ASTR 160: Frontiers and Controversies in Astrophysics

Lecture 17

 - Hubble's Law and the Big Bang (cont.)


Class begins with a review of magnitudes and the problem set involving magnitude equations. Implications of the Hubble Law and Hubble Diagram are discussed. Professor Bailyn elaborates on the Big Bang theory of cosmology and addresses controversial questions related to the age, development, and boundaries of the universe. The fate of the universe, and possibly its end (known as the Big Crunch) are addressed. Imagining an expanding three-dimensional universe is proposed. The lecture ends with a question-and-answer session during which students inquire about a variety of topics related to cosmology, such as the center of the universe, its current expansion, and hypothetical collapse.

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Frontiers and Controversies in Astrophysics

ASTR 160 - Lecture 17 - Hubble's Law and the Big Bang (cont.)

Chapter 1. Review of Magnitudes [00:00:00]

Professor Charles Bailyn: Okay, welcome back for more cosmology. What I want to do today is quickly review what we were doing about magnitudes and make a comment or two about the problem set, and then, go back and talk about the implications of the Hubble Law and the Hubble Diagram, which are formidable, to put it mildly.

Okay, magnitudes. There’s a couple of these magnitude equations. I’m just going to write them down. The first of them looks like this. And this equation is used–okay. So, this equation is used to relate magnitudes of two different objects to each other. So, we’ve got two different objects. And it can be used for either kind of magnitude–either absolute or apparent magnitude, just so long as you don’t mix them. So, it’s two different objects, but only one of the magnitudes. One kind of magnitude. And depending on which kind of magnitude you use, this brightness ratio–it’s either the ratio of how bright it looks or the ratio of how bright it is–whatever’s appropriate.

Now, on the help sheet on the web, I have this equation in a somewhat different form, and it’s important to realize that it’s the exact same equation. Watch this. Let’s see. Let’s multiply both halves by - 25 which is  - 0.4. So, this is -0.4 (M1 ‒ M2) = log (b1 / b2). And then, let’s take 10 to the power of that. That gets rid of the log. And this is now the form that it is on the help sheet on the web. So, it’s exactly the same equation, just expressed differently. And you can use either form, whichever is more convenient.

Okay. The other equation looks like this. 5 log (D/10 parsecs). And this relates one object, but it relates both kinds of magnitude to each other. So, the first one is two different objects, but only one of the magnitudes. The other is one object and it relates the two different kinds of magnitudes to each other, and to the distance to the object. And as you can see, this–both of these equations, actually, have three unknowns. One, two, three. That means you’ve got to know two things in order to find out the third.

And that brings me to the comment I want to make about problem 2a on the current problem set. You are asked in this problem to determine the difference between the absolute magnitude of one kind of star, called Type 1 Cepheid, so I label it C1. And another kind of star, Type 2 Cepheids, which I label C2. And if you’re asked to–and this difference is called, I don’t know, delta MC or something like that.

And having been asked to do this, the logical thing that you might try to do is say, all right, I’m going to use one or the other of these equations–I’m not sure, in advance, which, to compute this one. Then I’m going to compute this one. And I’m going to subtract the two, and that’s going to give me the answer. That approach will fail. Okay? That doesn’t work in this particular case, because you don’t actually have enough information to compute either one of these things. You do have enough information to compute the difference. And let me just give you a very brief hint on how you might go about doing that. Let’s see. Let me take a new piece of paper here.

Write down mC1 - MC1 = 5 log (DC1 / 10parsecs). And now write the exact same equation down for C2, where the two different distances are the distances you get by assuming one or the other kinds of these magnitudes. This is equation one. This is equation two. Now, let’s subtract.

1 ‒ 2.

So, then, you get mC1 - mC2 - [MC1 - MC2] = 5 (log DC1 - log DC2).

Okay, now. Here’s the trick. Turns out for reasons that you had better tell me–and TFs [teaching fellows], take note that we really want them to say why this is true, now that I’ve told them it is. This is zero. The two apparent magnitudes are the same. And so, that means that this side of the equation is what you want. It’s the difference between these two magnitudes. And then, over here, you have to use one of these log rules: log (x) - log (y), if you remember back to eleventh grade, is log (x / y).

And if you use that, it turns out that you have information elsewhere in the problem, which will tell you what you need to know about the distances. And so, in this way, you can solve for the difference without actually being able to determine either one of these two things.

So, we’ll just leave it at that for the moment. If you have problems there’s the usual forum, there’s the usual office hours, but ponder this. That’s basically how the problem has to go. Okay? Problems with magnitudes? Okay. If you do have some, let us know, because this is going to be critical for solving, basically, problems for the whole rest of the class. All right. Yes go ahead.

Student: What is the location–the last thing in green–

Professor Charles Bailyn: This?

Student: –5 log–is it over log?

Professor Charles Bailyn: This is log (DC1 / 10 parsecs) - log (DC2 / 10 parsecs). That’s just subtracting the two right-hand sides of this equation. But then, you get to do the log thing and divide them in stead, which is this to this on this log subtraction. Okay.

Chapter 2. Implications of Hubble’s Discoveries on the Aging Universe [00:07:38]

Let me remind you why we’re putting ourselves through this pain. Okay? Recall why we started doing this in the first place. The goal was to figure out how to measure the Hubble Diagram. The Hubble Diagram is this diagram of velocity, which you can measure by redshift versus distance. And the whole reason we embarked on this adventure in magnitudes was because that’s a critical component in how you determine the distance.

But if you’ve got a bunch of galaxies and you measure these two quantities for each one of them, what you discover–what Hubble discovered–what Edwin Hubble discovered many years ago, is that they line up. You get this perfect; well, not quite perfect, but close to it–this beautiful straight line, if you measure a bunch of these things. And the way you represent that straight line is with this equation, where H is Hubble’s Constant. And so, that’s the purpose of all this magnitude stuff, is to be able to determine the y-axis of this plot.

What I now want to do is talk about the implications of this observational fact that galaxies line up on this line. It turns out, this is one of the most profound plots in all of astrophysics, and possibly all of science. Because what this implies is, first of all, that the Universe is expanding, and hence, it’s the basis for the whole Big Bang Theory of cosmology. And by performing relatively simple calculations using this quantity H, you can determine the age of the Universe, and the ultimate fate of the Universe. Not bad for a relatively simple algebraic equation. Yes?

Student: [Inaudible]

Professor Charles Bailyn: Huh?

Student: [Inaudible]

Professor Charles Bailyn: Oh, fate. I’ll get back to that. The big question in cosmology is, you know–the Universe is now expanding. The question is, will it continue to expand, in which case, the Universe just gets sparse and cold and boring, and expands forever until there’s, you know, one pathetic hydrogen atom every cubic megaparsec of space. Or, alternatively, it could stop, slow down, and recollapse into something called a Big Crunch, which is sort of the Big Bang run backwards, and basically the whole thing turns into a massive black hole. These are–you know, this is ending in fire or in ice, I guess. And it can be computed in ways we’ll describe later.

Okay, so, here’s what I want to do. I want to start understanding how this plot and this little equation gives you all these wonderful things. I’m going to go on for a little while, then we’ll pause, and we’ll do one of these Q & A sessions, because this is sort of the heart of the Big Bang Theory. And so, we’ll do one of these things that we did when we were talking about relativity, where you talk to each other and come up with questions. So, if you’ve got questions along the way, by all means, ask them, but we will have a specific moment a little ways down the line where we actually pause and do this on purpose. So, everybody, keep thinking as we go along, what are your questions? What don’t you understand or what questions could you ask to understand more than what I’ve just told you?

Okay. Here we go. Imagine a one-dimensional Universe, just because it’s easy for me to write down. And here’s our one-dimensional Universe. It’s all strung out on a line. Here’s the line. And it’s got a bunch of galaxies on it. Let’s label these galaxies, ABCDE, and F. And these galaxies are spaced evenly, let us imagine. And we’ll give them coordinates. So this is at 0, 1, 2, and so forth. Okay?

Now, next thing we’re going to do: the Universe is going to double in size. So, we’re just going to stretch the thing. The whole thing is going to get stretched. So, here’s our Universe. And now ABCDE and F are further apart by a factor of two. AB, C, D, EF. And if A, we imagine stays at the same coordinate–if our coordinate system starts with A, this meansB is now at 2. C is 4–6, 8 and 10. Okay. And let us imagine that it takes one time unit–one year or something like that, for this doubling to take place.

Now, we’re going to ask, if you sit–if you live in galaxy A, if you live on planet A, and you observe the distance and velocity of all these other galaxies, what’s it going to look like? So, observer on A. And so, we’re going to observe a particular galaxy–one of these other galaxies. We’re going to write down the distance. We’ll choose the distance at the start, because it’s going to change. Then we’re going to evaluate how that galaxy has moved, and then the distance changed. And then, over here, we’re going to get the velocity. The velocity is going to be the change in distance divided by the change in time, which we’ve defined to be one time unit. Okay?

So, galaxy B. Galaxy B starts at a distance of 1 away from us, because it starts 1. A starts at 0. It moved from 1 to 2, and that gives it a change in distance of 1 and therefore its velocity over this time, it’s changed in distance by 1, it’s taken 1 time unit. The velocity is 1 divided by 1, equals 1. Okay? The algebra is easy. Okay, so, how about CC starts 2 away. Its motion–it goes from 2 away to 4 away, and so, the change in its distance is 2. And since it takes 1 time unit to do, its velocity is 2/1, which is equal to 2. See, the algebra is simple, but I screw it up. Bethany is laughing at me and well she might, but I caught myself. All right.

And so on down the line. I could repeat this simple exercise. DEF start at 3, 4, and 5. Goes from 3 to 6, from 4 to 8, from 5 to 10, and their velocities are 3, 4, and 5, respectively. And so, if I plot distance versus velocity, I’ll get points lined up just like this. And so, basically, what happens is this. If you take a set of points on a coordinate system and you simply stretch the coordinate system, what happens is that the further away you start, the greater the stretch is. And so, there is a correlation between how far away you start and how fast the thing recedes from you.

Now, this is true regardless of which point you sit on. Let’s imagine that we sit on point–that the observer is on point E. So, on point E, we’re now going to–which galaxy, initial distance, motion, velocity. So, this is the same plot–chart as before. E–let’s look at F, starts at a distance of 1 because it’s 1 unit away.

The motion–okay, at the start it’s–let’s take a quick look at how this is set up here. Yes, E is at 4. So, at the start, the distance is between 4 and 5, which is 1. And it goes to a distance of 8, which is where E ends up, and 10, which is whereF ends up. So that is a change of 2. No a change of 1, right? Because it’s gone from 1 away to 2 away, and the velocity is 1. If we look at D, it starts at a distance of 1, goes–starts at 4 to 3, that’s one separation. Goes to 8 to 6, that’s a separation of 2. Delta D is again equal to 1. Velocity is equal to 1.

Let’s look, for example, at B. That starts 3 away. And it starts from 4–BC, yeah–from 4 to 1. And it goes from 8 to 2. So, that’s a difference of 6 here, a difference 3 there. And so delta D is equal to 3. And so, the velocity is once again 3. And so, you get the exact same plot with different galaxies, because you’re sitting in a different place. So, it doesn’t matter which galaxy you’re sitting on. You see the exact same ratio of distance to velocity and you create the exact same Hubble flow no matter which galaxy you sit on in this little toy Universe.

So, that’s the key point–that if you take a coordinate system and you expand it, you naturally get this relationship between distance and velocity. Or to turn it around, if you observe this relationship between distance and velocity, then what you’re looking at is a system in which all the coordinates are–in which you’ve simply stretched the coordinate system.

Okay. Now, this gives rise to–this analogy, with these stretching one-dimensional lines gives rise to two questions people have, which I like to call un-questions, because they’re actually questions that arise because of the analogy, not because of the way the Universe works.

One question is, Q1: “Where is the center?” You know, here’s your line. It’s expanding. But somewhere in the middle here, around C or D in our thing, is the center away from which everything is expanding. So that’s one question. And the second question is, “What is it expanding into?” You know, here you have a little Universe and it’s moving outwards. And so, what’s going on over here? What was there before the Universe moved into it? These kinds of questions. And those kinds of questions come about because this is actually a bad analogy–this straight line Universe.

So, let me give you a slightly better one. We’ll stick with the one-dimensional Universe, but now we’ll do it this way. Here’s a one-dimensional Universe. You have to stay on the line. So here’s A, B, C, D, E, F, whatever. And it’s going to expand. And it’s going to expand into something that looks like this: A, B, C, D, E, F. And all of what we just did about the velocity, and so forth, remains the same.

But, notice that this system is unbounded. There’s no edge. There’s no edge. There’s no place where you can say, this is the end of the Universe, because if you traveled around it you’d just come back to where you were, and therefore, there’s also no center. And where does it expand into? It expands into a dimension that, if you’re a one-dimensional creature, you can’t experience because the whole thing is being pushed out. But if you’re forced to live on this circle you can’t even–you have no comprehension of what it expands into. It expands into a higher dimension. But all of this stuff about, you know, velocity and distance, remains basically the same.

Here’s a two-dimensional analogy. Let’s see, this is–so, I made a little diagram of the Old Campus [an area of the Yale campus]. Here’s Linsley‒Chittenden [a classroom building] where we’re sitting right now, yes? Here’s the statue of Abraham Pierson. This is the gate between Durfee and Wright [two undergraduate dorm buildings]. Here’s Phelps Gate [a classroom building and the entrance to Old Campus]. Here’s Vanderbilt [a dorm building]. This is Harkness Tower and here’s Starbucks. Okay, that’s all that’s important, right? So, you following me with that?

And what I did was, I took this little picture and I took it to the Xerox machine and I blew it up by 20%. So, here’s the exact same diagram blown up by 20%. So, now, supposing we’re sitting in Linsley-Chittenden, which we happen to be doing, and the Universe expanded by 20%–or, our little corner of the Universe expanded by 20%, here’s what would happen.

Now, notice what’s happening. Every object in the Universe is moving away from us. See? Here’s where Harkness was, and now it’s moved a little further, in a straight line away from us. Here is Pierson, and he’s moved a little further, straight away from us. And here’s Phelps, and it’s moved a little further, straight away from us.

And let me erase those lines, because what I want to demonstrate is that if you’re anywhere else in this Universe, the exact same thing happens. Here we’re now sitting on that statue, and Linsley-Chittenden is moving away from us. Harkness is moving away from us. Phelps is moving away from. Starbucks is moving away from us, and so forth. Similarly, if you’re sitting in Starbucks, waiting for students to come by or something, the exact same thing happens.

And now, because the distances are greater, you can see the effect that the velocity is greater at greater distances. If I’m looking down at Vanderbilt, it moves straight away from me, but only a little bit. If I’m looking all the way across Old Campus, this gate moves a lot away from me. And so, once again, you have a situation in which the further away some–everything is moving straight away from you, but the further away it is, the faster it’s moving away, right? And that’s just a consequence of the fact that you have taken this geometry and expanded it. And so, wherever you sit in an expanding geometry, every object you see will be moving directly away from you. And the further away it is, the faster it will be moving, which is Hubble’s law.

Oh, and one other thing about this nice analogy, here. Let us imagine for a second that this tiny piece of a tiny Universe is actually not a flat plane, but is sitting on a curved surface, which is curved all the way round into a big ball. That’s actually not so hard to imagine because it’s true. This sits on the surface of the Earth. And so, what is happening when this thing blows up by 20% is, basically, somebody has taken a valve to the Earth and has blown the Earth up by a factor of 20%. And that would have this effect. And it would have the exact same effect everywhere else on the surface of the Earth.

And the Earth, you know–where is the center of the surface of the Earth? You can answer the question: “Where is the center of the Earth?” But you can’t answer the question of where is the center of the surface of the Earth. Because wherever you sit, whether you’re sitting at Starbucks or in Phelps Gate or, you know, in Los Angeles somewhere or wherever, if they blow the Earth up by 20% you’re going to see this exact same effect. Everything will be moving away from you. The further away something is the faster it will be moving. So that’s the one–yes, go ahead.

Student: Someone in the back apparently [inaudible]

Professor Charles Bailyn: Yeah, talk.

Student: If something’s far enough away from you would it appear to be moving at the speed of light?

Professor Charles Bailyn: Yes, yes, good question. If something’s far away from you, will it appear to be moving at the speed of light? Yes, it will, and that’s one of the fundamental differences between the motion of an object due to what’s called the Hubble Flow, due to the expansion, and ordinary motion of objects. Now, if something’s moving faster than the speed of light, of course, you can’t see it, because the light from that is redshifted down to greater than infinite wavelengths. So, the photons don’t have any energy left. But, let me come back to the question after I do one more thing. That’s a good question.

Chapter 3. Conceptualizing a Three-Dimensional Universe [00:26:36]

Here’s the thing I want to do. So, we’ve had the one-dimensional case, the circle. We’ve had the two-dimensional case, the expanding sphere. Of course, what we want is the three-dimensional space. Okay, here we are in three dimensions. Someone is expanding the Universe, so everywhere we look, everything is going away from us, and the further away it is, the faster it’s going.

What’s it expanding into? Well, that, we have a little more trouble visualizing, right? Because in one dimension, you can visualize this circle expanding onto the plane. In two dimensions, you can imagine this spherical surface expanding. In three dimensions, we can’t imagine what it’s expanding into. That’s beyond us.

And so, having had this failure of the imagination, what do you do? You resort to mathematics. That’s what we always do. And so, imagine that every object has a position, which is denoted by three coordinates, three spatial coordinates x,y, and z. But now, let’s imagine that every object’s position has this coordinate system times a scale factor, which is a function of T. So, it’s a scale factor times a coordinate position.

And there are two ways that things can change their position. One is, they can move; they can change their x, y, zposition. This is the equivalent of somebody walking across the Old Campus. You walk from Starbucks to Phelps Gate, or something like that, and you change your x, y, z coordinate position by moving through space. So, changes in position, which is to say, velocity, can be accomplished in two ways. One is motion through the coordinate system–that is to say, changing your x, y, and z. This is called peculiar motion. That’s a jargon. And it’s called peculiar because one object can have a different peculiar motion from every other. It’s peculiar in the old-fashioned sense meaning specific to one object. So, you have a peculiar motion that’s all your own. I have one that’s all mine as we move through x, y, and z.

But the other is just the effect of the change. And in particular, in the case of the current Universe, the increase in the scale factor. And these two kinds of velocity are conceptually different from each other. Because you don’t have to do anything to change your position in this way. You just sit there. You don’t expend any energy. You don’t have any requirement to expend energy or to exert a force or to do any of these things that we ordinarily do to change our position. You just sit there and the Universe expands you, or expands your position.

And that’s why, going back to your question, that’s why it’s possible for this kind of velocity to turn out to be greater than the speed of light, if it’s far enough away, whereas, it’s not possible here. What the effect of having this kind of velocity be faster than the speed of light does, is it makes the object impossible to see, because photons coming off them would be redshifted into oblivion. And so, you can’t actually see them. And this imposes a kind of cosmic event horizon, similar in kind to the event horizons around black holes. You can’t see events on the other side of these horizons.

All right, let me go backwards in time. Back in time to when this scale factor A of T is equal to 0. So, if the scale factor is expanding, you know, the Old Campus, whatever, is expanding, and you reverse time and you think about what happened long ago, it must have been smaller. So this A factor must have been smaller. And if you go back a sufficient amount in time, you go back to the point where A is equal to 0. So then, what does the Old Campus look like? Looks like this, right? It’s all been–imagine I take a Xerox machine and I de-magnify the thing down to 0% of its original size, or 0 plus epsilon, perhaps. What does that little diagram look like? It looks like a tiny little dot. A single point, except it isn’t a single point. What it is, is it’s many points superposed on each other. And not only is the Old Campus in there–imagine you’ve taken this sphere that is the Earth and collapsed its radius down to 0. Not only the Old Campus is there, but so is the rest of New Haven, so is Connecticut, so is Los Angeles, so is the whole rest of the surface of the Earth. All of the points that will eventually make the surface of the Earth are superposed on one another.

And yet, the whole geometry of the Earth, of the Old Campus, whatever, is already somehow encoded in that point. Because, you know, you’re taking 0 times xyz for every point. But then, if you increase this so that it becomes epsilon or some non-zero number, times xyz, then you already get the geometry of the Old Campus and everything else on the surface of the Earth.

So, it is wrong to think of the Big Bang as starting at a point and expanding into space. That’s kind of the impression that word the Big Bang gives you. And you think, naturally enough, of an explosion, where something at a point explodes into empty space. But that’s not right. All the space, all the empty space is contained in that point. It’s all in there. It’s just, it’s all multiplied by 0, so it all comes out to be on top of each other in the same place.

How are we doing? Wonderful. Okay, that’s the essence of the Big Bang–the whole idea of the Big Bang–that what is happening is that the whole coordinate system of the Universe is multiplied by this constant. And that constant changes in time. It gets bigger. And that fact is inferred from the observations of these galaxies. That you observe galaxies, and that they’re moving away from us. But more than that–that there’s a linear relationship between how far away from us they are and how fast they’re moving. That implies this kind of coordinate expansion. And if you run it in reverse, it implies an origin to the Universe at some specific time in the past.

Chapter 4. Q&A: The Big Bang, the Expansion, and the Big Crunch [00:34:22]

All right. Let’s have questions. I’ll tell you what. Talk to each other. Come up with good questions, and remember how we do this. When you come up with a question–a question can be either to explain some of this or to expand upon it. When you’re ready with a question, put your hand up. I’ll answer a few of them while other people are getting their questions together. And then we’ll answer as many of them as we have time for in the remainder of the class. So, talk to each other. Talk amongst yourselves. Come up with a question, any question. All questions are good, and see what you can do. I’ll come around and try and answer some of these. Yeah ‒ so, talk to each other by all means. Yes?

Student: There is a center of the Universe, isn’t there?

Professor Charles Bailyn: No.

Student: Well, I mean, like, there’s got to be a point in the Universe at which, like, all other points–or all other points that are furthest away from that point are expanding equally fast, right?

Professor Charles Bailyn: No. Well, that’s true of every point in the Universe. That’s true of every point in the Universe. If you’re sitting on the statue of Pierson, everything at the same distance is moving away from you by the same amount. That’s true of every point.

Student: Oh, I see.

Professor Charles Bailyn: Yeah.

Student: So, wait, so, like, is it, like, in four dimensions, then?

Professor Charles Bailyn: Imagine a three-dimensional thing expanding into the fourth dimension the way the surface of a sphere expands into three.

Student: Okay, thank you.

Student: If you switched from the expansion to getting close together, would it precipitate the point at which the expansion [inaudible]

Professor Charles Bailyn: Oh we’ll talk about that later. You can imagine that gravity is going to slow the expansion down. But at the moment I’m just thinking about running backwards in time.

Student: Okay, and at some point you got to do expansion [inaudible]

Professor Charles Bailyn: Yeah, or could. Yeah?

Student: I don’t think–so have you had like one of those fake event horizons where something could be like–appear to be moving faster than the speed of light? Could you then like–in like a normal event horizon you can’t pass information between them [inaudible]

Professor Charles Bailyn: Right.

Student: But could you, like relay information?

Professor Charles Bailyn: Yes, but by the time you had done that–that is to say, the Universe is expanding faster than the information would get to you. So, you know, by the time you’ve moved your information from here to here, the distance from here to here has expanded. And so, it’s not actually getting any closer.

Okay, let’s have a few of these. Yes, go ahead.

Student: Well I have–first, just simply, I didn’t understand why it’s 0 times the coordinate, why there’s a point at all and why there’s just nothing. Why there’s not [inaudible]

Professor Charles Bailyn: Well, okay. So, if A is equal to 0 here, then all events are at 0, 0, 0. So, that point of 0, 0, 0 is occupied, if you want to think of it that way. It’s the point–the point in question is the one at 0, 0, 0. And if you have anything in here times 0, that’s where you’re going to end up. And as you go backwards and run this scale factor down to 0 as you think about going back in time, everything’s going to end up closer and closer to 0. And then, at the moment when this equals 0, it’s all piled on top of itself. Does that make sense? Yeah go ahead.

Student: Everything has to be equidistant [Inaudible]

Professor Charles Bailyn: No, no, no, no, no. Imagine–

Student: [Inaudible]

Professor Charles Bailyn: Oh, in the other dimension.

Student: Yes.

Professor Charles Bailyn: Yeah, okay. In the higher dimension, that’s true. Well, okay, let me back off. That’s true if you have constant curvature. You could imagine something that, you know, looks like this, that also expands in exactly the same way. And what creates the curvature of space-time? Gravity. And so, we’re going to end up in a situation where we’re going to be able to determine what the curvature of space is by seeing how much mass there is. So, this actually relates back to part two of the course.

Student: So, is that why time could move differently?

Professor Charles Bailyn: Huh?

Student: Time could move differently.

Professor Charles Bailyn: Time could move differently, too. Yep. Yes?

Student: Isn’t–in the one-dimensional circle and you said that thing’s expanding to a higher dimension–what does that mean?

Professor Charles Bailyn: Okay. So, imagine you’re a one-dimensional creature. You can only go on a line. But your line, unbeknownst to you, is curved, so that it ends on itself. That then–the line expands. The circle expands. What do you think is happening? You think suddenly all the distances are greater, but you can’t perceive what it’s expanding into, because that’s a second dimension, and you’re constrained to move on the line. Similarly, if you’re an ant, or something moving on the surface of the Earth, and you have no understanding that there’s up and down, and the Earth suddenly increases in size, you have no perception of the dimension you’re moving into.

Student: So then, they just keep moving into higher dimensions?

Professor Charles Bailyn: Well, our perception is of a three-dimensional Universe. So, if you say, okay, what happens when the scale factor gets bigger? You can write it down mathematically into all these calculations, and what you have to imagine, if you want to have a conceptual understanding of this, is that there’s some fourth spatial dimension into which this three-dimensional space is moving into. But we can’t perceive that. So, it’s better to do it mathematically, because it’s hard on the brain to imagine that fourth dimension.

And there’s basically two ways to think about this. One is by analogy. You think of the one-dimensional creature and what it thinks about the second dimension, or the two-dimensional creature–what it thinks about the third dimension. And then, you just, sort of, take this leap of faith into us moving into the fourth. Or, you have to write things down mathematically. Those are the only two options you have. Actually picturing this is not going to happen. Yes?

Student: If we were to speed up this process would we be able to see things getting farther away from us?

Professor Charles Bailyn: Ah, interesting question. Would we, if we cranked up this process some, would we actually, you know, be able to take out our tape measure and notice that the distance between here and Starbucks has increased?

No. Here’s why. We are not–we–that is to say, you, me, our bodies, whatever–are not participating in the expansion of the Universe at the moment. Why? Because the molecules in our body are being held together by other forces–chemical forces. We are being–the Earth is held together by gravitational forces and those forces have stopped the expansion of the Universe locally, but not globally. And that’s why you have to go out and measure this stuff with galaxies. Otherwise, you’d be able to, you know, measure it with this.

Now, of course, then, there’s the problem, the ruler also expands, all this kind of stuff. But, in fact, what happens is that locally–local objects, by which I mean, our own galaxy and anything smaller, are held together by other forces. Picture a balloon expanding, so that’s the two-dimensional case, this balloon expands. And put a bunch of leather patches on the balloon. So, those leather patches don’t expand, but the distance between–because they’re held together by something else–but the distance between them does. Yes?

Student: So, what causes the expansion?

Professor Charles Bailyn: What causes the expansion? Oh, that’s theology. No, no, seriously, it’s an initial condition. Something at the start, at T equals 0, where this A is 0. So, you’re starting with A of–with this scale factor of 0, but the derivative isn’t 0. It’s expanding, A is increasing.

Why? Why is it increasing by the particular amount it is? You have to think of that as being a parameter of the Universe. It’s one of the things about the–sort of like the speed of light. Why is that the quantity it is? Or why is the gravitational constant the value it is? It’s one of the parameters that governs our Universe. Why those particular parameters? That’s not quite a science question, and we don’t know.

You can turn it around. You can make the following interesting point. If it wasn’t expanding at approximately the rate it is, we wouldn’t be here to notice it. Because if it was expanding a whole lot slower, then it would never have gotten big enough for stars to condense out. If it was expanding a whole lot faster, all the atoms would be spread out so far that, again, stars could never form. So, you can make this interesting argument that while you don’t know why it has this particular value, it’s very important that it does, because otherwise we wouldn’t be here to observe it. This is a form of argumentation called the anthropic principle. It is highly debated among scientists whether this is a scientific argument or not. But it’s amusing either way. So–yes?

Student: So if derivative A is positive, then A should have been 0 for that, right?

Professor Charles Bailyn: Well, you go back before A and you’re in trouble. So -

Student: What’s that mean?

Professor Charles Bailyn: Okay. So, this is another one of these un-questions. It’s usually phrased as, well, what happened before the Big Bang, right? That’s the equivalent question. And, again, that’s a theological question. It’s like asking the question: “What’s going on inside an event horizon?” You can write down equations. You can talk about it. But it’s un-testable by its very nature. And so, something‒

This, by the way, is why the Catholics like the Big Bang Theory so much. In fact, the mathematics, the relativistic mathematics that describe the Big Bang and the expansion of the Universe, were worked out by a man named Lemaitre, who was a Catholic priest, who was a Jesuit. They love this because it gives you a creation moment. It kind of gives you a scientifically verified creation moment. John Paul was very enthusiastic about astrophysics. He used to throw big conferences in the Vatican, give after-dinner speeches. There’s a thing called the Vatican Astrophysical Observatory. They have a bunch of Jesuits. They run a telescope in Arizona. They do research into this.

And so, you know, if you want to see science and religion converge, you want to run away from biology as fast you can and talk to us about cosmology. And what I don’t understand is why the, sort of, fundamentalist type worry so much about biology, where all the science is dead-set against them. Whereas, in the case of astrophysics–now, of course, it is possible, if you take an atheistic point of view, to come up with all kinds of clever ways to avoid this creation event. But again, it stops being science at a certain point. It becomes another odd kind of theology, and we’ll talk about that, perhaps, a little bit. But if you want a place where science turned out to be the congruent to, at least, certain kinds of non-fundamentalist religious beliefs, you’re way better off in astrophysics than you are in biology. Yeah, go ahead.

Student: Back to the fourth dimension, is that a time dimension or a space dimension?

Professor Charles Bailyn: No, that’s a fourth spatial dimension that I’m talking about in this case.

Student: So, in other words, spatial dimension [Inaudible]

Professor Charles Bailyn: It’s expanding into a dimension which we don’t perceive. Or, at least, that’s one way of conceptualizing it. We don’t get to deal with that dimension. So, again, it’s a concept. It’s not necessarily a physical thing. Yes?

Student: Back to that one point, I was just wondering, is that like anything that will ever [Inaudible]

Professor Charles Bailyn: Right.

Student: [inaudible] and anything that will ever expand?

Professor Charles Bailyn: Yeah, yeah, yeah. It has to be. Think of all the points that now exist in the Universe. Multiply them by 0, and they’re all on top of each other. Yes?

Student: So if you were to go really, really fast in a straight line in any direction, would it eventually end at a [Inaudible]

Professor Charles Bailyn: So, this is the question of what is the curvature of the Universe. Is it, in fact, like this analogy, with the circle or the surface of the sphere. If you go forever, do you come back to where you are? That depends on what the overall curvature of the Universe turns out to be. That depends on how much mass it contains, because mass curves space. And so, that is a question that’s empirically answerable. We think the answer is no. In any case, though, it would take you longer, at the speed of light, than the age of the Universe to accomplish it. So, even if it were positively curved, in this sense that it rejoined itself, you wouldn’t actually be able to make that voyage.

Student: [Inaudible]

Professor Charles Bailyn: No.

Student: [Inaudible]

Professor Charles Bailyn: Well, you might, eventually, unless the Universe is expanding faster than you can move. This goes back to the question of can parts of it expand faster than the speed of light, and the answer is yes. So, yeah. You take off on this voyage to the back of your own head. Going in that direction, you’re going to end up here–except the Universe is expanding faster than you can go, and so, you don’t necessarily end up back there. Yes?

Student: How does the Big Crunch happen in the expansion?

Professor Charles Bailyn: Oh, one of the things that we haven’t talked about is the expansion rate constant. Might it be that the Universe is expanding, but perhaps slowing down? Why would it slow down? It’s full of mass. Gravity slows things down. And so, in principle, if you’ve got enough mass, the Universe will stop, turn around, and fall backwards. And you can compute from–we will compute, next period, how fast you have to go to make that happen. Or contrary, how dense the Universe has to be to stop it.

And so, what you expect is the Universe is slowing down. Because you know you don’t know how much it’s slowing down, whether it will stop the expansion or not, but you expect it to be slowing down.

The punch line of this whole part of the course, in order to anticipate where we’re going, is that, in fact, we can measure the change in the expansion rate, and it’s speeding up, not slowing down. This is very disturbing, because it means there’s some kind of pervasive cosmic anti-gravity, which we call dark energy, because we don’t know what it is. And that’s basically the punch line of this whole section of the course. But you could imagine that there is a change in the expansion rate, either positive or negative, and that, therefore, various different outcomes are possible. Yeah?

Student: Would a hypothetical of the Big Crunch be sort of different from Big Bang in that the Big Bang sort of expanded the whole geometry of the Universe where as the Big Crunch being motivated by gravity would only cause the object [inaudible]

Professor Charles Bailyn: No, no, no, no, no. It turns out it–you might imagine that, but you’d be wrong. It turns out that this scale factor–you can write down a differential equation for the scale factor, and that it’s the scale factor that stops, turns around and comes back. Now, it is the case that when it’s going out, the particular objects the Universe contains are different because of the peculiar motions. It starts out much more evenly spread, and then gradually stuff accumulates due to gravity, and you get–when you’re coming back, you get a lot of individual points, rather than a smooth distribution of matter. But that has to do with peculiar motions, not the overall motion of the Universe. One more and then we got to continue this next time. Yes, go ahead.

Student: Before the Big Bang, what happens when the minus–negative number did [Inaudible]

Professor Charles Bailyn: Right. So, before the Big Bang, can you put a negative number into that A? This gets you into exactly the same kind of mathematical difficulty as moving inside an event horizon. You can actually write down a metric for the whole Universe and you change all the signs when that happens. And you get into a very similar kind of problem. So, mathematically, you could compute a bunch of stuff. The physical meaning of that is not–first of all, the fact that that mathematics actually works in the real world is impossible to determine, and the physical nature of consequences of that are questionable. All right, stop.

[end of transcript]

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