Problems of General Relativity

0:00 Thank you. I think I was just talking about the primary of the future of some set S. It doesn't allow closure less, because it's not easy to pass from that point. And, yeah, we consider the possibility that there are two large pieces in the past, and we can't extend the future beyond that point, and make our boundaries. Now, try to sustain that proposition. So suppose we have a set B, which is bound to be a feature of some set S, I suppose that X is in B, but it's not in the closure of X. and that's the curve that x is an end point of two nalgesics. I'm not going to do that. Two nalgesics.

2:30 Now, first of all, it acts as a past 10.1 or above. Then it's implied that a union is an union to come to both Nautilus, it is next to the end of it. And once a continuation of the other World Assembly. or if the second possibility is that X might be a future end point on both of them. when either one of them is improving the other, in my particular case. Or else, when every extension by the one into the future beyond us must we be meant to be children, so, more in a great extension. Thank you.

5:00 I think this is not very hard to show, but it's not just to see the situation. Here's S here. There's the boundary of the future. We've got X. This is the boundary of the future. So, in the first case, x is a passed endpoint of one or of both. Well, there's an algebraic of x given up from x like this. Now, let's just see what happens. Well, there is, we know there is an Lgdc going to be passed from X by putting this on the length of the theorem we just had. And there was a pass from Lgdc going to be passed from X. Now, there are two Lgdc going to be passed from X. First of all, that if we have one going into the past and one going into the future, like this, then the one going into the future must be a continuation of the one in the past. This is clear because if it weren't, we could have two points here and here, which had a It's time-like separation. This is a novel, that's a novel. Time-like separation. It depends on the continuation there. I'll get one of our own results. Therefore, this boundary, if they're both on the boundary B, B would be incredible. So if B is incredible, this can't happen. Therefore, if you have a future endpoint of one and a past endpoint of the other, then they must, one must be continuation of the other. And then the other thing is, if it's a past endpoint of both, As we know from, again, the previous result of the period of the past, by what I just said, this must be the continuation of that, and this must be the continuation of that, and the union of the human rights, and the union of the peace. So that establishes. Note A. We've got B. If B is 6, we've got 2, and that's the correct side of the piece. We've got the situation of SX, and we've got two analogy V65s.

7:30 We now, if you consider a continuation, one of these in the future, like this, then you look at the other one, and just by the argument, I just give it again, must be a continuation of that, as well as be a continuation of that, if this lies on the boundary of B. So, that's the first situation. If this continues to lie on the boundary, on B in fact, then by what we just said about A, one must conclude with the other. The directions of these must be the same. So that's the first possibility. So the alternative is, if the the extension of this into the future is not with that line. So that's what it means. It means B. So we can show it inside plus S. But it means B, it goes into the future of B, and then it's the future of B. And then you must lie. That is, it's quite simple. Okay, any questions about that? Okay, the next thing. Anticdependence, I introduced the idea of anticdependence. In some ways, it's a sort of dual concept, futures and pasts, and it makes sense that if you have a set here, and you want to look at the future, then you'll look at it like that.

10:00 That is to say, at any point in which there exists a time-lapse curve to S, then it lies in the future of S. But the way the attendance is going the other way around, if you take a set S here, Well, it's the other way round in the sense that what you want is every time-like curve, which extends from this point in the past, into sect X. The domain dependence in a table looks like that. This point is in the domain dependence if every time-like curve extended, The idea of the main dependence, physically, is that if you're given some sort of space-like surface or something, and you're given some Cauchy data on that space-like surface, and you want to know what portions of the space-time does the information on this Cauchy data define what's going on in the space-time. Now, if you're over here, then the information on that surface doesn't fit well. Now this, of course, depends upon the right sort of differential equations to define what's going on. Then you have suitable partial differential equations which have a property that the like-comes are different characteristics. The differential equation is something which is local in the Lorentzian period. Basically, you're not going to go into the relation of differential equations. The way one uses these sets, the context that I'm interested in here, is much more just using them as a mathematical construction which you then prove things about the boundaries. The physical idea is that you have information on that surface,

12:30 and that this information propagates in a relativistic way, that's to say the information doesn't probably pass at night, then the main dependence is the region in which you can tell what's going on as long as you can tell what's going on on that surface, It's supposed to be some sort of equations which we should think of. And I don't talk about the correct equations. I can simply say that if I have a point which is upside, while talking about the future domain of dependence, if I take a point which lies outside the future domain of dependence of S, you take the point outside this domain of dependence, then information to get to this point which hasn't registered on S, if there's a time-like curve which extends into the past of that. So we can say at this point it's outside of the major dependence. If there exists a timeline curve, or I'd say trip, which extends indefinitely in the past, which doesn't exist. So that's what makes it very open, right? The major dependence is to the union of all these timeline curves, and what's left over is the major dependence. You can similarly define the path to the end of dependence, E minus of S, and the E two together, in the interpretive form. That's right now the definition of formal. We suppose that S is some set, the matter of a number of S would be taken to be a chrono. I can define it whether it's equal or not, but I can't forget what I would say, but if there is a difference, I would assume the next is to be equal to the set.

15:00 And d plus this is equal to set point x such that every past endless trip through x And this, in fact, is the complement of the union of all pastors, pastors, pastors, states. If you look the other way, let's see. If S is a closed step, then D plus of S will be necessarily the closed step as well. Not much C in that. Start with S. Suppose that's closed. And then you look at the union of all time-like curves of the union of all We should pass down this, and we should only say yes. Well, it's pretty clear that's an open set, because at this point, the desk is closed, and this point doesn't mind this. Then, we can take something at that point, if it doesn't mean yes, and we can just move it around. Thank you. We'll just be using this, too. That's the set. X is actually an embryo. End the script.

17:30 Invoking an LSA actually gives a slightly different definition. They allow these things to be cautious as well, so that they don't have the... It doesn't make that much difference, actually. Let me indicate what sort of difference it can make. First of all, if s is t equals more to the Pusky space, then, of course, v plus of s would include the whole region, the t-grading region, which would include s itself. Because if you take the point on S, then that point, every time line or otherwise, would have a point in space. So it includes S. So the D part of S is space for B. So you wait for B. And then the D part of S, The information on the space-by-surface defines what goes on in normal space. But in the country's place, first of all, you have to do the portions. So whether or not you have the endpoints in this portion doesn't make very much difference. If the endpoints are in so that S is closed, then it simply means that these two points out of fact indeed must be minus S, C, S. Whereas if you take these points out, set S, it's still in space, set S, the very difference is that these actual points themselves will not line the main dependence.

20:00 But everything I'm here will still land on my independence, because I'm only talking about trips, that is the time-like curves, and it doesn't make a difference if I'm not here at all, because it would have to be in all of that. It does make a difference, if you shake the Hawking and Everset definition, because they're talking about, not trips, but times of time, causal curves, and if you talk about causal curves, then it does make a difference to this point. This point was in here. I was not in this, but this is for one of the minimum number. And according to Hawking Lewis's definition, that part of the boundary would not be in the middle of the sentence. The students here would not be in the middle of the sentence. It used to be positive what's going down in the middle of the sentence. Whereas everything on here would be a step that's in there. I prefer to take trips, or I do positive trips, for this kind of reason, because it depends on whether you want them or not. If you take that definition, I suppose the idea is that the computation here reflects in the structure of the structure of the structure of the structure of the structure of the structure. It's simpler, and these trips, and that's the top of the structure of the structure. So if I could one point out here and one point in here, then the main dependence will include that boundary, that boundary, this point. And I said, the only thing to be out of it, in this picture, will be that actual point here, which is missing. If you take the point out of the middle of this, if you just take that point out, take this when we said that, as I said, then it does make quite a difference, because if you take any point here, then there will be a trip, just to behold, and therefore, that's not going to be the end of this. For some purposes, it's useful to restrict S to be not only a cron, but also close.

22:30 You might argue that if you were talking about closure days or something, then you may expect what goes on in endpoints to be defined by closure. If you have some c-finity function, I'm going to describe it as some field of space. We're going to know the value of that function. How is your function to find that value? On the other hand, we might be talking about delta functions or something. In which case, delta functions. You might know what you really do. It doesn't make much difference, I don't think it's... ...many terms are something else that just stick to closed A-chronousness. Certainly A-chronousness. Do you want to read about it? Well, it still works. See, A-chronous includes null. So, it's perfectly good. I'm not desisting on space right now. That's, that's, uh... I want it to be possible. In fact, if we like to say like, we might look at the past like general point, and that would be S. If you want to look at the domain of dependence here, for future domain of dependence, in fact, it's simply S itself. Because if you take a point out here, there will in fact be trips which do that sort of thing. Nevertheless, no point in the future of this cycle could in fact be good indeed. So, in this case, these must be the same as the other divine is best, and through all, there is another in the future, first. so, if it would deserve the tip of the light for the... That one, yeah, I can see them. And then, I think this one, So, in this case, if you simply take the tip off,

25:00 Well, let's see. space, and it will not be a kernel, because you have a picture of such a screen. There's a cylinder where you're going to have a problem. Um, no set can be a kernel, because it's so important to talk, so it's supposed to have a curve in a while. But that doesn't, the definition of food there, I think you just found d plus of s, because, uh, I'm just right, it depends on what you're saying. If s has got a hole in it somewhere, whether it's a light term or not, if it's got a hole in it somewhere, then you keep going through these, so you will have passed that history of tremendous problems. So, probably, the test as a whole, in some way, is the T plus of S equals S. Why is it defective? Yes, it's not skyline. So this is... Oh, I'm so sorry. You said the Pax Ply. Yeah. I was thinking about the event, but the one... Yeah, yeah. That's what you meant, but the one... That's one of the best... I mean, I'm not looking for the Bama. It's back one. But it's not actually space time, so... It's a man-proved Bama. Well, it's not even a man-proved Bama. It was Bama. Because of that point. So, did we take that point off? Yes, but strictly speaking, as far as the man's dependence makes sense, these points make a pretty difference. Because you will have trips, which go through the public around the world. And so, if you have a past-endist trip, that will be a past-endist trip.

27:30 And so this one might be in the future. I haven't actually talked about these ideal points. There's one thing I do want to talk about, and later on I'm going to discuss. First of all, how one defines these ideal points by generally. And then secondly, talk about the mental. I'll just like to do that with this kind of use of the best way to do it. At the time being, and I'm not talking about the situation because it's the best time for the time. They're rather similar situations taking actually the class-like thing. We could take, simply, all these hyperboloids, given by constant distance. I don't know how to do a space-like surface. Space-like surface, which is in fact intrinsically complete, as if you look at the intrinsic metric of this space-like surface, It's in the volume, it's a GD6 line of inlet surface, that's the state, which is the matrix we mentioned, extend to the length of space. It has the property that the domain of dependence is, again, not the whole of the cosmic space. So you might think it's perfectly a good surface in which to specify the new data. It is the property. But nevertheless, it does have the property that the D plus of it is only this with the what you can do. In passive S, it's the same for this. For the same reason before, you take a point outside of the and you find a situation where there are parts connected to the 3.3, which never intercepted. But there would be to say a minus of that.

30:00 The D of S then If you don't consist on Aspen either closed or achronal, then it actually makes a difference whether you talk about trips or time-like curves. And when I started here, I talked about trips, and the concept of trips the time-like curve are more or less interchangeable. I mean, the time-like curve is smooth, and it's really going to layer between these segments. It doesn't make any difference, provided S is either a closed set or an incremental set, either one of those suspicions to show that there's no difference. But if you remove both those descriptions, you'll write into the set of sets S, which look like this, This is a very artificial situation. Yes, here. But these S is open just in the net because that's . If you're looking at the point here, if you're looking at trips, I mean, there are trips which will work their way too long, so this one doesn't have any many things for the trips. When you just come to do that, then, as we do, they don't very much occur, you see. Time like students, time like girls, we make sure that one makes a difference. That, of course, doesn't matter. And you don't consider such things either if you insist on it first, or if you insist on it being a criminal. And I'm going to insist on it being a criminal from now on. Or the definitions I tend to use this. one point. So if you have a little hole out there, then that little hole, that would be

32:30 one point. And when you get through that hole, it will be true for a time like that. I'm just going to see it removed. I'm not going to talk about the time I've covered in the name, so I'm not going to prove it. That's a big thing. I'm going to see what's going on. Okay. I want to talk about the financial crisis now. So the Kochi-Horizon is essentially the boundary of T+, not the boundary which is given by S itself, but the rest of it. Let's just say S here, and this part here will be the Kochi-Horizon. That is the future Kochi-Horizon, H+, and S, and the past Kochi-Horizon will be that part there. So it's sort of the future boundary of the future, and it's the past boundary of the past. And you want to formulate... A plus of S is a set of X, perhaps the X is a plus of S, but the future of X is a set of X. Let's see. Here's D-plus. What I want is the points out here. So they are in D-plus. because it's closed, it's quite a bit less than a second. But i plus an x, it's up here.

35:00 It doesn't interfere with the universe, so that means it's not going to put inside it, but it's a half of it. We can also put these definitions in a different way. way as simply h plus or minus of s equals d plus or minus of s with i minus or plus of of D plus or minus S of A and F. Just definitely, that's saying the same thing as this. With a plus, that means it's D plus, like this one here, but it doesn't lie in the past of P plus or S. So the points which lie in the past of P plus or S are these points down there. And as long as I look at D minus his class, so I put on the B class of the S test, this thing here. Minus the class of the B class of the S, and I'm just left with this class, this is B. of the edge of the set S. Incidentally, before I looked at some of these examples again, I was talking about, say, in healthy space, where I remove the points from the set S, you might consider removing the points from the manifold itself. It actually looks rather like

37:30 what happens when you remove the points from S. If I remove the points from the whole manifold here, then again, the magnetic points In the previous case, if this point simply moved from s, then the matrix goes into there. If I move this point from manifold, that matrix goes into there, and it's still past endless. I mean, four heads are right through, it can pass endless. And I see how it's past endless seems to be moving into the other way. It doesn't make any difference to the local matrix. There is a distinction between one case, S can be opposed to that, whereas one is on the other. If this is removing bound mode, S can go right up to that hole, and close. Whereas the other case, S can be a whole other S, and that's what we're supposed to do. Well, we've noticed that these sort of end points of the set S in a fair role. So, one way to clear the world is if you look at the future, the future of Coaching Horizons, then one of the properties that we're going to have is rather like future sets, the future of S looking like that, we look at the boundary of the future of S, and that's the property that you have these malgies that's going on with us. Well, coaching horizons also have this property, and if you take a point on the coaching horizon of S, then again, there wouldn't be none of the basics going down each pass. And here we had to keep away from the closure of S, that is probably the problem. Here what we had to keep away from was not so much the closure of S, although that we do, but we couldn't have a following situation where S consisted, say, of a piece of space like and a piece of a null surface like that. So now we look at the future domain of 10s, then it will contain that part and that part, and it will also contain these points of S as well. Now, if you take the point, which is all this part here, is h plus of s.

40:00 The s, the s in the other, h plus of s. Now, if we take a point away from the closure of s, the point right here, then it will be true, there will be analogy pieces going into the past with that point. But if it would still be true, it might take a point here. So it can be actually on ends itself. And if I take a point there, there will be largely pieces going on in the past. But it won't be true if I haven't seen that part of that bottom end here. If I take a point there, then I'm really... ...in a coaching horizon, it's not true. So what I want to know is what's so special about that point. It isn't only here, either here or here. It is also happening here. And for this, what I need is the concept of the edge of the A-code set. I can't really talk about the boundary of the A-code set. I talk about the boundary of S. The boundary of S is simply S itself. If S acts as well, this is an episode. Then, what is the A-code set? Then the boundary of S is simply, So what we want is a sort of concept of a of a boundary, but from a kind of two-dimensional point of view. What you want is, you think of S, to say that we're doing this space, or public space. You want to be able to say, well, the path is going around the edges. So we call that the edge of S, and we get the definition of the edge of S. The sort of intuitive meaning is that it's like the boundary, except that we're really talking about one dimension after that.

42:30 Yes, we're trying to exclude the point of the interior as regards their trigonometric structure, but non-interior is important. Instead of S. The edge of s equals the center point x, such that every neighborhood of x and two trips from Elijah's end. There's one of which he says. It's a little strange. It's a little bit tricky to get this just right in a way which works when we allow close time like Kermit and the Lionel service. So, it's allowed to be the definition which works in most cases, and you see it doesn't give a little random. in some of the anthropological situations. See, if you take, for example, you try to take a script of the cosmic space and make an identification Make an identification where we slide down This is two-dimensional cosmic space here. We start down, so at this point, so I didn't quite work with this, but this is actually no curve there. So we do have close null curves in this picture. This is close null curves. No curves sound like this. And if I consider this set here, that's an acronym set.

45:00 We want to make sure we get a definition, which doesn't give it any edge. because the music is such an advantage. Be careful. Anyway, this is the CQO sort of thing. And roughly what it means, just to set S. Now, this point here is supposed to be on the edge of S. And that also will be on the edge of this step. Like this. But if it's called a manifold, and that's actually how the manifold is to hold instead of S. Well, that one might be in. That's how the manifold works. So then these points would be all included in the edge of s, but not that one. And that's what the point is in the edge of s. That's right. But whether it's in s or not, it's still in edge of s. But this point is not in the edge of the closure of S. If you took the closure of S first, you'd lose that one. But you wouldn't lose these ones. You can see it makes a difference if you take the closure first and then take the edge. Between the edge first and the closure. Okay, now let's have a look at these points. See, I take a point there, that's X. But I consider neighborhoods at that point. It contains points Y and Z, and two trips from Y and Z, just one of the details. Every neighborhood. So what we want to find is two trips, well, one of them to meet us, so we're just going to find the trip to meet us. Let's say here. OK. Two points. Well, there's one triplet between two and one triplet, doesn't it? See, there's nine z.

47:30 So you can see it's satisfied. Now, how small this label is, all you have to do is go just a bit above x, just a bit below it. It's like this, and then the one-driven kids, if they haven't heard of the kids, which is interesting. Incidentally, if S weren't a hydrosurface, because it was just a space-like curve, it might be a space-like curve, in three and four space-time dimensions, then the whole thing would be in the edge of the Earth. Because you can find, if you're right in the middle here, you can find one trip. You take some of them from here, one trip which means that in the other one we simply just come to the side. And if you're here, that's in the edge too, because you take any two points, you take a small neighborhood, you take one just as a future, one just as a past, and there will always be trips which go to the hull and other trips which hit them. So again, that will be the answer to this. But if you take a point here, if it's smaller than the neighbourhood, then you have a space-like surface which goes right across, and if we take a trip which intersects it, then any other trip with those two same endpoints will also intersect. Or if you take it all the way out here, then that would be intersected. So I think you can find this. Would you take the S together with the edge and think it's closer with the same thing and closer with the same thing? Would you take the S together with its edge and take its closer to the S? I think S together with its edge is this question. S-U-E-N-H-S. Thank you.

50:00 Oh, yes, it's good to see you. S will always be a problem, as I say. Well, that jazz is good. Well, let me just take this easy to go over. One way of characterizing the edge is the following. If you take point P, and it's not in edge of S, if only if it exists, open set Q such that the intersection of S with Q is an acromal boundary in that of Q. This is the cube. This is the integral of my grid. Manipal. The cube. This is including the empty set. The idea of this is, you see, if you go away from the edge of s, this s, then you go somewhere inside it. Then, you take a small level of the cube, then, if you simply regard that as your space-time, because that was a bit there. Then, this portion of S in Q is made for an advantage, and in fact, therefore, it's... by something else, two pieces, and it's locally at a couple of three months,

52:30 which is really a hyperservice, so... Points on edge of S are those places where, in the neighborhood of those points, because S fails to be a hyperspace, where it fails to be locally naked in the ground. So the edge of S is just those odd pieces where S doesn't apply to space like hyperspace. There are places where it doesn't apply to space like hyperspace. That's quite a useful word. So, I'm going to prove this one, okay? Now, what do you do? And once you have this, you can actually establish quite well, results in sending image quite easily. There are a number of things that I want you to switch. Closure. Closure, S, S equals S union, and S, and S, and S, and S, and S, and S, and S, and S, and S, and S, and S. Thank you. Thank you.

55:00 Thank you.