that complete covering of that spacetime will contain both parts. The regional spacetime and its time reversal. So that's basically the vague explanation why we have simultaneously for the static. In the next lectures, we will see that for the difference observation there will be present only some part of this space time and for what reason we will explain. But this fact that I described to you, what can be observed, also, with the pictures that we have encountered in the previous lecture. The end of the previous lecture, we used an ingoing Eddington-Finkelstein coordinates, and in those ingoing Eddington-Finkelstein coordinates we had. This is r = 0 singularity, and so we had the falling sedation, we had the horizon which is one of the, this is r equals 2rg so it's similar to this sedation. So r equals 2rg which is one of the light rays similar to this guys this light rays also light like surfaces and now in going light rays if you remember were like this. In going light rays were like this in the picture that we have drawn at the very end of the previous lecture and outgoing light rays have this behavior. So you see this kind of situation that I have described here can be seen also in this picture. For example, if somebody found himself here, somewhere, say here, it's future directed cone is like this. So you inevitably go to the singularity. So this is a black hole region. So light rays can not escape from here. You see there is no light like geodesic which goes from this part to outside. So this is a similar situation to this. During the next lecture, we will describe quantitatively these effects that we're now going to explain to you qualitatively on the Penrose diagram. Let me draw it again. The Penrose diagram. Let me draw it here. So, the Penrose diagram is like this. Again. So this a like like and this is like like, scry plus scry minus. And now suppose there is an observer which is fixed above the black hole, it's world line as I just explained is like this or any observer which says always the black hole has a world line something like this. So let me call this observer as number 3. So which always throughout all it's life history stays outside of he black hole. And now suppose this guy drops off at some point, something going inside. Object number 1. Now one can see immediately, well we're going to show explicitly in the next lecture that it takes a finite propatine for this thing to cross the horizon. Finite propotine, this line is at finite lengths. And the lengths of this guy is proper time spent from here to here, there's a finer links but at the same time, one can see that light ray. So, this absorb which stays always outside of the black-hole will never see this object number 1, crossing the horizon. Indeed, the left light ray, which is emitted by this guy at the crossing of the horizon, will be reached, will reach this observer number 3 only at the few chain infinity. So that never happens, never happens. Although this guy takes a finite proper time to cross the horizon and even actually to reach the singularity. So indeed, this is very similar to what we are have encountered in the first lecture in the description of the space time, remember that this surface is future event horizon, this is actually past event horizon so this is future event horizon, it corresponds to u equals to 0 at the same time it corresponds to r equal r g but also it corresponds to t. Equals plus infinity. And remember, this T is the time as measured by observers which are fixed above a black hole. Non-inertial observers which are fixed, but also it takes infinite time for the observer which are not necessarily fixed, as I have shown to you. To you here. So in fact it's similarly to what we have encountered in linear space. Remember that Rho equals to zero was the linear horizon and as it was so it's T, Tao the length of time So, finite amount of proper time, in regular space time, as rho approach to infinity, to 0, was corresponding to bigger, longer and longer amounts of coordinate time, regular time, tau. Here we encounter the same situation ds = 1- rg divided by r one half dt. So this is a relation between proper time and coordinate time in Szekeres spacetime. And as r approaches rg, finite proper times correspond to round bigger and bigger amounts of the coordinate time. This is another way to see the same. Well, the same statement that I have said. But what is interesting, on the other hand, that if someone was falling after this body number one, someone was falling accompany just a little bit later. Absorb the number two. Absorb a number two was going right after this guy tot he black hole, here this guy would never lose this body from his sight because he would receive always light rays coming from this guy but what is interesting that it would see, it will see this buddy when the absorbent number two stays outside of the horizon it sees this body also outside of the horizon but. When it's closest to the horizon, then it will start receiving the light rays refracted or emitted by the body number one after the crossing of the horizon. But it never loses this thing from his sight. So we can qualitatively see properties of the black hole with the use of the pendrils karta diagram, so you see, they are closely disconnected region. So one never going to need something from here that can be receive here. So this is the reason this guy is called black hole. And so this is the advantage of the Penrose Carter diagrams. The disadvantage is that to draw it one has to know the entire space time which is frequently in the real physical situation happens to not always be possible. But otherwise if it is possible then you can draw a boundary, select relevant two dimensional plot and draw the Penrose Carter diagram and understand causal properties of the spacetime. That we will do actually in the next lectures for different sorts of spacetimes. [MUSIC]