General Relativity | Lecture 8
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Susskind, Leonard
(November 12, 2012) Leonard Susskind develops the coordinate transformations used to create Penrose diagrams, and then uses them to describe the physics of black hole creation. This series is the fourth installment of a six-quarter series that explore the foundations of modern physics. In this quarter Susskind focuses on Einstein's General Theory of Relativity.
 Okay, the last time I explained to you Kruskal coordinates for a black hole, let's go over it. And tonight I want to show you how you can think about the formation of a black hole, a real black hole, and how much of this Kruskal story, the diagram we drew on the blackboard last time, really is physically meaningful. Not all of it is. Okay, so the last time I showed you that the geometry, the metric of a black hole, of a Schwarzschild black hole, has the property that very near the horizon, that's the horizon over there. It's like flat space. The horizon is a kind of polar coordinate, a hyperbolic polar coordinate point, similar to the behavior of accelerated coordinate frames. Somebody standing outside, and this wedge over here, this quadrant over here, is outside. The rule is, first of all, that 45-degree angles represent the motion of light rays either going in that direction or going in this direction. Somebody outside the black hole means out here, and this quadrant, then quadrant, let's label them, quadrant one, quadrant two, quadrant three, and quadrant four. And as you're going to see, quadrant three and quadrant four really don't have any physical significance for a real black hole. It's quadrant one and quadrant two, which have real significance. Outside the black hole, somebody at rest, outside the black hole, somebody at rest, of course, is experiencing a sense of acceleration. This is I'm experiencing a sense of acceleration right here. I've gotten used to it, so I don't notice it, the fact that I'm being accelerated. But if somebody who had been in outer space too long and free fall too long lost the sense of what a gravitational field is like, and you plunked them on the surface of the earth, they would be experiencing a sense of upward acceleration. And in fact, in this diagram, somebody at rest does appear to be accelerating away from the horizon. Come in and go out, and it's that acceleration, which is what we experience as gravitational acceleration near the surface of the earth, but in this case, near the horizon of the black hole. So this is a family of observers, further and further out, very far from the black hole, way out here. In close to the horizon here, the closer you get to the horizon, the more you have to accelerate, that's the curvature of these lines here, the more you have to accelerate to keep out of the black hole. And in any case, this is the exterior, the lines of constant time, they look like this, red lines are lines of constant position, the green lines are lines of constant time, and they accumulate up near this blue line up here. So this is t equals 0, t equals 1, t equals 2, t equals 3, and t equals infinity is way, way up on this 45 degree surface here. So that's the outside of the black hole, and the inside of the black hole is quadrant number two. In quadrant number two, for a real black hole geometry, not for flat space, but for a real black hole geometry, there's something nasty, and the nasty is the singularity, and the singularity looks like this. I don't know what it looks like, because I've never seen one, but it's a bad place with lots and lots of tidal force. And the problem with it is once you find yourself in quadrant two, there's no way out. The only way to get out, even if quadrant three was meaningful, which it is not, but even if it were meaningful, you can't escape from here to out here or to out here without exceeding the speed of light. Your trajectory somewhere has to have a direction which is shallow within 45 degrees, and that means exceeding the speed of light. So that means if you don't exceed the speed of light, the only place you can wind up is somewhere on the singularity, so you're doomed. If you're outside and you make a bad mistake, you could fall in, but if you don't make a mistake, you can jump back out and get out. So people on the outside or light sources on the outside, flashlights, whatever you like, they can produce signals which go in or they can produce signals which go out, and that's the character of a black hole. So that's all there is to black holes. The geometry is somewhat warped. It's not just flat space. It's somewhat warped and curved, but this is basically it. Could you say what the axes of that diagram are? Time and space. Time and space as seen by an in-falling observer. And so the lines of constant position that you drew for what stationary observer outside the... Yeah, that's correct. So for somebody falling in who doesn't accelerate, the coordinates that they see which are not too different from ordinary flat space in free fall, at least until they get near the singularity, time and x, those are the coordinates that such a person would use. So to them, the space looks flat pretty much, pretty much, especially near the horizon over here. To somebody on the outside, they experience this acceleration. And the acceleration is basically the same as the accelerated reference frame in flat space. Okay, now, it's very convenient to redraw pictures like this. Space is infinite here. It goes on and on and on and ends up somewhere in the next county or further than that. And time may come to an end at the singularity over here, but these directions offer 45 degrees. These light light directions, they are also infinite. And so I can't draw the entire space time on the blackboard. For many purposes, it is useful, convenient, provides some visual tools, provides some intuition to be able to draw the entire space time on a finite piece of the blackboard. Well, you can imagine why. You want to see everything that's going on. You want to get a complete overview of space time. You'd like to draw it on the blackboard in some finite region. Let's start doing that, but let's start with flat space. Let's start with good old ordinary flat space and take all the flat space time. When I say flat space, I mean flat space time. And let's redraw it or re-coordinateize it so that we do a coordinate transformation which pulls the whole thing into some finite region on the blackboard. I'll show you how to do that. This is useful, incidentally, for geometries and space times which have rotational symmetry. Rotational symmetry means that the same in every direction. They mean the same thing, rotational symmetry is when we say a gravitational source has rotational symmetry, means the same in every direction. And when a system is the same in every direction and has rotational symmetry, it's usually useful to describe it. Now I'm talking about ordinary polar coordinates. When a system has rotation symmetry, it's the same in every direction. We don't care so much about the angle. Everything is the same independent of angle. What we care about is how things vary as we move away from the center, the radial direction. And so for many, many purposes, we don't need to think too much about the angular direction. And we just think of space time, flat space time, ordinary flat space time as having a time axis. That's the time axis. It goes from minus infinity to plus infinity. And it has a radial direction. The radial direction is like that. The radial direction never gets negative. R is never negative. And polar coordinates, R is always positive. And so the entire space time is to the right of the time axis here. Only because being to the left of the time axis would mean negative R, and R is always positive. It's the distance from the origin. So here's everything, the entire space time. So let's put some markers on it to be able to visualize some landmarks. First of all, this horizontal axis is t equals zero. And it's t equals zero right over there. Let's put in t equals one. t equals one looks like this. t equals two looks like this. t equals minus one is like this. t equals minus two is like this. And if I wanted to draw the entire set of possible times, it would take me up to heaven and down to hell. So this is all possible times. And we could also plot all possible radial distances. And the radial distances would just look like this. This would be R equals one. R equals two. R equals three. And in all cases, I will imagine using units so that the speed of light is one. C equals one. That simply means that I work in, if I'm going to use seconds for time, I use light seconds for distance. And then light rays. Light rays will move at 45 degree angle. Well, that's not quite right. Let's think about the motion of light rays. If a light ray, let's draw another picture over here. This is R equals zero. This is polar coordinates. This is the radius here. It's just one instant of time looking and including the angle now. If a light ray comes in radially toward the origin, it hits the origin. And then what does it do? It goes back out. I meant to draw it straight. But this is just saying that a light ray in polar coordinates that happens to be aimed directly at the origin from far away. Imagine it's coming in from very far away. If it happens to be aimed right to the origin, it will go to the origin and then go back out. On this diagram, it will appear to bounce, to reflect off the origin. Here's R equals zero. This is R equals zero over here. And so a light ray coming in from long in the past would look like, I think I'll use another color, a light ray coming in from the past will look like this. Moving at 45 degree angle, it will get to the origin at some time. And when it gets to the origin, it will simply start back out. It hasn't really reflected off anything, of course. It has just gone through the origin, came in, and went back out. So a light ray starts out in the very remote past coming in at 45 degree angle from a place that we call light, like infinity, past light, like infinity. But all it means is the place where all light rays come from in the infinite past. And they go out to the infinite future. Of course, there are other light rays that do the same thing. This one comes in a little bit earlier and goes out a little bit earlier. Another one up here. But they all look pretty much the same. They come in at 45, they bounce off the origin, and they go out at 45 degrees. Now just for fun, let's think about a light ray which doesn't pass through the origin. A light ray which doesn't pass through the origin will miss the origin. It comes in and it misses the origin. When it's very, very far away, you can't really tell that it isn't done, unless you have very good precision, angular precision, extremely fine angular precision, watching that light ray way, way back there, it's hard to tell whether it's going to hit the origin or not. So it starts out coming in pretty much the same way. But it never gets to the origin. In fact, it almost appears to be repelled away from the origin and then goes back out. That repulsion is not a real repulsion, but it is what we call centrifugal force. It really is centrifugal force that keeps the light ray from hitting the origin. Or you can just say it doesn't hit the origin because it wasn't aimed at the origin. Same thing. But what does it look like? It comes in pretty much the same way at first. And instead of going through the origin, it gets a point of closest approach. The closest approach is over here. At that point, it starts going back out. So a light ray which is perfectly radially tuned to hit the origin looks like a 45 degree ingoing and outgoing line. But a light ray which misses the origin looks like, in fact, sort of like, it looks like a hyperbola. But whatever it looks like, that's it. I'm just telling you these things here to give you some landmarks and to give you some perspective on what we're going to do next. Next thing we're going to take this diagram and we're going to squeeze it and stretch it in various ways to get the whole thing on the blackboard. That's going to deform it. It's not going to look the same. But we're going to keep one thing fixed. The one thing fixed is that light rays will always be assumed to move on 45 degrees. In other words, any squeezing and squashing and stretching that we do should preserve the fact that light rays move in 45 degree angles. That's a useful thing to do because then we can see how light rays move. And we can see what going faster than the speed of light, what going slower than the speed of light looks like. So I'm going to show you how to do that mathematically. Excuse me. Yeah. There's nothing special about the origin. There's not a black hole there or anything. No, no, no. At the moment there's nothing special. It's just a particular point that we may want to focus on is there may be something there, but no need for it. Okay. So light rays in this diagram are bent just because of the particular way that we chose the coordinates. Yeah, that's right. It's just a choice of coordinates at the moment. Okay. Now, there may be good physical reasons to want to use a particular choice of coordinates. When studying the sun, there is no black hole, but the symmetry of the problem makes it convenient to think about the sun as being the center of something. And right. So that's all we're doing. All right. Next step. Let's assign some coordinates. I'm not going to call this T. I'm going to give it another name. I'm going to call a vertical axis capital T. Capital T and the horizontal axis capital R. I don't want to confuse them with other T's and R's that we're going to use and that we have been using. So we're going to call it capital T and capital R. But there is another way to coordinate this, and that's to use coordinates at 45 degrees angle. In other words, to introduce axes along 45 degree angles here and here. Of course, I don't mean to imply that there's anything behind R equals 0. We're only interested in this region here. But we're going to introduce new coordinates, and this coordinate over here, that's not going to be T or R. It's going to be T plus R. T plus R varies as you go from here to here to here. If I were to plot constant T plus R, constant T plus R would be these lines. Oops, I didn't draw it well. Let's draw another picture over here. R T, constant T plus R will look like this. These will be the lines of constant T plus R. OK, so this T plus R, and of course, there's also T minus R. T minus R, the lines look this way. But really we're only interested in this half over here. So the lines of T minus R, constant T minus R, look like that. These are useful coordinates, as we'll see. T plus R, that's called T plus. T minus R is called T minus, just as you might expect. OK, so we have a different way of describing the same spacetime, the same flat, good old flat spacetime. Instead of T plus R and T minus R, we have T plus and T minus. Now, what about this line here? This, what was this line? This line was just plain R equals zero. R equals zero. So what is that? What is R equals zero? That's T plus equals T minus. Let's write T plus equals T minus. T plus equals T minus, that's the same thing as T plus R equals T minus R, or R equals minus R. The only solution to R equals minus R is R equals zero. So this line over here, which was R equals zero, is also just T plus equals T minus. That's a way to describe it. OK. Now we're going to change coordinates, and the way we are going to change coordinates is to introduce two new coordinates, these are called light like coordinates. T plus and T minus is called light like coordinates. We're going to change light like coordinates. We're going to introduce two new light like coordinates. One of them is called U plus, and one of them is called U minus, and U plus is a function of t plus. So we're going to write u plus is some function of t plus. It's not going to involve t minus at all, and u minus is going to be, in fact, I think the same function of t minus. All right, but this function is just a coordinate transformation in which u plus and u minus independently, sorry, t plus and t minus just independently transformed to some new coordinate. Good. What kind of function am I going to use? I'm going to use a function of t plus, this is f or u, this way u equals f of t plus, and the function is going to look like this. When you go out to t, it doesn't ever become absolutely flat. It's always got a slight positive slope, also down here it also has a slight slope, but it'll go to one over here, u plus, u plus, u plus of infinity is one and, or just f, let's call it f. f of infinity is one and f of minus infinity is minus one. Now, anybody know a function that does that? There are many functions which do that. What's that? Yeah, a hyperbolic tangent is the one that's usually used. Hyperbolic tangent goes to one asymptotically, it's the usual one that's used but there's nothing special about it. Hyperbolic tangent, hyperbolic tangent, f of t plus is, and if you don't know what a hyperbolic tangent is, don't worry about it, it's just a function which looks like this, tanh of t plus. Okay, now think about what happens as we start moving on this graph further and further outward. T plus is getting bigger and bigger and bigger. What is u plus doing? It's never going to get bigger than one, right? It's never going to get bigger than one. So, let's draw our same space time with these new coordinates, I think I erased them but let's put them back. U plus equals hyperbolic tangent of t plus and same thing for minus. As we move out, let's redraw our picture. As we move out, let's start over here and move out, how big does u plus ever get? U plus never gets bigger than one. So, this diagram, if this is the u plane, not the t plane but the u plane, u plus never gets bigger than one, here it is, right there, it never gets bigger than one. So, that means all of the space time has been mapped somehow down into here. What about u minus? u minus is equal to hyperbolic tangent of t minus. What this says is as we move down in this direction, t minus never gets bigger or less negative than minus one. t minus is getting very, very big and negative as you go down here. t minus is getting bigger and bigger but negative as you go down here but u minus sort of maxes out again at minus one and here's t minus equals minus one. t minus never gets too far, t plus never gets too far. This is, I'm sorry, I said t plus and t minus never get very far, I meant u plus and u minus. u plus equals one, this is u minus equals minus one in fact. And what about this vertical line here? The original vertical line over here was t plus equals t minus, right? t plus equals t minus, that's the same as r equals zero. If t plus equals t minus, then that says that u plus equals u minus. So, in fact, this line happens to be u plus equals u minus and it looks like a vertical line. All we've done, if you had, don't follow the details, all we've done is squished the geometry this way, squished the geometry that way and squished it onto a finite triangle. You'll understand this better when I draw in these coordinates. Now, what is the trajectory of a light ray at outgoing, let's say, an outgoing light ray in terms of the t pluses and the t minuses? Well, an outgoing light ray is at a constant value of which, t minus, I guess, t minus would be constant and fixed on here, t plus would be constant and fixed for an ingoing light ray. But if t plus is constant and fixed along some trajectory, it means that u plus is constant and fixed along that trajectory, likewise for t minus and u minus. The upshot is that if you make a coordinate transformation like this, it is still true that radial light rays move on straight 45 degree angles. These are light rays that come in from the remote past, bounce off the origin and go back out. Again, bouncing off the origin is just a figure of speech, you just pass through the origin. So we've succeeded in taking all of space time and mapping it to a triangle that we can draw the whole thing on the blackboard. Let's draw in some more coordinates, let's draw in some more detail. In particular, let's take these horizontal slices which correspond to different times. Let's imagine starting at the origin at where is t equals zero? Here's t equals zero and moving outward, outward, outward will allow ourselves even to exceed the speed of light and just move outward in our imagination. What does that look like? That of course looks like starting at a point over here and just moving outward. And where do you eventually get to? You get to there. So this point must be spatial infinity, r equals infinity. It's where you get to if you just walk along this line till you get out to infinity. But what about this line over here? t equals one. If you plot that in these coordinates, it'll start about over here, but it will go to that same point. Likewise, t equals minus one. So what we've done is we've squished all of spatial infinity, we've squished it into a point right over here. And it's called space-like infinity. Yeah. No, they're not exactly straight. It depends of course on the particular choice of function, but they won't be straight. No, let's draw them more carefully. One of them will be straight. The one will be straight will be the t equals zero. This one will be straight. Then the next one will look like this. It'll have some curvature. What about way, way, way up there? t equals a hundred billion. t equals a hundred billion is going to be very close to this edge. So as we keep putting in more of these, they accumulate and get closer and closer to this edge up there. You've got to get an infinite number of them in there. Time really does go on and on forever, at least in flat space. And so this is what the series of timelines would look like, different times. On the other hand, now let's look at fixed spatial position. Fixed spatial position, we can mark off different locations. Here's r equals one, here's r equals two, r equals three, r equals four, r equals five, tata tata, and they again get all squished up near r equals infinity. Where do they all go? They all go vertically upward, and on this diagram, they all go up right up to this point up here. You can check that by doing the transformation, and that's what you would see. More and more and more of them. It's the same diagram, just in different coordinates. And this coordinates, as I said, everything gets mapped to a finite region of spacetime. Again, I emphasize, light rays move at 40, radial light rays move at 45 degrees, they come in and they go out, and so forth. Okay, this is called, this process of squeezing everything onto a finite piece of the blackboard without changing the speed of light. It's called compactification. You take everything and you compactify it. But it's also called making a Penrose diagram. This is a Penrose diagram of flat spacetime, Roger Penrose. Sometimes it's called a Carter Penrose diagram because a physicist named Carter, I think actually invented it first, invented Carter, but it's usually called a Penrose diagram. Good. So that's flat spacetime laid out on the blackboard. Now, the question is, supposing you took the black hole, is the black hole geometry? It also goes, oh, let's put some names of points. This is called spatial infinity or r equals infinity. This one up here, what would you call that? That's what you get when you just go vertically upward. That's time equals infinity. Right, these green, t equals infinity. That's this point. What about this one? That's t equals minus infinity. All right, so this, I'll give you the name. This is called future time-like infinity. This is called past time-like infinity. This is called spatial infinity or space-like infinity. And what about these lines here? These 45 degree angles. These 45 degree angles are where all light rays come from. Light rays come in and go out. They don't really go out because it takes an infinite amount of time for them to get out. But a light ray beginning in the infinite remote past starts out its life coming in along one of these light light trajectories here. Bounces off and goes back out. The standard terminology is to call this script i minus, sometimes called scri minus, script i, scri minus. And I don't know why, I don't know where the i came from. I know what the plus and minus means. This is called i plus or script i plus. And these are called past light-like infinity and future light-like infinity. The place where, the place where light rays begin and the place where light rays end. And that's the terminology. And that's what flat space looks like when compactified. Now, what about the black hole? The Schwarzschild black hole. What does that look like? We can do exactly the same thing. We can do exactly the same thing. Take this whole diagram. Introduce again, I guess we can call this R, R. R. R. The original Kreskal coordinates. And we can again introduce light-like coordinates, T plus R and T minus R. Do exactly the same operation. And what will we get? What it will look like, I'm going to draw it over here. And then I'll show you how it works. Okay, that's everything and the whole thing drawn on the finite blackboard. And I think you can probably see what goes where. The exterior of the black hole, that's this quadrant over here. The square. It's square if drawn properly. And we can again draw coordinates. Here's T equals zero. Here's T equals one. Here's T equals two. Here's T equals three. T equals four. That corresponds to these lines which all pass through this point over here. They also pass through this point over here. The origin over here, which is the horizon, is right over here. Okay, what about the vertical lines, the red vertical lines? Those are constant position. Those are observers who are hovering above the black hole at a constant radius. They look like this. Way out, even further out, even further out. In closer, here's somebody hovering close to the horizon. Here's somebody hovering very close to the horizon. The closer you hover close to the horizon, the more you have to accelerate to keep out. So it's not a comfortable thing to be hovering above the horizon because you're being pulled or jerked or whatever, you're being accelerated. This is the exterior of the black hole. What about quadrant two? Quadrant two is over here. Quadrant two is over here and it is the place where you're doomed. Here's the singularity. Now, the singularity is not off at infinity. It takes a finite amount of proper time to fall from the horizon to the singularity. It's a finite excursion from here to here. And so the singularity is a finite time away. Somebody who falls through doesn't get to experience an infinite time. They will hit the singularity. And this is it. Here it is. It's the whole thing, the whole black hole geometry laid out in a Penrose diagram. If you're over here, you're doomed. If you're out here, you can escape or you may go in. Now, you should ask yourself now, what is this other half? If this is the outside of the black hole, what is this other half over here and what's down here? What's down here in particular, what is this thing I drew down here? That's quadrant three and quadrant four. Quadrant three and quadrant four have no real meaning for a real physical black hole. As we're going to see, we're going to work out a real physical black hole, but they really have no meaning. They're not really on the diagram of a real black hole. Nevertheless, you can ask what kind of geometry is described by this full, what it's called as the extended Kruskal diagram. It seems to have two exterior regions. It seems to have two exterior regions, a second exterior region over here where you can escape outward in this direction or you can fall in from this side. It seems to have two exterior regions connected together at the horizon. That's the horizon. That's the place where r is equal to 2mg. r equals 2mg, Schwarzschild r. Remember, it's the place where 1 minus 2mg over r change sign. Whereas you came in, you suddenly changed sign and went off in a light-light direction. All right, so that's what it looks like, but supposing now we take a fixed slice here that looks like this, go right through from here to here. What does that space look like? It's space now. It's not space time, space on this green slice, so this green sheet surface. What does it look like? I'm going to tell you what it looks like. You start out way out here, you start out way out here and space is very big. Namely, the celestial sphere is very, very big. Way, way out here at large r. So you see a very big region of space. I'm just drawing it as a plane. As you come in, you're moving closer and closer to this horizon. What happens to the radius of the sphere? It shrinks down to r equals 2mg. So you could make, you could draw it like this. It's getting smaller and smaller. That's not very good. I don't like that. I don't want to. Let's try to do better. Let's look almost edge on. Then as we come in, it shrinks down these circles. Okay, let's go back a second. Each point on this diagram represents a two-dimensional sphere. Do you understand why it represents a two-dimensional sphere? Remember, we're using polar coordinates. Right. So just as in polar coordinates on a plane, each radial distance represents a circle. Each radial distance in polar, three-dimensional polar coordinates represents a sphere. At each point here, there's a sphere that somebody could move around on, a two-dimensional sphere. And the two-dimensional sphere is very big way out here. As you come in, the two-dimensional sphere gets smaller and smaller until it gets to some minimum size over here. And that's the horizon. That's r equals 2mg. Now, what happens if you just keep going on this diagram, on this sort of fake diagram, half of which has no real meaning? That we'll see. Well, it starts expanding out again. It starts expanding out again until it gets very big. It looks like you can pass through from one side to the other by going through what people call a wormhole. Okay. That's also called an Einstein-Rosen bridge. It connects to what appear to be external regions to the black hole, which get bigger and bigger and bigger as you move away from it. And it looks like the black hole is connecting to two universes, two asymptotic regions. You might think, well, you could pass through going from here to here. You can't. Let's think about somebody who wants to start on this side and wants to pass to this side. If they start out here, anywhere's on this side, and they want to pass to this side, their trajectory has to exceed the speed of light. There's no way to get from this quadrant to that quadrant, even if you start very, very long in the past and you're willing to get way up into the future here, you still have to exceed the speed of light. So, in fact, this diagram is a little bit misleading. It's everything at an instant of time, but there simply isn't really time to go from one side to the other. You cannot go from one side to the other. And so the Einstein-Rosen bridge is not really a bridge. One way to think about it is that it opens up and closes again before anything can pass through it. But the best way to think about it is just to look at this figure and say, yeah, if you could exceed the speed of light and move horizontally, yes, you would move from here to here and going through the neck at the center there, but you're not allowed to do that. You can only move on 45-degree angles or less, and so you really can't get from one side to the other. This has been a source of all sorts of science fiction, passing from one universe to another through the center of a black hole, but as you can see, it doesn't happen. These kind of wormholes, or whatever you want to call them, which you cannot get through, they're called non-reversible wormholes, meaning to say you can't traverse them. You can't go from one side to the other. I'm sorry to disappoint you, but that is the way it is. But in fact, as we're going to see in a few minutes, the meaning of the left-hand side, there's no real meaning to the left-hand side anyway. There's no place to go. There is no real left-hand side. The left-hand side is really absent from here. So let's talk about now creating a black hole. Creating a black hole, not in a laboratory that's too hard, but creating a black hole in infinite space by having some in-falling matter. We're going to take very special kind of in-falling matter. What we're going to do is imagine making a black hole. We start with no black hole. There's no black hole. But what we have is a very, very distant shell. It's not a shell made out of iron or anything like that. It's a shell of incoming radiation. Somebody way, way, far away created an incoming shell of radiation, a thin shell of radiation. Radiation carries energy. Radiation carries momentum. And the shell is coming in. It's coming in with the speed of light. At some point, as we will see, this shell gets close enough together that there's so much energy in a small region that a black hole forms. So this is the simplest version of black hole formation. A star isn't really made up out of things which fall in with the speed of light. This is the problem of a black hole being formed by stuff coming in with the speed of light on a thin shell. It's the simplest of all relativity problems. So let's see if we can figure out how it works. You need to know really only one important thing, two important things. The shell moves in with the speed of light. And the other thing is a version of Newton's theorem. Anybody know what Newton's theorem is? About gravity? For a spherically symmetric gravitational region? In particular, for a mass distribution which forms a shell. Supposing you have a mass distribution which forms a shell, it doesn't matter what whether it's moving or doing anything. If you have a mass distribution which forms a shell for which there's nothing on the interior, then there's no gravitational field on the interior. The interior is completely without gravitational field. So if you have a shell of material spherically symmetric with nothing on the interior, there's a gravitational field on the outside. This is Newtonian. There's a gravitational field on the outside which is identical to a gravitational field of a point mass at the center, same amount of mass. But on the inside of the shell, there is nothing. So outside, it looks like a point mass. Inside, it looks like there's nothing. That's Newton's theorem or a special case of Newton's theorem that on the interior of a distribution of mass which forms a shell, you see no gravitational field. On the outside, you see the gravitational field as it would be if all of that mass was concentrated at a point. That's even true if the shell is moving. As the shell moves in, more and more space will look like the field of the point mass and less and less will look like empty space. But interior to the shell, it always looks like nothing's there. Exterior to the shell, it looks like a point mass. This theorem is also true in general relativity. The statement of the theorem is if you have a shell of infalling or a shell of mass of any kind, then the interior region is just flat spacetime, just as if there was no source. The exterior region, it doesn't look like the Newtonian point source. What does it look like? It looks like the Schwarzschild solution. So if you have in general relativity, on the inside of a shell of mass, flat spacetime, on the exterior of it, it looks like the Schwarzschild black hole, a C-H black hole. So if you had a static non-moving shell, what you would do to construct the actual solution is you would sort of paste together an interior here which was flat with an exterior which was, which is Schwarzschild. This would do the same thing in Newtonian physics. Paste together no gravitational field on the inside with standard gravitational point source on the outside, and that's the way you find solutions of Newtonian physics. You do exactly the same thing in general relativity. Okay, so now we can, let's redraw the rest of this black hole here for a minute. Let's put it back. I erased part of it, but I want to put it back. Okay, so let's go back up to here. Flat spacetime. In flat spacetime, I'm going to redraw it. I want to get rid of all these decorations and just put in nothing, nothing in spacetime. But now I want to add the idea of an incoming shell of radiation. You can think of the incoming shell of radiation as a sort of pulse of incoming photons, a pulse distributed on a nice sphere of incoming photons. In this picture here, they come in from light like infinity. So here's the pulse. Let's draw the pulse of incoming photons, arranged around. Let's put it a little bit later. Let's make it a little bit later. Put it up here. There's the pulse of incoming radiation. It forms a shell. Where is the interior of the shell? At any given instant, where is the interior and where is the exterior of the shell? So that's easy. Here's a line at constant time, space. Where is the interior of the shell? The interior of the shell, in other words, the interior region, interior to the shell, that's the part of the spacetime that's inside the shell near r equals 0. The part of the spacetime that's outside the shell is out beyond the blue line here, and it's out here. Where is the incoming shell, and what does Newton's theorem or the general relativity version of it say? It says that on the interior of the shell, that's everything in here. Everything in there is flat spacetime. In other words, it's correctly represented by the spacetime that we drew here, which was just a representation of flat spacetime. So first of all, on the interior of the shell, which is to the lower left here, the spacetime of just the flat Penrose diagram is the correct representation of everything that's going on. Now what about out here? Out here, we're told that there's a gravitational field. The gravitational field does not look like flat spacetime, so this region out here is wrong. This is not the correct representation of the physics or the geometry of this infalling shell out beyond the shell. What is the correct representation? The correct representation is the Schwarzschild black hole. So let's redraw the Schwarzschild black hole to get rid of all the junk on it. Just redraw it. I lost my black pen. Okay, so here it is. And now somebody on the outside throws in the shell. The shell comes riding in. And what does it look like? It looks like this. Oh, I used blue before. Well, all right, let's use red. Here's the incoming shell. It comes in. Experience is nothing special at this point over here. It just keeps going, but eventually it's a light ray. It's a light ray, a radially incoming light ray. It must move on a 45-degree angle. It moves in and just eventually hits the singularity. All right, so that's what the light ray would look like on the Schwarzschild geometry. But now which part of the diagram is correctly representing the physics that we're doing? The interior of the shell is not correctly represented by the black hole. The interior of the shell is correctly represented by flat spacetime. It's the exterior which is correctly represented. The exterior is out here. We can make it a little bigger. Just I didn't mean to slice it so thin. Let's make it a little bit bigger. That just means the shell starts in earlier. The orange part here is the correct description of the spacetime of the in-falling shell. And the interior in here is not correctly represented. So how do we put these two together? How do we put these two together in order to make a single geometry? And it's pretty easy. You just take the wrong part of this and throw it away. And you take the wrong part of this and you throw it away. And you take the two right parts and sew them together, or glue them together along the joining line where the in-falling shell is. You take this, you put it up there, redraw it, and it looks like this. Let's see if we can see what's going on. You see I've taken this diagram over here and simply placed it on top of the flat spacetime drawing so that the shell matches across them. The shell is the thing they have in common. The in-falling shell is the region that the two of them have in common. And on one side you plaster the black hole geometry. On the other side you put in the flat space geometry. That is the geometry of a black hole that is formed by an in-falling shell. Now excuse me while I consult with Newton. Any questions about this? No discontinuity. But we're going to put that in and see where it is. No, nothing special happens. Okay, where is the horizon? Where is the point of no return? Where on this diagram is somebody doomed? What you do is you start up in that corner there. That's the corner where the singularity appears to meet scri plus. That's infinitely far away so don't worry about it. Scribe plus is very, very far away, very, very late time. But now run a 45 degree line from there down. It's not a place where anybody gets hurt if they fall through it. It's just a place where if you do fall through it you cannot escape from the singularity. You will hit the singularity. If you're on the outside of it, here you can get away. You can escape to scri plus for example. But notice something very, very curious. Even over here you're doomed. You're still in flat space time. You're still in flat space time. The shell hasn't even gotten in yet. You can't see the shell. If you look backward on your light cone you'll not see the shell. The shell is coming in at you at the speed of light. Light hasn't gotten to you yet. You don't know the shell is coming. And yet if you're standing over here, in other words if you were somebody who was just falling over here, just in free fall over here, you're already behind the horizon even though the shell hasn't even gotten in yet. You're still in flat space time. Not only don't you feel anything drastic. You don't feel anything. Nothing coming at you but nevertheless you're doomed. You're doomed. You cannot get out. So the horizon extends all the ways back to here. This is inside the horizon. What happens when the horizon is inside the shell? It hasn't collapsed down to the source of the rays. When the shell hasn't collapsed? It hasn't reached the source of the radius. Right over here is where the shell crosses the source of the radius. Okay, so let's go back and draw a picture. Here's the incoming shell. It has a certain amount of mass. Associated with that mass is a Schwarzschild radius. All right, so there's some massive shell, mass meaning energy. It's made of light but it still has energy, therefore mass. And here is the Schwarzschild radius of a black hole with that mass. If we had a black hole of that mass, it would have a Schwarzschild radius that's that big. As long as the mass is on the outside, well we can say that the mass is on the outside. That's all there is to say. But once the mass, once the light crosses past the Schwarzschild radius, it's trapped. It cannot get out. It has formed a black hole for sure. Where does that form on here? Okay, right over here. Here's the in-falling shell. It crosses the horizon right over here and then it's on the inside and there's no way for that shell to change its mind and get back out. Out here, the shell could change its mind and accelerate back out. Well, I suppose shells of photons don't have minds, but if they had a mind, the photons could turn around over here. Once they get past here, they cannot turn around. They can try to turn around, but they won't get very far. They'll hit the singularity. So this is the point at which the shell crosses its own Schwarzschild radius. Yeah. Is there a meaning of the word horizon until it gets to that point? Yeah. It still means something? The definition of the horizon is the place which separates those people who can get out from those people who can't get out. Can't they get out before a collapsing shell crosses? No. Look, we'll get a picture of it. If you're over here, the shell hasn't crossed. What do you mean before? Pick a time coordinate. This diagram you drew over here. If I'm inside the horizon, its flat space has no gravitational field. Why can't I just walk across here or shine a light? Because by the time you get to here, the shell will have- You can't get to the top of that. Yeah. I think the only answer is to look at this diagram. You look at this diagram and you say, I'm over here. I'm going to try to get out. But by the time you've gotten out, by the time you can't get out, you simply can't get out. You have to exceed the speed of light. The diagram is the whole story. There's no more story than that. If you're here, you can't get out. If you're here, you can't get out. So the horizon has materialized before the shell has actually gotten into its own Schwarzschild radius. So what's the explanation of this? At least what I think is a apparent paradox. Inside the shell there's no gravitational field in this glass space time. Why can't I just walk across the radius? What do you mean walk? Here's walking. You're walking with the speed of light. It's pretty fast walking. I mean the shell is collapsing with the speed of light. If it's a lot farther away from the horizon than you are, you ought to be able to be at the big top. Here it is. What's that saying is that if we're sitting here and a light shell is a million miles away, we're doomed. If the light shell is a million miles away, we are still doomed. It's coming this way. That's right. You can't walk fast enough to get out of its way. If that light is coming in at you, you may be trapped even though you don't know that there's a light ray coming in. So the proof is the diagram. But with this analogous to radiation coming towards us, what we can't get past it, is not a shell coming in. It's not a shell coming in. The horizon is a funny thing. It actually forms before the material gets past the Schwarzschild radius. Its definition is it's the point of no return. It is the point which if you're on this side of it, you can't get out. If you're on that side of it, you can't get out. It's not a point. It's a cone. So if there was a mirror right at the outside of the Schwarzschild radius, and the light comes in, hits the mirror and turns around, bounces back out, what would that diagram look like? Well, not likely you can really do this. You're saying supposing there was a mirror right outside here. I think the point is you can't have a mirror that has no weight. If you could, now surely you can do this to some extent. You can build a mirror around a region like this, and the light ray would reflect off. If you really succeeded in building a mirror, the light ray would reflect off, and it wouldn't... Yeah. Yeah, you wouldn't have a black hole. Yeah, so we can draw it on here. Here's the mirror. The mirror is at a certain distance away from the horizon, away from the center. There's the mirror. The light comes in and bounces off, and that's all that happens. So it's kind of not until it actually crosses the horizon that you're committed to being inside the mirror. Yeah, that's right. Question? Yeah. Just to test my understanding. This light could decide to turn around at this point, but there's no way that this gentleman over here could tell that light to turn around over there. Not even make a mirror really fast. Not over here he can't. If he's over here, he can't get the material out for a mirror over here. He could have made a mirror in the past. He could have made a mirror in the past and protected him. Well, let's see. Can it? Yeah, maybe so. If he was over here coming in, he could decide to shoot out a mirror and have that light ray reflect off the mirror over there. But once he gets past this point, once he gets past that point, he cannot shoot out the mirror out to this point. He can't get it out there in time to rescue himself. Of course, in practice, he wouldn't even know that that light ray was coming in. Well, he might know. He might have Bob could be, or let's see if I can get Alice. Alice is the one who always goes in the black hole. It's just a convention. I didn't invent it. Alice goes in the black hole, but up till this point over here, she's together with Bob and they're having a conversation. And she tells Bob, look, I got an idea. I want you to go out there and then shoot in a shell. And this is going to be a lot of fun. I'm going to get trapped behind the horizon and annihilated at the singularity. And in fact, I know that that's going to happen even before I can see this shell coming at me. So isn't it interesting? So she could arrange with Bob to contrive, to send in that shell, know that it was going to come in, be over here. But once she gets over there, it is too late for her to do anything about it. She can't get a mirror out there. She can get a mirror over to here. But getting it over to here is no good. Everything's behind the horizon already. Anything back here will hit the singularity. She's got to get the mirror out to here. Okay. So. I'm curious about what happens if the mirror is inside the horizon. The mirror seems to, it has to get annihilated. Yeah. The mirror will just hit the singularity. The mirror is just another thing which is doomed. Like the overwhelm of any structure that has been built there when it was flat-spaced top. You could build a structure over here. Well, let's say a structure over here which reflected light. So that light coming in here would hit the structure, reflect off, and go into the singularity. Mirror, Alice, light, all of it winds up on the singularity. So, see there's actually a particular time or is it like infinitely far away? No, it's not infinitely. It's not infinitely far away. There's a finite amount of proper tide between here and here. What about the corner there? The corner there is infinitely far away. But only the corner. Only the corner. Right. But the implication is that somebody falling through the horizon has a finite amount of time to survive. Well, they could try to accelerate like mad once they pass the horizon over here and try to save themselves by sticking to that corner. The problem with that is that it would take an infinite amount of acceleration and an infinite amount of energy to make that much of an acceleration. You basically have to accelerate yourself up to the speed of light. So, once you cross there, you're doomed. But you could last a long time if you could accelerate hard enough. Actually, that doesn't work. The more you accelerate, the less time you experience. That's the time valuation. It doesn't work. It doesn't work. But yeah, I take it back. Your best strategy is to stand still and just accept your fate. Just to make sure I understand. So, if there's a shell, let's say, a million light years lost, no matter what its mass is or its energy equivalent to mass, that thing, if it's going inward, it's going to form a black hole at some point. And if I'm inside that thing, but I'm far enough away from its eventual center, then I could just hang out here and then I can get a problem. Is that right? Sorry, you're at the center of it? No, I'm not at the center. I'm inside it. Inside the shell? No, I'm inside the shell far away from the center, the eventual center. There's no problem for me to interview you. No, sorry, you're inside the shell before the shell crosses the shell. Yeah, you can just hang out and let the shell cross you. Yeah, absolutely. Yeah. Of course, the shell might hit you and make some damage, but that's a separate issue. Translate, transforming that coordinate system to this. Would this be analogous here in a particular point in time, whatever that means? That's a point in time. This is a point of time, this picture. Would... Some point of time. Would it relate that the distance from the shell to the event horizon is greater than the event horizon to the person in between? Is that a different way of looking at this picture? Or how fast he accelerates to the eventual event horizon? He can't get through in time before the shell comes in. All right, so let's ask about what the relationship between these pictures are. Is... Or, I don't know, maybe I should draw it bigger. Let's draw it bigger and understand. No, we can do it over here, right over here. All right, so here's our Penrose diagram for the formation of the black hole. Here's the horizon. Here's the informing shell. The informing shell hits right over here. Okay. Now, in order to speak of an instant of time, which time do we use? Time is just a coordinate. We can make coordinate transformations. An instant of time really means you pick a surface. You pick a surface which is everywhere's space-like. You pick a surface, here's a surface, here's another surface, but you could pick it, you know, in various kinds of ways. Here's another surface. And these series of surfaces define a time variable. Time is one value on here, another value on here, another value on here, and another value on here. If you follow the time, at this time over here, there's a shell far away. The shell is far away. No horizon has formed yet. So, this point has not been reached yet. A little bit later, the shell is still outside the horizon. All right? You're inside, you're in here. This point over here corresponds to the position of the shell, and so far, nothing has formed on the, well, there's a horizon over here. There is a horizon. At this point, right over here, at that time, right over there, that's the time at which the shell passes its own Schwarzschild radius. It's the time that the shell passes into the interior of the horizon. Here's the horizon. The shell passes into the interior, and once that happens, it can't get out. Okay? The Schwarzschild radius grows larger and larger. What's that? The Schwarzschild radius grows larger and larger, right? It goes up that way. Oh, yeah. You're talking about this. Yeah. Yeah. It's not the Schwarzschild radius. Schwarzschild radius is by definition always the same thing, but the size of the horizon, right at this point here, the horizon starts, okay, the horizon starts very small. Only if you're in right near the center here are you doomed. If you're out here, you still get a chance to get away. That corresponds to being here or here. The little tiny horizon has formed. The little tiny horizon has opened up. If you're in here, you're dead. If you're out here, you got a chance. All right? Same thing here. If you're in here, you're dead. If you're out here, you got a chance. And then without the shell haven't even gotten to you yet, the horizon starts to expand until it gets to this point over here. At that point, the shell crosses it. At that point, the incoming shell crosses the growing horizon. And past that point, the shell is on the inside. The horizon is on the outside. And it should stop growing afterwards. It should stop growing. It does stop growing. It stops growing right at that point. Above it is constant radius line. Yeah, this is a constant radius line. That's correct. It doesn't look like it, but it is. Yeah. Did you draw an observer who's sitting at half of the Schwarzschild radius on that diagram? Yeah. Stationary at half of the Schwarzschild radius? Yeah. Well, all right. Yeah, at some point, it just crosses the horizon. It's half the Schwarzschild radius and crosses the horizon. I thought he wouldn't experience any gravity until it passes to the shell. He doesn't experience any gravity until he crosses the shell. But nevertheless, he's dead. He's not dead yet. He's not dead yet. Nevertheless, it's too late for him to escape. There's no gravity until that. Right. Crossing the horizon does not mean that you sense any sudden gravity taking place. That only happens when you cross the shell. But when you cross the horizon, you know what that means. That means you simply don't get away. You can't get away. So in that sense, the horizon is a little bit ghostly. It doesn't have a material significance in the way the shell does. Somebody falling through it doesn't experience it. In fact, they're down here. Somebody falling through it. There's nothing there but flat space. Nevertheless, from that point, cannot travel fast enough to escape the fact that that shell is coming in. This says then that if you start out early enough, and you move faster than the shell grows, then the horizon grows. That's the horizon, though, then you can escape. That's right. Your velocity has to be the same. That's right. From here, you can escape. From here, you can't escape. So your velocity has to be greater than the velocity of the horizon. Well, yes, yes, yes, yes, yes. But that's correct. But you don't see any horizon opening up at you. That's right. You'll have to, you'll have to, you'll have to, that's right. But there's a much simpler way of kind of envisioning it. You have two shells, if the distance between the observer to the inner shell is larger than the inner shell, which is the horizon shell to the radiation shell, no matter how fast he goes, he can't get to that point before they crossed. So he's, he's doomed. That's a much simpler way of envisioning it, I think. That seems third of me. It's a, it's a coordinate, it's warped my brain. You try to do a calculation with that. You find out where, for example, the horizon forms. Try to do that. You'll find, you'll find you can't do it, and you wind up drawing the picture, taking this line, going backward to that point and saying, that's where it is. I don't know. I mean, I assure you, if you want to get a good, get a good picture of what's going on, the way to do it is to become familiar with these diagrams. If you try to reason it out without a good picture in front of you that accurately represents the relationships between the different parts here, and when the relationship, the important relationship, incidentally, is not the relative size of things. Here, there's a huge region of space and time up in this corner here. It's huge, infinite. The same little circle over here might be a tiny little region. So these diagrams are not correctly or faithfully representing size scales. What are they representing? They're representing the motion of light rays. And in Motman, in representing light rays, they're representing what things are in causal relationship to other things. What signals can propagate? Who can send a signal to whom? Can a signal from here get from Alice in here get the bob out here? It reflects, the technical language would be that it reflects the causal structure, cause and effect. Cause and effect mean what can influence what? And the rule is a thing can only influence the things in front of it where light can get to. It cannot influence the things outside the light cone. So these diagrams were intended. They were built, first of all, to be able to draw everything on the blackboard. And second of all, to reflect faithfully the causal relations. What causes an effect? Who can send signals to whom? Who is trapped if they're stuck having to go slower than the speed of light? Those things are correctly reflected here. And so for that reason, these diagrams are very valuable. I can't think without them. If I have to answer a question of the kind you're asking, I have to draw the diagram. And the diagram says things much more clearly than trying to picture, I can't do it. I simply can't. I can't guess whether my guess is a right or I can't tell whether my guess is a right or wrong until I've drawn the diagram. Okay. How long did it take you to be able to be comfortable and intuitive with these diagrams? Not long. Just play with them for a while. Yeah. So this is sort of the opposite of what you said in quantum mechanics. Instead of quantum mechanics, you can't think of it in terms of pictures. You have to go to the mathematics to understand it. Yeah. But yeah, keep in mind, I mean, these are, in a certain sense, these are abstract diagrams. But you're right. I mean, you're right. That's absolutely right. General relativity is classical physics and it can be represented. It can be drawn. It can be even pictured to some extent. All right. It's not so easy to draw, to picture four dimensional curved spacetime. But that's hard. Yeah. But still, you can go a lot further in visualizing it than you can quantum mechanics. I think that's right. I think that's an interesting fact. Quantum mechanics is much more abstract in that sense. Okay. We're going to quit a little early tonight.
 Okay, the last time I explained to you Kruskal coordinates for a black hole, let's go over  it.  And tonight I want to show you how you can think about the formation of a black hole,  a real black hole, and how much of this Kruskal story, the diagram we drew on the blackboard  last time, really is physically meaningful.  Not all of it is.  Okay, so the last time I showed you that the geometry, the metric of a black hole, of a  Schwarzschild black hole, has the property that very near the horizon, that's the horizon  over there.  It's like flat space.  The horizon is a kind of polar coordinate, a hyperbolic polar coordinate point, similar  to the behavior of accelerated coordinate frames.  Somebody standing outside, and this wedge over here, this quadrant over here, is outside.  The rule is, first of all, that 45-degree angles represent the motion of light rays either  going in that direction or going in this direction.  Somebody outside the black hole means out here, and this quadrant, then quadrant, let's  label them, quadrant one, quadrant two, quadrant three, and quadrant four.  And as you're going to see, quadrant three and quadrant four really don't have any physical  significance for a real black hole.  It's quadrant one and quadrant two, which have real significance.  Outside the black hole, somebody at rest, outside the black hole, somebody at rest,  of course, is experiencing a sense of acceleration.  This is I'm experiencing a sense of acceleration right here.  I've gotten used to it, so I don't notice it, the fact that I'm being accelerated.  But if somebody who had been in outer space too long and free fall too long lost the sense  of what a gravitational field is like, and you plunked them on the surface of the earth,  they would be experiencing a sense of upward acceleration.  And in fact, in this diagram, somebody at rest does appear to be accelerating away from  the horizon.  Come in and go out, and it's that acceleration, which is what we experience as gravitational  acceleration near the surface of the earth, but in this case, near the horizon of the  black hole.  So this is a family of observers, further and further out, very far from the black hole,  way out here.  In close to the horizon here, the closer you get to the horizon, the more you have to accelerate,  that's the curvature of these lines here, the more you have to accelerate to keep out  of the black hole.  And in any case, this is the exterior, the lines of constant time, they look like this,  red lines are lines of constant position, the green lines are lines of constant time,  and they accumulate up near this blue line up here.  So this is t equals 0, t equals 1, t equals 2, t equals 3, and t equals infinity is way,  way up on this 45 degree surface here.  So that's the outside of the black hole, and the inside of the black hole is quadrant number  two.  In quadrant number two, for a real black hole geometry, not for flat space, but for a real  black hole geometry, there's something nasty, and the nasty is the singularity, and the  singularity looks like this.  I don't know what it looks like, because I've never seen one, but it's a bad place with  lots and lots of tidal force.  And the problem with it is once you find yourself in quadrant two, there's no way out.  The only way to get out, even if quadrant three was meaningful, which it is not, but  even if it were meaningful, you can't escape from here to out here or to out here without  exceeding the speed of light.  Your trajectory somewhere has to have a direction which is shallow within 45 degrees, and that  means exceeding the speed of light.  So that means if you don't exceed the speed of light, the only place you can wind up is  somewhere on the singularity, so you're doomed.  If you're outside and you make a bad mistake, you could fall in, but if you don't make a  mistake, you can jump back out and get out.  So people on the outside or light sources on the outside, flashlights, whatever you like,  they can produce signals which go in or they can produce signals which go out, and that's  the character of a black hole.  So that's all there is to black holes.  The geometry is somewhat warped.  It's not just flat space.  It's somewhat warped and curved, but this is basically it.  Could you say what the axes of that diagram are?  Time and space.  Time and space as seen by an in-falling observer.  And so the lines of constant position that you drew for what stationary observer outside  the...  Yeah, that's correct.  So for somebody falling in who doesn't accelerate, the coordinates that they see which are not  too different from ordinary flat space in free fall, at least until they get near the  singularity, time and x, those are the coordinates that such a person would use.  So to them, the space looks flat pretty much, pretty much, especially near the horizon over  here.  To somebody on the outside, they experience this acceleration.  And the acceleration is basically the same as the accelerated reference frame in flat  space.  Okay, now, it's very convenient to redraw pictures like this.  Space is infinite here.  It goes on and on and on and ends up somewhere in the next county or further than that.  And time may come to an end at the singularity over here, but these directions offer 45 degrees.  These light light directions, they are also infinite.  And so I can't draw the entire space time on the blackboard.  For many purposes, it is useful, convenient, provides some visual tools, provides some  intuition to be able to draw the entire space time on a finite piece of the blackboard.  Well, you can imagine why.  You want to see everything that's going on.  You want to get a complete overview of space time.  You'd like to draw it on the blackboard in some finite region.  Let's start doing that, but let's start with flat space.  Let's start with good old ordinary flat space and take all the flat space time.  When I say flat space, I mean flat space time.  And let's redraw it or re-coordinateize it so that we do a coordinate transformation  which pulls the whole thing into some finite region on the blackboard.  I'll show you how to do that.  This is useful, incidentally, for geometries and space times which have rotational symmetry.  Rotational symmetry means that the same in every direction.  They mean the same thing, rotational symmetry is when we say a gravitational source has  rotational symmetry, means the same in every direction.  And when a system is the same in every direction and has rotational symmetry, it's usually  useful to describe it.  Now I'm talking about ordinary polar coordinates.  When a system has rotation symmetry, it's the same in every direction.  We don't care so much about the angle.  Everything is the same independent of angle.  What we care about is how things vary as we move away from the center, the radial direction.  And so for many, many purposes, we don't need to think too much about the angular direction.  And we just think of space time, flat space time, ordinary flat space time as having a  time axis.  That's the time axis.  It goes from minus infinity to plus infinity.  And it has a radial direction.  The radial direction is like that.  The radial direction never gets negative.  R is never negative.  And polar coordinates, R is always positive.  And so the entire space time is to the right of the time axis here.  Only because being to the left of the time axis would mean negative R, and R is always  positive.  It's the distance from the origin.  So here's everything, the entire space time.  So let's put some markers on it to be able to visualize some landmarks.  First of all, this horizontal axis is t equals zero.  And it's t equals zero right over there.  Let's put in t equals one.  t equals one looks like this.  t equals two looks like this.  t equals minus one is like this.  t equals minus two is like this.  And if I wanted to draw the entire set of possible times, it would take me up to heaven  and down to hell.  So this is all possible times.  And we could also plot all possible radial distances.  And the radial distances would just look like this.  This would be R equals one.  R equals two.  R equals three.  And in all cases, I will imagine using units so that the speed of light is one.  C equals one.  That simply means that I work in, if I'm going to use seconds for time, I use light seconds  for distance.  And then light rays.  Light rays will move at 45 degree angle.  Well, that's not quite right.  Let's think about the motion of light rays.  If a light ray, let's draw another picture over here.  This is R equals zero.  This is polar coordinates.  This is the radius here.  It's just one instant of time looking and including the angle now.  If a light ray comes in radially toward the origin, it hits the origin.  And then what does it do?  It goes back out.  I meant to draw it straight.  But this is just saying that a light ray in polar coordinates that happens to be aimed  directly at the origin from far away.  Imagine it's coming in from very far away.  If it happens to be aimed right to the origin, it will go to the origin and then go back  out.  On this diagram, it will appear to bounce, to reflect off the origin.  Here's R equals zero.  This is R equals zero over here.  And so a light ray coming in from long in the past would look like, I think I'll use  another color, a light ray coming in from the past will look like this.  Moving at 45 degree angle, it will get to the origin at some time.  And when it gets to the origin, it will simply start back out.  It hasn't really reflected off anything, of course.  It has just gone through the origin, came in, and went back out.  So a light ray starts out in the very remote past coming in at 45 degree angle from a place  that we call light, like infinity, past light, like infinity.  But all it means is the place where all light rays come from in the infinite past.  And they go out to the infinite future.  Of course, there are other light rays that do the same thing.  This one comes in a little bit earlier and goes out a little bit earlier.  Another one up here.  But they all look pretty much the same.  They come in at 45, they bounce off the origin, and they go out at 45 degrees.  Now just for fun, let's think about a light ray which doesn't pass through the origin.  A light ray which doesn't pass through the origin will miss the origin.  It comes in and it misses the origin.  When it's very, very far away, you can't really tell that it isn't done, unless you  have very good precision, angular precision, extremely fine angular precision, watching  that light ray way, way back there, it's hard to tell whether it's going to hit the origin  or not.  So it starts out coming in pretty much the same way.  But it never gets to the origin.  In fact, it almost appears to be repelled away from the origin and then goes back out.  That repulsion is not a real repulsion, but it is what we call centrifugal force.  It really is centrifugal force that keeps the light ray from hitting the origin.  Or you can just say it doesn't hit the origin because it wasn't aimed at the origin.  Same thing.  But what does it look like?  It comes in pretty much the same way at first.  And instead of going through the origin, it gets a point of closest approach.  The closest approach is over here.  At that point, it starts going back out.  So a light ray which is perfectly radially tuned to hit the origin looks like a 45 degree  ingoing and outgoing line.  But a light ray which misses the origin looks like, in fact, sort of like, it looks like  a hyperbola.  But whatever it looks like, that's it.  I'm just telling you these things here to give you some landmarks and to give you some  perspective on what we're going to do next.  Next thing we're going to take this diagram and we're going to squeeze it and stretch  it in various ways to get the whole thing on the blackboard.  That's going to deform it.  It's not going to look the same.  But we're going to keep one thing fixed.  The one thing fixed is that light rays will always be assumed to move on 45 degrees.  In other words, any squeezing and squashing and stretching that we do should preserve  the fact that light rays move in 45 degree angles.  That's a useful thing to do because then we can see how light rays move.  And we can see what going faster than the speed of light, what going slower than the  speed of light looks like.  So I'm going to show you how to do that mathematically.  Excuse me.  Yeah.  There's nothing special about the origin.  There's not a black hole there or anything.  No, no, no.  At the moment there's nothing special.  It's just a particular point that we may want to focus on is there may be something there,  but no need for it.  Okay.  So light rays in this diagram are bent just because of the particular way that we chose  the coordinates.  Yeah, that's right.  It's just a choice of coordinates at the moment.  Okay.  Now, there may be good physical reasons to want to use a particular choice of coordinates.  When studying the sun, there is no black hole, but the symmetry of the problem makes it convenient  to think about the sun as being the center of something.  And right.  So that's all we're doing.  All right.  Next step.  Let's assign some coordinates.  I'm not going to call this T. I'm going to give it another name.  I'm going to call a vertical axis capital T. Capital T and the horizontal axis capital  R.  I don't want to confuse them with other T's and R's that we're going to use and that we  have been using.  So we're going to call it capital T and capital R. But there is another way to coordinate  this, and that's to use coordinates at 45 degrees angle.  In other words, to introduce axes along 45 degree angles here and here.  Of course, I don't mean to imply that there's anything behind R equals 0.  We're only interested in this region here.  But we're going to introduce new coordinates, and this coordinate over here, that's not  going to be T or R. It's going to be T plus R. T plus R varies as you go from here to here  to here.  If I were to plot constant T plus R, constant T plus R would be these lines.  Oops, I didn't draw it well.  Let's draw another picture over here.  R T, constant T plus R will look like this.  These will be the lines of constant T plus R.  OK, so this T plus R, and of course, there's also T minus R. T minus R, the lines look  this way.  But really we're only interested in this half over here.  So the lines of T minus R, constant T minus R, look like that.  These are useful coordinates, as we'll see.  T plus R, that's called T plus.  T minus R is called T minus, just as you might expect.  OK, so we have a different way of describing the same spacetime, the same flat, good old  flat spacetime.  Instead of T plus R and T minus R, we have T plus and T minus.  Now, what about this line here?  This, what was this line?  This line was just plain R equals zero.  R equals zero.  So what is that?  What is R equals zero?  That's T plus equals T minus.  Let's write T plus equals T minus.  T plus equals T minus, that's the same thing as T plus R equals T minus R, or R equals  minus R.  The only solution to R equals minus R is R equals zero.  So this line over here, which was R equals zero, is also just T plus equals T minus.  That's a way to describe it.  OK.  Now we're going to change coordinates, and the way we are going to change coordinates  is to introduce two new coordinates, these are called light like coordinates.  T plus and T minus is called light like coordinates.  We're going to change light like coordinates.  We're going to introduce two new light like coordinates.  One of them is called U plus, and one of them is called U minus, and U plus is a function  of t plus. So we're going to write u plus is some  function of t plus. It's not going to involve t minus at all,  and u minus is going to be, in fact, I think the same function of t minus.  All right, but this function is just a coordinate transformation in which u plus and u minus  independently, sorry, t plus and t minus just independently transformed to some new coordinate.  Good. What kind of function am I going to use?  I'm going to use a function of t plus, this is f or u, this way u equals f of t plus,  and the function is going to look like this. When you go out to t, it doesn't ever become  absolutely flat. It's always got a slight positive slope, also down here it also has a slight  slope, but it'll go to one over here, u plus, u plus, u plus of infinity is one and, or  just f, let's call it f. f of infinity is one and f of minus infinity is minus one.  Now, anybody know a function that does that? There are many functions which do that.  What's that? Yeah, a hyperbolic tangent is the one that's  usually used. Hyperbolic tangent goes to one asymptotically, it's the usual one that's  used but there's nothing special about it. Hyperbolic tangent, hyperbolic tangent, f  of t plus is, and if you don't know what a hyperbolic tangent is, don't worry about  it, it's just a function which looks like this, tanh of t plus.  Okay, now think about what happens as we start moving on this graph further and further outward.  T plus is getting bigger and bigger and bigger. What is u plus doing? It's never going to  get bigger than one, right? It's never going to get bigger than one. So, let's draw our  same space time with these new coordinates, I think I erased them but let's put them back.  U plus equals hyperbolic tangent of t plus and same thing for minus.  As we move out, let's redraw our picture. As we move out, let's start over here and  move out, how big does u plus ever get? U plus never gets bigger than one. So, this  diagram, if this is the u plane, not the t plane but the u plane, u plus never gets  bigger than one, here it is, right there, it never gets bigger than one. So, that means  all of the space time has been mapped somehow down into here. What about u minus? u minus  is equal to hyperbolic tangent of t minus. What this says is as we move down in this  direction, t minus never gets bigger or less negative than minus one. t minus is getting  very, very big and negative as you go down here. t minus is getting bigger and bigger  but negative as you go down here but u minus sort of maxes out again at minus one and  here's t minus equals minus one. t minus never gets too far, t plus never gets too  far. This is, I'm sorry, I said t plus and t minus never get very far, I meant u plus  and u minus. u plus equals one, this is u minus equals minus one in fact. And what about  this vertical line here? The original vertical line over here was t plus equals t minus,  right? t plus equals t minus, that's the same as r equals zero. If t plus equals t minus,  then that says that u plus equals u minus. So, in fact, this line happens to be u plus  equals u minus and it looks like a vertical line. All we've done, if you had, don't follow  the details, all we've done is squished the geometry this way, squished the geometry  that way and squished it onto a finite triangle. You'll understand this better when I draw in  these coordinates. Now, what is the trajectory of a light ray at outgoing, let's say, an  outgoing light ray in terms of the t pluses and the t minuses? Well, an outgoing light  ray is at a constant value of which, t minus, I guess, t minus would be constant and fixed  on here, t plus would be constant and fixed for an ingoing light ray. But if t plus is  constant and fixed along some trajectory, it means that u plus is constant and fixed  along that trajectory, likewise for t minus and u minus. The upshot is that if you make  a coordinate transformation like this, it is still true that radial light rays move on  straight 45 degree angles. These are light rays that come in from the remote past, bounce  off the origin and go back out. Again, bouncing off the origin is just a figure of speech,  you just pass through the origin. So we've succeeded in taking all of space time and  mapping it to a triangle that we can draw the whole thing on the blackboard. Let's draw  in some more coordinates, let's draw in some more detail. In particular, let's take these  horizontal slices which correspond to different times. Let's imagine starting at the origin  at where is t equals zero? Here's t equals zero and moving outward, outward, outward will  allow ourselves even to exceed the speed of light and just move outward in our imagination.  What does that look like? That of course looks like starting at a point over here and just  moving outward. And where do you eventually get to? You get to there. So this point must  be spatial infinity, r equals infinity. It's where you get to if you just walk along this  line till you get out to infinity. But what about this line over here? t equals one.  If you plot that in these coordinates, it'll start about over here, but it will go to that  same point. Likewise, t equals minus one. So what we've done is we've squished all of  spatial infinity, we've squished it into a point right over here. And it's called space-like  infinity. Yeah. No, they're not exactly straight. It depends of course on the particular choice  of function, but they won't be straight. No, let's draw them more carefully. One of them  will be straight. The one will be straight will be the t equals zero. This one will be  straight. Then the next one will look like this. It'll have some curvature. What about  way, way, way up there? t equals a hundred billion. t equals a hundred billion is going  to be very close to this edge. So as we keep putting in more of these, they accumulate and  get closer and closer to this edge up there. You've got to get an infinite number of them  in there. Time really does go on and on forever, at least in flat space. And so this is what  the series of timelines would look like, different times. On the other hand, now let's look at  fixed spatial position. Fixed spatial position, we can mark off different locations. Here's  r equals one, here's r equals two, r equals three, r equals four, r equals five, tata  tata, and they again get all squished up near r equals infinity. Where do they all go?  They all go vertically upward, and on this diagram, they all go up right up to this point  up here. You can check that by doing the transformation, and that's what you would see. More and more  and more of them. It's the same diagram, just in different coordinates. And this coordinates,  as I said, everything gets mapped to a finite region of spacetime. Again, I emphasize, light  rays move at 40, radial light rays move at 45 degrees, they come in and they go out,  and so forth. Okay, this is called, this process of squeezing everything onto a finite piece  of the blackboard without changing the speed of light. It's called compactification. You  take everything and you compactify it. But it's also called making a Penrose diagram.  This is a Penrose diagram of flat spacetime, Roger Penrose. Sometimes it's called a Carter  Penrose diagram because a physicist named Carter, I think actually invented it first,  invented Carter, but it's usually called a Penrose diagram. Good. So that's flat spacetime  laid out on the blackboard. Now, the question is, supposing you took the black hole, is  the black hole geometry? It also goes, oh, let's put some names of points. This is called  spatial infinity or r equals infinity. This one up here, what would you call that? That's  what you get when you just go vertically upward. That's time equals infinity. Right,  these green, t equals infinity. That's this point. What about this one? That's t equals  minus infinity. All right, so this, I'll give you the name. This is called future time-like  infinity. This is called past time-like infinity. This is called spatial infinity or space-like  infinity. And what about these lines here? These 45 degree angles. These 45 degree angles  are where all light rays come from. Light rays come in and go out. They don't really  go out because it takes an infinite amount of time for them to get out. But a light ray  beginning in the infinite remote past starts out its life coming in along one of these  light light trajectories here. Bounces off and goes back out. The standard terminology  is to call this script i minus, sometimes called scri minus, script i, scri minus.  And I don't know why, I don't know where the i came from. I know what the plus and minus  means. This is called i plus or script i plus. And these are called past light-like  infinity and future light-like infinity. The place where, the place where light rays  begin and the place where light rays end. And that's the terminology. And that's what  flat space looks like when compactified. Now, what about the black hole? The Schwarzschild  black hole. What does that look like? We can do exactly the same thing. We can do exactly  the same thing. Take this whole diagram. Introduce again, I guess we can call this R, R. R.  R. The original Kreskal coordinates. And we can again introduce light-like coordinates,  T plus R and T minus R. Do exactly the same operation. And what will we get? What it will  look like, I'm going to draw it over here. And then I'll show you how it works. Okay,  that's everything and the whole thing drawn on the finite blackboard. And I think you  can probably see what goes where. The exterior of the black hole, that's this quadrant over  here. The square. It's square if drawn properly. And we can again draw coordinates. Here's  T equals zero. Here's T equals one. Here's T equals two. Here's T equals three. T equals  four. That corresponds to these lines which all pass through this point over here. They  also pass through this point over here. The origin over here, which is the horizon, is  right over here. Okay, what about the vertical lines, the red vertical lines? Those are constant  position. Those are observers who are hovering above the black hole at a constant radius.  They look like this. Way out, even further out, even further out. In closer, here's  somebody hovering close to the horizon. Here's somebody hovering very close to the horizon.  The closer you hover close to the horizon, the more you have to accelerate to keep out.  So it's not a comfortable thing to be hovering above the horizon because you're being pulled  or jerked or whatever, you're being accelerated. This is the exterior of the black hole. What  about quadrant two? Quadrant two is over here. Quadrant two is over here and it is the place  where you're doomed. Here's the singularity. Now, the singularity is not off at infinity.  It takes a finite amount of proper time to fall from the horizon to the singularity.  It's a finite excursion from here to here. And so the singularity is a finite time away.  Somebody who falls through doesn't get to experience an infinite time. They will hit the singularity.  And this is it. Here it is. It's the whole thing, the whole black hole geometry laid  out in a Penrose diagram. If you're over here, you're doomed. If you're out here, you can  escape or you may go in. Now, you should ask yourself now, what is this other half? If  this is the outside of the black hole, what is this other half over here and what's down  here? What's down here in particular, what is this thing I drew down here? That's quadrant  three and quadrant four. Quadrant three and quadrant four have no real meaning for a real  physical black hole. As we're going to see, we're going to work out a real physical black  hole, but they really have no meaning. They're not really on the diagram of a real black hole.  Nevertheless, you can ask what kind of geometry is described by this full, what it's called  as the extended Kruskal diagram. It seems to have two exterior regions. It seems to have  two exterior regions, a second exterior region over here where you can escape outward in  this direction or you can fall in from this side. It seems to have two exterior regions  connected together at the horizon. That's the horizon. That's the place where r is equal  to 2mg. r equals 2mg, Schwarzschild r. Remember, it's the place where 1 minus 2mg over r change  sign. Whereas you came in, you suddenly changed sign and went off in a light-light direction.  All right, so that's what it looks like, but supposing now we take a fixed slice here that  looks like this, go right through from here to here. What does that space look like? It's  space now. It's not space time, space on this green slice, so this green sheet surface. What  does it look like? I'm going to tell you what it looks like. You start out way out here,  you start out way out here and space is very big. Namely, the celestial sphere is very,  very big. Way, way out here at large r. So you see a very big region of space. I'm just  drawing it as a plane. As you come in, you're moving closer and closer to this horizon. What  happens to the radius of the sphere? It shrinks down to r equals 2mg. So you could make, you  could draw it like this. It's getting smaller and smaller. That's not very good. I don't  like that. I don't want to. Let's try to do better. Let's look almost edge on. Then as  we come in, it shrinks down these circles. Okay, let's go back a second. Each point on  this diagram represents a two-dimensional sphere. Do you understand why it represents  a two-dimensional sphere? Remember, we're using polar coordinates. Right. So just as  in polar coordinates on a plane, each radial distance represents a circle. Each radial  distance in polar, three-dimensional polar coordinates represents a sphere. At each point  here, there's a sphere that somebody could move around on, a two-dimensional sphere.  And the two-dimensional sphere is very big way out here. As you come in, the two-dimensional  sphere gets smaller and smaller until it gets to some minimum size over here. And that's  the horizon. That's r equals 2mg. Now, what happens if you just keep going on this diagram,  on this sort of fake diagram, half of which has no real meaning? That we'll see. Well,  it starts expanding out again. It starts expanding out again until it gets very big. It looks  like you can pass through from one side to the other by going through what people call  a wormhole. Okay. That's also called an Einstein-Rosen bridge. It connects to what appear to be  external regions to the black hole, which get bigger and bigger and bigger as you move  away from it. And it looks like the black hole is connecting to two universes, two asymptotic  regions. You might think, well, you could pass through going from here to here. You can't.  Let's think about somebody who wants to start on this side and wants to pass to this side.  If they start out here, anywhere's on this side, and they want to pass to this side,  their trajectory has to exceed the speed of light. There's no way to get from this quadrant  to that quadrant, even if you start very, very long in the past and you're willing to  get way up into the future here, you still have to exceed the speed of light. So, in  fact, this diagram is a little bit misleading. It's everything at an instant of time, but  there simply isn't really time to go from one side to the other. You cannot go from  one side to the other. And so the Einstein-Rosen bridge is not really a bridge. One way to  think about it is that it opens up and closes again before anything can pass through it.  But the best way to think about it is just to look at this figure and say, yeah, if you  could exceed the speed of light and move horizontally, yes, you would move from here to here and  going through the neck at the center there, but you're not allowed to do that. You can  only move on 45-degree angles or less, and so you really can't get from one side to the  other. This has been a source of all sorts of science fiction, passing from one universe  to another through the center of a black hole, but as you can see, it doesn't happen.  These kind of wormholes, or whatever you want to call them, which you cannot get through,  they're called non-reversible wormholes, meaning to say you can't traverse them. You can't  go from one side to the other. I'm sorry to disappoint you, but that is the way it is.  But in fact, as we're going to see in a few minutes, the meaning of the left-hand side,  there's no real meaning to the left-hand side anyway. There's no place to go. There is no  real left-hand side. The left-hand side is really absent from here.  So let's talk about now creating a black hole. Creating a black hole, not in a laboratory  that's too hard, but creating a black hole in infinite space by having some in-falling  matter. We're going to take very special kind of in-falling matter. What we're going to  do is imagine making a black hole. We start with no black hole. There's no black hole.  But what we have is a very, very distant shell. It's not a shell made out of iron or anything  like that. It's a shell of incoming radiation. Somebody way, way, far away created an incoming  shell of radiation, a thin shell of radiation. Radiation carries energy. Radiation carries  momentum. And the shell is coming in. It's coming in with the speed of light.  At some point, as we will see, this shell gets close enough together that there's so  much energy in a small region that a black hole forms. So this is the simplest version  of black hole formation. A star isn't really made up out of things which fall in with the  speed of light. This is the problem of a black hole being formed by stuff coming in with  the speed of light on a thin shell. It's the simplest of all relativity problems. So let's  see if we can figure out how it works. You need to know really only one important thing,  two important things. The shell moves in with the speed of light. And the other thing is  a version of Newton's theorem. Anybody know what Newton's theorem is? About gravity? For  a spherically symmetric gravitational region? In particular, for a mass distribution which  forms a shell. Supposing you have a mass distribution which forms a shell, it doesn't matter what  whether it's moving or doing anything. If you have a mass distribution which forms a  shell for which there's nothing on the interior, then there's no gravitational field on the  interior. The interior is completely without gravitational field. So if you have a shell  of material spherically symmetric with nothing on the interior, there's a gravitational field  on the outside. This is Newtonian. There's a gravitational field on the outside which  is identical to a gravitational field of a point mass at the center, same amount of  mass. But on the inside of the shell, there is nothing. So outside, it looks like a point  mass. Inside, it looks like there's nothing. That's Newton's theorem or a special case  of Newton's theorem that on the interior of a distribution of mass which forms a shell,  you see no gravitational field. On the outside, you see the gravitational field as it would  be if all of that mass was concentrated at a point. That's even true if the shell is  moving. As the shell moves in, more and more space will look like the field of the point  mass and less and less will look like empty space. But interior to the shell, it always  looks like nothing's there. Exterior to the shell, it looks like a point mass. This theorem  is also true in general relativity. The statement of the theorem is if you have a shell of  infalling or a shell of mass of any kind, then the interior region is just flat spacetime,  just as if there was no source. The exterior region, it doesn't look like the Newtonian  point source. What does it look like? It looks like the Schwarzschild solution. So if you  have in general relativity, on the inside of a shell of mass, flat spacetime, on the  exterior of it, it looks like the Schwarzschild black hole, a C-H black hole. So if you had  a static non-moving shell, what you would do to construct the actual solution is you  would sort of paste together an interior here which was flat with an exterior which  was, which is Schwarzschild. This would do the same thing in Newtonian physics. Paste  together no gravitational field on the inside with standard gravitational point source on  the outside, and that's the way you find solutions of Newtonian physics. You do exactly the same  thing in general relativity. Okay, so now we can, let's redraw the rest of this black  hole here for a minute. Let's put it back. I erased part of it, but I want to put it  back. Okay, so let's go back up to here. Flat spacetime. In flat spacetime, I'm going to  redraw it. I want to get rid of all these decorations and just put in nothing, nothing  in spacetime. But now I want to add the idea of an incoming shell of radiation. You can  think of the incoming shell of radiation as a sort of pulse of incoming photons, a pulse  distributed on a nice sphere of incoming photons. In this picture here, they come in from light  like infinity. So here's the pulse. Let's draw the pulse of incoming photons, arranged  around. Let's put it a little bit later. Let's make it a little bit later. Put it up here.  There's the pulse of incoming radiation. It forms a shell. Where is the interior of the  shell? At any given instant, where is the interior and where is the exterior of the shell?  So that's easy. Here's a line at constant time, space. Where is the interior of the shell?  The interior of the shell, in other words, the interior region, interior to the shell,  that's the part of the spacetime that's inside the shell near r equals 0. The part of the  spacetime that's outside the shell is out beyond the blue line here, and it's out here.  Where is the incoming shell, and what does Newton's theorem or the general relativity  version of it say? It says that on the interior of the shell, that's everything in here. Everything  in there is flat spacetime. In other words, it's correctly represented by the spacetime  that we drew here, which was just a representation of flat spacetime. So first of all, on the  interior of the shell, which is to the lower left here, the spacetime of just the flat  Penrose diagram is the correct representation of everything that's going on. Now what about  out here? Out here, we're told that there's a gravitational field. The gravitational field  does not look like flat spacetime, so this region out here is wrong. This is not the  correct representation of the physics or the geometry of this infalling shell out beyond  the shell. What is the correct representation? The correct representation is the Schwarzschild  black hole. So let's redraw the Schwarzschild black hole to get rid of all the junk on  it. Just redraw it. I lost my black pen. Okay, so here it is. And now somebody on the outside  throws in the shell. The shell comes riding in. And what does it look like? It looks like  this. Oh, I used blue before. Well, all right, let's use red. Here's the incoming shell.  It comes in. Experience is nothing special at this point over here. It just keeps going,  but eventually it's a light ray. It's a light ray, a radially incoming light ray. It must  move on a 45-degree angle. It moves in and just eventually hits the singularity. All  right, so that's what the light ray would look like on the Schwarzschild geometry. But  now which part of the diagram is correctly representing the physics that we're doing?  The interior of the shell is not correctly represented by the black hole. The interior  of the shell is correctly represented by flat spacetime. It's the exterior which is correctly  represented. The exterior is out here. We can make it a little bigger. Just I didn't  mean to slice it so thin. Let's make it a little bit bigger. That just means the shell starts  in earlier. The orange part here is the correct description of the spacetime of the in-falling  shell. And the interior in here is not correctly represented. So how do we put these two together?  How do we put these two together in order to make a single geometry? And it's pretty  easy. You just take the wrong part of this and throw it away. And you take the wrong  part of this and you throw it away. And you take the two right parts and sew them together,  or glue them together along the joining line where the in-falling shell is. You take this,  you put it up there, redraw it, and it looks like this. Let's see if we can see what's  going on. You see I've taken this diagram over here and simply placed it on top of the  flat spacetime drawing so that the shell matches across them. The shell is the thing  they have in common. The in-falling shell is the region that the two of them have in  common. And on one side you plaster the black hole geometry. On the other side you put in  the flat space geometry. That is the geometry of a black hole that is formed by an in-falling  shell. Now excuse me while I consult with Newton. Any questions about this?  No discontinuity. But we're going to put that in and see where it is. No, nothing special  happens. Okay, where is the horizon? Where is the point of no return? Where on this diagram  is somebody doomed? What you do is you start up in that corner there. That's the corner  where the singularity appears to meet scri plus. That's infinitely far away so don't  worry about it. Scribe plus is very, very far away, very, very late time. But now run  a 45 degree line from there down. It's not a place where anybody gets hurt if they fall  through it. It's just a place where if you do fall through it you cannot escape from  the singularity. You will hit the singularity. If you're on the outside of it, here you can  get away. You can escape to scri plus for example. But notice something very, very curious.  Even over here you're doomed. You're still in flat space time. You're still in flat  space time. The shell hasn't even gotten in yet. You can't see the shell. If you look  backward on your light cone you'll not see the shell. The shell is coming in at you at  the speed of light. Light hasn't gotten to you yet. You don't know the shell is coming.  And yet if you're standing over here, in other words if you were somebody who was just falling  over here, just in free fall over here, you're already behind the horizon even though the  shell hasn't even gotten in yet. You're still in flat space time. Not only don't you feel  anything drastic. You don't feel anything. Nothing coming at you but nevertheless you're  doomed. You're doomed. You cannot get out. So the horizon extends all the ways back to  here. This is inside the horizon. What happens when the horizon is inside the shell? It  hasn't collapsed down to the source of the rays. When the shell hasn't collapsed? It  hasn't reached the source of the radius. Right over here is where the shell crosses  the source of the radius. Okay, so let's go back and draw a picture. Here's the incoming  shell. It has a certain amount of mass. Associated with that mass is a Schwarzschild radius.  All right, so there's some massive shell, mass meaning energy. It's made of light but  it still has energy, therefore mass. And here is the Schwarzschild radius of a black hole  with that mass. If we had a black hole of that mass, it would have a Schwarzschild radius  that's that big. As long as the mass is on the outside, well we can say that the mass  is on the outside. That's all there is to say. But once the mass, once the light crosses  past the Schwarzschild radius, it's trapped. It cannot get out. It has formed a black hole  for sure. Where does that form on here? Okay, right over here. Here's the in-falling shell.  It crosses the horizon right over here and then it's on the inside and there's no way  for that shell to change its mind and get back out. Out here, the shell could change  its mind and accelerate back out. Well, I suppose shells of photons don't have minds,  but if they had a mind, the photons could turn around over here. Once they get past  here, they cannot turn around. They can try to turn around, but they won't get very far.  They'll hit the singularity. So this is the point at which the shell crosses its own Schwarzschild  radius. Yeah. Is there a meaning of the word horizon until it gets to that point? Yeah.  It still means something? The definition of the horizon is the place which separates  those people who can get out from those people who can't get out. Can't they get out before  a collapsing shell crosses? No. Look, we'll get a picture of it. If you're over here,  the shell hasn't crossed. What do you mean before? Pick a time coordinate.  This diagram you drew over here. If I'm inside the horizon, its flat space has no gravitational  field. Why can't I just walk across here or shine a light?  Because by the time you get to here, the shell will have-  You can't get to the top of that. Yeah. I think the only answer is to look at this  diagram. You look at this diagram and you say, I'm over here. I'm going to try to get  out. But by the time you've gotten out, by the time you can't get out, you simply can't  get out. You have to exceed the speed of light. The diagram is the whole story. There's no  more story than that. If you're here, you can't get out. If you're here, you can't get out.  So the horizon has materialized before the shell has actually gotten into its own Schwarzschild  radius. So what's the explanation of this? At least  what I think is a apparent paradox. Inside the shell there's no gravitational field  in this glass space time. Why can't I just walk across the radius?  What do you mean walk? Here's walking. You're walking with the speed of light. It's pretty  fast walking. I mean the shell is collapsing with the speed of light. If it's a lot farther  away from the horizon than you are, you ought to be able to be at the big top.  Here it is. What's that saying is that if we're sitting  here and a light shell is a million miles away, we're doomed. If the light shell is  a million miles away, we are still doomed. It's coming this way. That's right. You can't  walk fast enough to get out of its way. If that light is coming in at you, you may be  trapped even though you don't know that there's a light ray coming in. So the proof is the  diagram.  But with this analogous to radiation coming towards us, what we can't get past it, is  not a shell coming in. It's not a shell coming in.  The horizon is a funny thing. It actually forms before the material gets past the Schwarzschild  radius. Its definition is it's the point of no return. It is the point which if you're  on this side of it, you can't get out. If you're on that side of it, you can't get out.  It's not a point. It's a cone.  So if there was a mirror right at the outside of the Schwarzschild radius, and the light  comes in, hits the mirror and turns around, bounces back out, what would that diagram  look like?  Well, not likely you can really do this. You're saying supposing there was a mirror right outside  here.  I think the point is you can't have a mirror that has no weight.  If you could, now surely you can do this to some extent. You can build a mirror around  a region like this, and the light ray would reflect off. If you really succeeded in building  a mirror, the light ray would reflect off, and it wouldn't...  Yeah.  Yeah, you wouldn't have a black hole. Yeah, so we can draw it on here. Here's the mirror.  The mirror is at a certain distance away from the horizon, away from the center. There's  the mirror. The light comes in and bounces off, and that's all that happens.  So it's kind of not until it actually crosses the horizon that you're committed to being  inside the mirror.  Yeah, that's right.  Question?  Yeah.  Just to test my understanding.  This light could decide to turn around at this point, but there's no way that this gentleman  over here could tell that light to turn around over there.  Not even make a mirror really fast.  Not over here he can't. If he's over here, he can't get the material out for a mirror  over here. He could have made a mirror in the past. He could have made a mirror in the past  and protected him. Well, let's see. Can it? Yeah, maybe so. If he was over here coming  in, he could decide to shoot out a mirror and have that light ray reflect off the mirror  over there. But once he gets past this point, once he gets past that point, he cannot shoot  out the mirror out to this point. He can't get it out there in time to rescue himself.  Of course, in practice, he wouldn't even know that that light ray was coming in.  Well, he might know. He might have Bob could be, or let's see if I can get Alice. Alice  is the one who always goes in the black hole. It's just a convention. I didn't invent it.  Alice goes in the black hole, but up till this point over here, she's together with Bob  and they're having a conversation. And she tells Bob, look, I got an idea. I want you  to go out there and then shoot in a shell. And this is going to be a lot of fun. I'm  going to get trapped behind the horizon and annihilated at the singularity. And in fact,  I know that that's going to happen even before I can see this shell coming at me. So isn't  it interesting? So she could arrange with Bob to contrive, to send in that shell, know  that it was going to come in, be over here. But once she gets over there, it is too late  for her to do anything about it. She can't get a mirror out there. She can get a mirror  over to here. But getting it over to here is no good. Everything's behind the horizon  already. Anything back here will hit the singularity. She's got to get the mirror out to here.  Okay. So.  I'm curious about what happens if the mirror is inside the horizon. The mirror seems to,  it has to get annihilated.  Yeah. The mirror will just hit the singularity. The mirror is just another thing which is  doomed.  Like the overwhelm of any structure that has been built there when it was flat-spaced top.  You could build a structure over here. Well, let's say a structure over here which reflected  light. So that light coming in here would hit the structure, reflect off, and go into  the singularity. Mirror, Alice, light, all of it winds up on the singularity.  So, see there's actually a particular time or is it like infinitely far away?  No, it's not infinitely. It's not infinitely far away. There's a finite amount of proper  tide between here and here.  What about the corner there?  The corner there is infinitely far away. But only the corner. Only the corner. Right.  But the implication is that somebody falling through the horizon has a finite amount of  time to survive.  Well, they could try to accelerate like mad once they pass the horizon over here and try  to save themselves by sticking to that corner.  The problem with that is that it would take an infinite amount of acceleration and an  infinite amount of energy to make that much of an acceleration. You basically have to  accelerate yourself up to the speed of light.  So, once you cross there, you're doomed. But you could last a long time if you could  accelerate hard enough.  Actually, that doesn't work. The more you accelerate, the less time you experience.  That's the time valuation. It doesn't work. It doesn't work. But yeah, I take it back.  Your best strategy is to stand still and just accept your fate.  Just to make sure I understand. So, if there's a shell, let's say, a million light years  lost, no matter what its mass is or its energy equivalent to mass, that thing, if it's going  inward, it's going to form a black hole at some point.  And if I'm inside that thing, but I'm far enough away from its eventual center, then  I could just hang out here and then I can get a problem. Is that right?  Sorry, you're at the center of it?  No, I'm not at the center. I'm inside it.  Inside the shell?  No, I'm inside the shell far away from the center, the eventual center.  There's no problem for me to interview you.  No, sorry, you're inside the shell before the shell crosses the shell.  Yeah, you can just hang out and let the shell cross you.  Yeah, absolutely.  Yeah.  Of course, the shell might hit you and make some damage, but that's a separate issue.  Translate, transforming that coordinate system to this.  Would this be analogous here in a particular point in time, whatever that means?  That's a point in time. This is a point of time, this picture.  Would...  Some point of time.  Would it relate that the distance from the shell to the event horizon is greater than  the event horizon to the person in between?  Is that a different way of looking at this picture?  Or how fast he accelerates to the eventual event horizon?  He can't get through in time before the shell comes in.  All right, so let's ask about what the relationship between these pictures are.  Is...  Or, I don't know, maybe I should draw it bigger.  Let's draw it bigger and understand.  No, we can do it over here, right over here.  All right, so here's our Penrose diagram for the formation of the black hole.  Here's the horizon.  Here's the informing shell.  The informing shell hits right over here.  Okay.  Now, in order to speak of an instant of time, which time do we use?  Time is just a coordinate.  We can make coordinate transformations.  An instant of time really means you pick a surface.  You pick a surface which is everywhere's space-like.  You pick a surface, here's a surface, here's another surface, but you could pick it, you  know, in various kinds of ways.  Here's another surface.  And these series of surfaces define a time variable.  Time is one value on here, another value on here, another value on here, and another value  on here.  If you follow the time, at this time over here, there's a shell far away.  The shell is far away.  No horizon has formed yet.  So, this point has not been reached yet.  A little bit later, the shell is still outside the horizon.  All right?  You're inside, you're in here.  This point over here corresponds to the position of the shell, and so far, nothing has formed  on the, well, there's a horizon over here.  There is a horizon.  At this point, right over here, at that time, right over there, that's the time at which  the shell passes its own Schwarzschild radius.  It's the time that the shell passes into the interior of the horizon.  Here's the horizon.  The shell passes into the interior, and once that happens, it can't get out.  Okay?  The Schwarzschild radius grows larger and larger.  What's that?  The Schwarzschild radius grows larger and larger, right?  It goes up that way.  Oh, yeah.  You're talking about this.  Yeah.  Yeah.  It's not the Schwarzschild radius.  Schwarzschild radius is by definition always the same thing, but the size of the horizon,  right at this point here, the horizon starts, okay, the horizon starts very small.  Only if you're in right near the center here are you doomed.  If you're out here, you still get a chance to get away.  That corresponds to being here or here.  The little tiny horizon has formed.  The little tiny horizon has opened up.  If you're in here, you're dead.  If you're out here, you got a chance.  All right?  Same thing here.  If you're in here, you're dead.  If you're out here, you got a chance.  And then without the shell haven't even gotten to you yet, the horizon starts to expand until it gets to this point over here.  At that point, the shell crosses it.  At that point, the incoming shell crosses the growing horizon.  And past that point, the shell is on the inside.  The horizon is on the outside.  And it should stop growing afterwards.  It should stop growing.  It does stop growing.  It stops growing right at that point.  Above it is constant radius line.  Yeah, this is a constant radius line.  That's correct.  It doesn't look like it, but it is.  Yeah.  Did you draw an observer who's sitting at half of the Schwarzschild radius on that diagram?  Yeah.  Stationary at half of the Schwarzschild radius?  Yeah.  Well, all right.  Yeah, at some point, it just crosses the horizon.  It's half the Schwarzschild radius and crosses the horizon.  I thought he wouldn't experience any gravity until it passes to the shell.  He doesn't experience any gravity until he crosses the shell.  But nevertheless, he's dead.  He's not dead yet.  He's not dead yet.  Nevertheless, it's too late for him to escape.  There's no gravity until that.  Right.  Crossing the horizon does not mean that you sense any sudden gravity taking place.  That only happens when you cross the shell.  But when you cross the horizon, you know what that means.  That means you simply don't get away.  You can't get away.  So in that sense, the horizon is a little bit ghostly.  It doesn't have a material significance in the way the shell does.  Somebody falling through it doesn't experience it.  In fact, they're down here.  Somebody falling through it.  There's nothing there but flat space.  Nevertheless, from that point, cannot travel fast enough to escape the fact that that shell is coming in.  This says then that if you start out early enough, and you move faster than the shell grows,  then the horizon grows.  That's the horizon, though, then you can escape.  That's right.  Your velocity has to be the same.  That's right.  From here, you can escape.  From here, you can't escape.  So your velocity has to be greater than the velocity of the horizon.  Well, yes, yes, yes, yes, yes.  But that's correct.  But you don't see any horizon opening up at you.  That's right.  You'll have to, you'll have to, you'll have to, that's right.  But there's a much simpler way of kind of envisioning it.  You have two shells, if the distance between the observer to the inner shell is larger than the inner shell,  which is the horizon shell to the radiation shell, no matter how fast he goes,  he can't get to that point before they crossed.  So he's, he's doomed.  That's a much simpler way of envisioning it, I think.  That seems third of me.  It's a, it's a coordinate, it's warped my brain.  You try to do a calculation with that.  You find out where, for example, the horizon forms.  Try to do that.  You'll find, you'll find you can't do it, and you wind up drawing the picture, taking this line,  going backward to that point and saying, that's where it is.  I don't know.  I mean, I assure you, if you want to get a good,  get a good picture of what's going on, the way to do it is to become familiar with these diagrams.  If you try to reason it out without a good picture in front of you that accurately represents the relationships between the different parts here,  and when the relationship, the important relationship, incidentally, is not the relative size of things.  Here, there's a huge region of space and time up in this corner here.  It's huge, infinite.  The same little circle over here might be a tiny little region.  So these diagrams are not correctly or faithfully representing size scales.  What are they representing?  They're representing the motion of light rays.  And in Motman, in representing light rays, they're representing what things are in causal relationship to other things.  What signals can propagate?  Who can send a signal to whom?  Can a signal from here get from Alice in here get the bob out here?  It reflects, the technical language would be that it reflects the causal structure, cause and effect.  Cause and effect mean what can influence what?  And the rule is a thing can only influence the things in front of it where light can get to.  It cannot influence the things outside the light cone.  So these diagrams were intended.  They were built, first of all, to be able to draw everything on the blackboard.  And second of all, to reflect faithfully the causal relations.  What causes an effect?  Who can send signals to whom?  Who is trapped if they're stuck having to go slower than the speed of light?  Those things are correctly reflected here.  And so for that reason, these diagrams are very valuable.  I can't think without them.  If I have to answer a question of the kind you're asking, I have to draw the diagram.  And the diagram says things much more clearly than trying to picture, I can't do it.  I simply can't.  I can't guess whether my guess is a right or I can't tell whether my guess is a right or wrong until I've drawn the diagram.  Okay.  How long did it take you to be able to be comfortable and intuitive with these diagrams?  Not long.  Just play with them for a while.  Yeah.  So this is sort of the opposite of what you said in quantum mechanics.  Instead of quantum mechanics, you can't think of it in terms of pictures.  You have to go to the mathematics to understand it.  Yeah.  But yeah, keep in mind, I mean, these are, in a certain sense, these are abstract diagrams.  But you're right.  I mean, you're right.  That's absolutely right.  General relativity is classical physics and it can be represented.  It can be drawn.  It can be even pictured to some extent.  All right.  It's not so easy to draw, to picture four dimensional curved spacetime.  But that's hard.  Yeah.  But still, you can go a lot further in visualizing it than you can quantum mechanics.  I think that's right.  I think that's an interesting fact.  Quantum mechanics is much more abstract in that sense.  Okay.  We're going to quit a little early tonight.
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