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ASTR 160: Frontiers and Controversies in Astrophysics
Lecture 9
 Special and General Relativity
Overview
The discussion of black holes continues with an introduction of the concept of event horizon. A number of problems are worked out to familiarize students with mathematics related to black hole event horizons. In a longer question and answer session, Professor Bailyn discusses the more mystifying aspects of the nature of black holes and the possibility of time travel. Finally, the issues of reconciling Newton’s laws of motion with Special Relativity, and Newton’s law of gravity with General Relativity, are addressed.
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htmlFrontiers and Controversies in AstrophysicsASTR 160  Lecture 9  Special and General RelativityChapter 1. Event Horizon [00:00:00]Professor Charles Bailyn: Okay, let’s start–let’s see, we started talking about black holes last time, and there’s now going to be a problem set that is going to be available later today, due next week. And I should also point that I’ve put together this website on black holes. Actually, some of the help sheets already send you to that website. And I’ll put a link on the classes server as a whole, but here it is as an actual URL, cmi.yale.edu/bh for black holes. And that website kind of serves as a sort of online textbook for this part of the course. And so, you have something written down to look at for this whole section of the course. And, as you’ll see, many of the things we’ll be talking about are actually discussed on that website, as well. So, you can find out more information there. Okay, so I–last time, I defined a black hole. This is simply something where the escape velocity is faster than the speed of light. Or, alternatively, and this is–amounts to the exact same thing, the radius of the object is less than the Schwarzschild radius, which is defined for an object of any given mass. And this isn’t particularly extraordinary or interesting, as long as the speed of light isn’t particularly extraordinary or interesting. And one of the things that happens when you start talking about relativity is that it turns out the speed of light is a very important quantity. So this–these are interesting because the speed of light is interesting. And, we’ll talk in a minute about how all this arises, but one of the interesting things about the speed of light is that you can’t go any faster. It’s the speed limit, and no physical process can make you go faster than the speed of light. So, c is fastest velocity possible. And the consequence of that is, supposing you have one of these black hole things–so, here’s some object, and it’s got some radius. And its radius and its mass are such that the escape velocity, here, is greater than the speed of light. You could imagine that around it is a kind of imaginary sphere. Let’s put it in dots. And this is the sphere where the escape velocity is equal to the speed of light. And you remember what the formula for the escape velocity–V_{esc} is equal to 2GM / R, the square root of 2GM / R. And so, if on the surface of this object, the escape velocity is greater than that of light, if you keep moving out, R will keep getting bigger. And so, eventually you’ll come to a point in space where the escape velocity is equal to the speed of light, and this is called the event horizon. And the reason it’s called that is because, if nothing can go faster than the speed of light, what it means is that any event that takes place inside this imaginary sphere can’t radiate any information about what’s going on to the outside, because this–the escape velocity’s greater than the speed of light. Light can’t escape, and because nothing else can go faster than the speed of light, nothing else can escape. And so, no information of any kind can come–comes from inside the event horizon to the outside. And so, in a certain philosophical sense, this event horizon sort of constitutes an edge of the Universe, because nothing that happens inside can affect what happens outside in any way. You can’t see in. You can’t detect anything that goes on inside. No event inside this event horizon can tell us what’s–about itself or–and we–there’s no–in principle, there’s no way we can find out anything about what happens inside an event horizon. It turns out that this rather grandiose statement, that the event horizon constitutes an edge of the Universe, has a mathematical meaning as well. The great physicist Stephen Hawking–you may have heard about him. He’s the guy in the wheelchair. His –shorthand–his–one of his great discoveries was to demonstrate that the mathematics of the event horizon is actually very similar to the mathematics of another edge of the Universe–namely The Big Bang, which is the start of the Universe in time. And so, it turns out that this statement that this constitutes an edge of the Universe is not just a philosophical statement having to do with defining the Universe as something we might eventually, possibly, know something about, or be influenced by, but also has a mathematical meaning, as well. And so, the way you have to think about the Universe is as a kind of Swiss cheese. There are holes in it. There are holes in it, which aren’t part of our Universe, where these little event horizons cut pieces out. And it’s this kind of thing that gives black holes the kind of extraordinary aura of mystery that they sort of have in everyday life. Various strange things happen inside the event horizon. And now I better pause, because I just contradicted myself, right? Because I said a minute ago that you can’t find out anything about what happens in the event horizon, and now I’m about to tell you what happens inside an event horizon. Okay, so it–we’re ten–we’re eight minutes into class and already I’m contradicting myself. And so, let me explain exactly what I mean by this. Here’s what you do if you want to make statements about what’s going on the inside of one of these things. What you do is you take the equations that apply outside the event horizon. You assume that the same equations are going to apply inside the event horizon. Turns out, the mathematics of these things, and I’ll show you an example of this next week, the mathematics of these things do very bizarre things at the event horizon. But mathematicians are smart people. One can cope with these kinds of problems. It tends–you tend to have problems like, everything gets divided by zero right at the event horizon. But, as you may recall, if you’ve done a good precalculus course, there are ways you can get past those kinds of infinities–but everything goes haywire. But, if you make the assumption that the same set of equations apply inside the event horizon as outside, you can then manipulate those equations, interpret the results as if the symbols mean the same thing inside as they do outside, and then make statements about what happens inside the event horizon. And this looks a lot like physics, right? Because you’ve got a bunch of equations and, you know, some of them stand for time, and some of them stand for space, and energy, and all this good stuff. And you can do manipulations of those equations, and then you can make statements about what happens. But, it looks like physics–you can make an argument that it isn’t actually physics, because you’re making an assumption that those equations apply inside there, and that’s an assumption that is untestable. Even in principle, untestable, because you can’t ever get any information about what’s going on in there to see if those really are the right equations. And so, here’s something where it looks like physics but, depending on what you define the limits of physics to be–and this is kind of a deep philosophical question that we won’t go too far into–it might actually be something that, while interesting, it is really something else. So, just a couple of examples of the kinds of things that you can deduce from the mathematics about what goes on inside event horizons. One–here’s something. All matter with R less than the Schwarzschild radius–oh, this distance, here, will always be the Schwarzschild radius, by definition, because that’s the radius away from an object where the escape velocity is equal to the speed of light. So, all matter inside the Schwarzschild radius collapses to a single point in a finite amount of time; so, it doesn’t take forever to do this. And that points–since it’s got a lot of matter but no volume, that’s what a point is–that’s a point of infinite density. And this is sometimes called a singularity. And you can see some of the problems with doing mathematics inside event horizons. If you try and calculate the density for this–density is mass over volume. If you’ve got a bunch of mass in a single point, the volume is zero, and you’ve just divided by zero, and you’re in all kinds of trouble. So it–everything collapses down to a point of infinite density. There’s no force strong enough to stop this from happening. You can’t set up a rocket system to keep yourself in orbit or keep yourself from falling down. Partly that–one way of understanding that is that the energy you would exert in trying to hold yourself up has a mass equivalent. You’ll recall that E = mc^{2}, so for a given amount of energy there’s a mass equivalent. The extra mass just pulls you down more. So you can’t–by expending energy, you can’t hold yourself up inside a singularity, so everything collapses. It’s also true–this is the kind of thing you see in popular accounts of black holes. The statement is sometimes made: the properties of space and time are reversed. That’s a weird thing to say. What could this possibly mean? Well, let’s see. It’s related actually to what I just said, that everything falls down to a singularity. In an ordinary situation, we can move around in space. I can expend energy and move all around in space. I can move back and forth. I could jump up and down, see–but you don’t have a lot of choice of where you go in time. You move forward in time at the rate of one day per day, right? And you can expend a lot of energy trying to avoid that, and it won’t actually do you any good. Chapter 2. Singularity and Clarifications on Black Holes [00:11:47]So, inside an event horizon you fall down to a singularity. And, as I said, you can expend a lot of energy but it doesn’t change your trajectory in space. So, what happens when you fire off your rocket? Well, in principle, providing you believe in all these equations, you could, in principle, move around in time. And, this is why the science fiction writers like black holes so much, because it’s a kind of scientificallycertified time machine. You go into the event horizon. You turn on your energy pack and move all around in time. And then, what they forget to tell you is that you’re inside an event horizon, so it does you no good, because you can’t come out again. And so, then, the effort is to try and figure a way that you could have this nice effect of the time machine without having an event horizon. And, in particular, there’s some interest in whether one can have what’s been described as a naked singularity. A naked singularity is a singularity with no event horizon around it, so that you could get in down to where all the weird physics happens, and then come back out again. Hawking proposed what he called the Cosmic Censorship Theorem. That is the statement that naked singularities are not allowed. And so, hence, censorship can’t have naked singularities. And there is, to this day, some dispute over whether that will turn out to be true, in general, or not. And you can go on in this vein. You can talk–you can talk on and on about the extraordinary physical events and properties that these things might actually have. And what you have to keep in mind, as you’re doing this, is that there’s a limit to how much this could ever, even in principle, be realized in real life. You know, you fall into one of these wormholes that they like to have on the science fiction–in the science fiction stories, and you’d better have some very tricky physics if you’re going to be able to come back out again. And that’s not something that the science fiction world typically dwells on. Although, it is interesting–you know the movie Contact? So, this is based on Carl Sagan’s novel of the same name. And, one of the things that happens late in this–this is the first use, I think, in science fiction, of the concept of a worm hole, where you fall into a black hole and you fall out again somewhere totally different in space and time. And there’s an interesting story about this–that what happened, supposedly, was that Sagan, when he was writing his novel, got to the point in the novel and he needed something to happen for plot reasons, in order to move the plot forward. And he–Sagan himself was not an expert in relativity, so he called up one of his friends, who was a guy named Kip Thorne, one of the most famous general relativity people. He said, look, here’s what has to happen in my novel. Give me some science jargon so that–that will make this work. And Thorne, who was a very smart guy, thought about this for a while, and came up with this concept of the wormhole that, you know, you fall into a black hole here and you fall out of it again on the other side. Then, Thorne went away and thought about this for a while. At least, this is what he claims in his book, and said, you know, this might work. And a whole series of scientific papers flowed from that. So, here is a place where art pushed science. Art in the form of science fiction, if you consider that art, pushes science–pushes science forward, rather than the other way around–quite unusual, in that case. All right, so, all of this is great. But I want to focus a little bit on what does it actually mean? These grand statements that one makes about, you can’t go faster than the speed of light, space and time are reversed, and so forth. What actually underlies those statements? And to what extent are these sort of spectacular kinds of phenomena actually realized in the real world? So, I’m an observational astronomer. What right do I even have to talk about black holes, the number one characteristic of which is that you can’t see them at all? So, that’s where we’re going to go in a minute, and to do that, we have to talk about relativity–special relativity and general relativity–because those are the theories that underlie all of this. So, before we go plunging into that let me pause for questions at this point. Yes? Student: [Inaudible] Professor Charles Bailyn: Yes, you should be. Yes, so the–a little confused about why space and time are reversed. Let me not answer that now, because there’s, there’s math next week, which will make that a little bit clearer. I think it is a legitimate question, not just to ask why are space and time reversed, but what that even could mean. And we’ll get back to that in a little while. Let me–yes, go ahead. Student: [Inaudible] Professor Charles Bailyn: Right, and doesn’t–so, supposing you have, for example, a neutron star that’s a little bit smaller than its event horizon, why wouldn’t it just stay there inside its event horizon, rather than collapsing down? And again, this is a calculation that I don’t think I can demonstrate for you, but it comes about because the energy, the pressure forces that would have to hold it up, actually also contribute to the mass of the thing. And if you then compute the balance, the hydrostatic equilibrium that we were talking about last time, it never works. It can’t work, because the amount of energy required to hold it up is so great that the mass equivalent is going to generate additional gravitational forces. And if this were a different kind of course, I’d do the calculation, but let’s not go there. Yes? Student: Time originally goes forward, right? Professor Charles Bailyn: Yeah, I would say. No, no that’s fine–yeah. Student: It can go backward? Professor Charles Bailyn: Well, the thing is, in our normal situation, we have no choice about where we go in time. And the statement that I made, which I didn’t back up in any way to–that a rational person would really want to believe me–but let me make this statement again. Inside the event horizon, you can move around in time. You don’t necessarily have to go forward. And that is the thing–that is the piece of the picture that the science fiction writers tend to hone in on. And so, you know, the statement is made about relativity. And I’ll make this explicit in a little while. Time is the fourth dimension. There are three space dimensions, one time dimension, and you can do all sorts of calculations that way. But it’s perfectly clear, just from everyday life, that the time dimension is not the same as the space dimensions, because we don’t have the opportunity to move around in them. So, it must be true that in a certain sense, time–the time dimension, if you want to call it that, of the spacetime continuum, must somehow behave differently from the spatial components. And they do, and I’ll show you why in a little while. And that behavior is reversed between space and time inside the event horizon. Things that are positive terms in the equation become negative, and vice versa. And as I say, we’ll actually–I’ll actually write some of this down. Yeah? Student: Since, like, nothing could escape–can escape from a black hole, is one of, like, the possible, you know, scenarios for the end of the Universe, be–if –all matter being in the form of black holes? Professor Charles Bailyn: All–okay, so since nothing can escape from a black hole, is that a possible end of the Universe? Yes. One of the potential ways the Universe could evolve–the Universe is expanding right now. Again, this is now in the third part of the course, we’ll talk about this. The Universe is expanding, but it’s got a lot of matter in it. So, in principle, it could stop expanding and collapse. You could ask the question, “Does the Universe expand faster than its own escape velocity, or not?” And this is a question that cannot only be asked, but answered. It could expand, slow down, stop, fall back together again. And, if that happened at a certain point when it fell together, the whole Universe would be inside its event horizon. Now, how that would affect those of us who are inside is a little less clear. Eventually, you’d get crushed, right? But–because nothing would be able to stop the inexorable collapse down to a singularity at that point. But yeah, that is a potential end of the Universe, is– we all end up inside a big event horizon. We all collapse down to a singularity in the middle. It turns out, that isn’t what’s going to happen–at least, not from current observations. The current observation is that the Universe is expanding faster than its own escape velocity. So, you have the opposite ending point where it just keeps getting bigger and bigger, and more and more diffuse and sparse, and cold and boring, and the Universe ends that way. But again, that’s part three of the course. We’ll get there. Yes? Student: [Inaudible] Professor Charles Bailyn: Where does all the mass go when it collapses? The mass is still there. The mass is still there. And this is actually a very important point, because things can still be in orbit around a black hole and not fall in. Because, when you’re far enough away, when you’re outside the event horizon, it just behaves like any other mass. Black holes can move through space. They can orbit around other things. Things can orbit around them. If the Sun turned into a black hole right now, well, we would freeze. But aside from that, the orbit of the Earth–if the Sun turned into a black hole of the same mass as the Earth, the orbit of the Earth wouldn’t change, because we’re far enough away that it just seems like any other mass down there. So, a solar mass black hole could have a planet in exactly the same orbit as the Earth is around the Sun. This is one of the differences between how black holes actually behave and the metaphorical use of black holes. You know, this is one of those words that gets used in common culture for various things, and it usually carries a connotation of a kind of great sucking thing that pulls everything in, inexorably. That’s true inside the event horizon, which is where this idea arises from. But, if you’re outside the event horizon, you can just keep going around in circles around it, in exactly the same way that you can be in orbit around any other kind of mass. Student: [Inaudible] Professor Charles Bailyn: Why is there no volume? Ah, okay. So, why is there no volume with this stuff at the center? If there’s nothing to–think about what we were talking about, hydrostatic equilibrium, the other day. If there’s nothing to stop the collapse, every–if there’s no energy or pressure force sufficient to stop the collapse, everything has to end up in the same place. It all falls to the center. And, you can calculate how long that will take. And after that happens, all the mass is down at the center. What does it mean to be the center? It’s the central point. Now, in fact, there’s got to be–this is how it works in relativity. Relativity does not incorporate quantum mechanics. Quantum mechanics does funny things when things get small. And so, it is likely that there are physical effects that we don’t understand at the quantum level before you get to precisely the same point–but we don’t know what they are, because that would require a theory that encompasses special–general relativity and quantum mechanics at the same time. And until the string theorists finish their work, we don’t have such a theory. So, probably something happens to prevent a pure point, but we don’t know what it is. And so, from the point of view on the outside, you just imagine everything falling down toward a central point, and kind of landing there in some way. But it is a mystery. Yes? Student: [Inaudible] Professor Charles Bailyn: Is the Universe finite? Yes. It’s got to–in fact, you know it has to be finite, because the sky is dark at night. Let me explain that. If the Universe were infinite and had stars distributed at random throughout the Universe, then any direction you looked at, you’d eventually see the surface of a star. You could do this in a calculus kind of way by integrating over the total light of the Universe. And so, it can’t be infinite in extent, and filled with stars–the same as stars like the Sun, or any kind of star, really, that is distributed evenly. So, one of several things must be the case. Either the Universe is inhomogeneous–that is to say, we live in a very special place where there are stars–after a while there are no stars anymore. Or, it has to be finite in extent. It only goes out a certain distance and stops. Or, it has to be finite in time. That is to say, it began a certain number of years ago, and didn’t exist before that. That helps you, because then you only–if it’s–if the Universe is 13 billion years old, you only see things that are within 13 billion light years, because the light hasn’t had a chance to travel. It turns out that it’s pro–the easiest way to understand this is, it’s the latter. The Universe had a beginning. And this is the whole idea of The Big Bang–that it started as some kind of very small object, and then expanded at some finite and indeed calculable time ago. The Universe is now thought to be, I think it’s 13.7, plus or minus 0.4 billion years old. And so, the finiteness of the Universe is–at the moment, is–the primary reason it’s finite is that it’s finite in time, not so much that it’s finite in space. But, it has to be limited in some way, or you’d see an infinite amount of starlight. Student: [Inaudible] Professor Charles Bailyn: Excellent. If the Universe is finite, what’s outside the Universe? That is exactly the right question. I can’t answer that, because–although I will try in section three of the course. We’ll learn much more about this stuff. I can’t answer that because, you know, what is the Universe? The Universe is what we can study, scientifically. And so, the very first question that a sensible person has about The Big Bang and the finite Universe is what’s outside it. And that’s actually not a science question, because it’s intrinsically unanswerable by scientific means. Now, there are things you can say about it, just as there are things you can say about the inside of an event horizon. And you can base them on scientific ideas and scientific thinking, and you can say these nice things about space and time. And one could say similar things about what’s outside the Universe and, you know, how it might have come to be and so forth. But you’ve done a little epistemological shift when you start talking about those kinds of things, because you’re no longer talking about things that, even in principle, can be verified by observation or experiment. So, I do have some things to say about that. I’ll say it in the third part of the course when we talk about the Universe. But you should think about what I have to say about that in a different way from what you–from what you think about what I have to say about how planets orbit around the Sun. We’ll come back to this later. Yes, go ahead. Chapter 3. Locating Black Holes [00:27:46]Student: I was just curious about the black holes. How do they find them? Professor Charles Bailyn: How do they find them? That’s the next three weeks of the course. That–and, in fact, that is subject of my research. This is what I do for a living. I find black holes, and I’ll tell you how it’s done. We can see if we can find another one over the next few weeks or so. Yes? Student: [Inaudible] Professor Charles Bailyn: Ah, is it possible that we are inside an event horizon and we don’t know it? Yes, in that–again, this relates to the question that was asked over here. Could it be that the Universe is going to ultimately collapse and be inside a black hole? Supposing that happened. Supposing we were in the collapsing phase of the Universe, gradually, over many billions of years, falling down to some singularity; we would not necessarily know that. Now, if you approach a black hole that is of the size of a star, or even the size of a galaxy–when you got close, you’d feel it. And the way you’d feel it is, is if you’re falling in, your feet would be closer to the center than your head. And the gravitational force on your feet would be stronger than that on your head, and you’d stretch. And so, if we were really near to a black hole of a size of objects we understand, then we’d know that, because the tidal force is on our bodies would mean that we couldn’t keep ourselves together. But, you could imagine a sufficiently big black hole, kind of encompassing the whole Universe, that wouldn’t affect us in that way. So, it’s not inconceivable. Yeah? Student: [Inaudible] Professor Charles Bailyn: Yeah, so the word “reverse” in this “reverse space and time”–how about this: I agree that might–that wording might not be–I’ve seen this wording used, which is why I used it. But I agree that it might not be the best wording one could use. How about, the properties of space and the properties of time are interchanged with each other? That’s–then it doesn’t have this kind of directional thing. It’s about how these things behave. Yeah? Student: Do we get–does the Earth get any heat from other stars than the Sun? Professor Charles Bailyn: Yes. Well, of course, we do. If you look out in the sky, you see stars. We see the radiation from those stars. And so, of course, there’s radiation in all forms, heat and otherwise. The fraction of the radiation that we get from sources other than the Sun is extremely small. The– Student: [Inaudible] Professor Charles Bailyn: No. No, because–and here’s why: because, the second most prominent source of heat that we have is actually geothermal heat from the center of the Earth. And that’s vastly more than the sum of all contributions by stars. And it’s vastly less than what we get from the–from our own Sun. And so, you know, radioactivity and the latent heat in the core of the Earth, that’s, by far, more than anything that comes to us from the stars. But, we certainly do get energy radiation from the stars. If we didn’t, astronomers would be out of business, because this is how we–this is how we study them, by looking at the radiation. Yeah? Student: Okay, so let’s say that you’re inside the event horizon of a black hole and–as I remember, the circular velocity of any object is less than the escape velocity? Professor Charles Bailyn: Uhhuh. Student: So couldn’t you theoretically orbit the black hole inside the event horizon after the circular velocity? Professor Charles Bailyn: Ah, excellent question. Could you orbit–could you get yourself into a stable orbit inside an event horizon? If everything were in–worked the way Newtonian mechanics says it works, you could. But, it turns out, there are relativistic corrections. There is something called the smallest circular–smallest–the Innermost Circular Stable Orbit. IC–ISCO, Innermost Stable Circular Orbit. That’s important, because that actually can be observed in various ways, and we’ll talk about this in two weeks’ time. The Innermost Stable Circular Orbit turns out to be outside the event horizon. But, you’re right, if you were only using Newtonian calculations, there would be that square root of 2, and you could imagine being in a circular orbit inside the event horizon. But that’s one of the things that is different about the relativistic calculation. It turns out that for nonrotating black holes the Innermost Stable Circular Orbit is three times the Schwarzschild radius. Yes? Student: [Inaudible] Professor Charles Bailyn: Ah, yes. So, is there a potential for a black hole to accumulate an infinite amount of mass, which is the sort of vacuum cleaner scenario? First of all, there isn’t an infinite amount of mass in the Universe, necessarily. The Universe is finite. Second of all, even if there was, if there was a huge black hole here, and something else outside the event horizon that’s moving faster than the escape velocity of this thing, at that point, then it doesn’t fall in. So, it’s not necessarily true that it will pile up an infinite amount of mass. In fact, black holes–black holes ought not to have infinite mass. They ought to have a particular amount of mass. They’ll gain more as stuff falls, but if stuff doesn’t fall in, if it’s moving faster than the escape velocity, then they won’t accumulate it. Student: [Inaudible] Professor Charles Bailyn: Well it depends what you have–we observed extremely massive holes. It depends what you mean by “extremely massive.” The mass–most massive black holes that we know anything about are about three billion times the mass of the Sun. So, that’s pretty massive, but it’s about the same as an ordinary galaxy. And we–what we don’t have evidence for is black holes that are substantially more massive than whole galaxies of stars. We don’t–there’s no evidence for that. So they’re, basically–just to jump ahead, they’re basically two kinds of black holes that we have empirical evidence for. One are black holes the–about ten times the mass of the Sun. So, masses of individual stars. Then there are the socalled supermassive black holes, which are the masses of whole galaxies, and those are the kinds of black holes for which we have direct evidence at the moment. Yes? Student: [Inaudible] Professor Charles Bailyn: Okay, so, is it possible to detect the presence of a black hole by looking at things around it? Yes. That’s exactly what is done, and we’ll–I’ll describe this in the next couple of weeks. But, it doesn’t happen the way you suggest. It’s not like something suddenly disappears, so we know it fell into the event horizon. What you do is you see these things in stable orbits around–that are perfectly happy going around in circles, around the black hole. For example, you can see binary stars, double stars, in which there’s a black hole and something else. And these things are in nice stable orbits, and you infer the properties of one object by looking at the motion of the other object. That’s exactly what we did for the planets. And so, you can do that again for black holes. Student: [Inaudible] Professor Charles Bailyn: If you actually saw something disappearing? Yeah, and in fact there is–there is indirect evidence that there’s material falling through–that in some systems there is material currently falling through the event horizon of black holes. And, in fact, that’s one of the things that is a current topic of active research, is, can you demonstrate that? If there were a black hole in our own Solar System, anywhere near us, we’d know it. And the reason we’d know it is, that’s a lot of mass. It would disrupt the orbits of the planets. And as long as the planets–and so, we have a kind of early warning system. If a black hole cruises by, Pluto’s orbit gets disturbed first. And so, all of those nice objects out there in the Kuiper Belt, their orbits would go all haywire if something as massive as a stellar mass black hole were accidentally to come nearby, in exactly the same way they would be as if an ordinary star of the same mass were to come by. You wouldn’t be able to see it, but the orbits of everything would be changed rather drastically. So, the odds of us actually falling into a black hole are extremely small, and if it were to happen, we’d get a great deal of advance notice. So, not to worry. There are astronomical things you maybe should worry about, asteroids crashing into the Earth, this isn’t one of them. Actually, Bethany Cobb and I were on a TV show that explored this particular possibility. This is a wonderful thing. It happened last summer, where ABC News decided to have a twohour special on all the different ways the Earth could be destroyed. And they called us up, and we went down to New York. It was very entertaining, and you know–but, I was too much of a scientist. I kept saying it isn’t going to happen. You shouldn’t worry about it. You know, you should worry about the asteroids and global warming, and all this kind of stuff. Don’t be worrying about black holes coming rolling in. And, of course, the consequence of that is that I didn’t get a lot of screen time. And the guys from the American Museum of Natural History, who were better at popularizing science than I am, got all the screen time. So, you know, I know better now. We’ll speak in better sound bites next time around. Yeah, that was fun. But we should screen it, you know, or at least the first part. Other questions? All will become clear in due time. Actually, that’s not true. But some things will become clear in real time. Gosh, this was great, excellent. All sorts of questions. Chapter 4. Introduction to Special and General Relativity [00:37:51]So now, let me at least start talking about special relativity. There are two kinds of relativity. There’s special relativity–and special relativity, which was devised in 1905 by Einstein–this is essentially a set of laws of motion. So, it’s the same thing as Newton’s Laws of Motion, only it’s a more elaborate version of it. So it substitutes, if you will, for Newton’s Laws of Motion, and in fact, incorporates Newton’s Laws of Motion. Then there’s general relativity, discovered about ten years later, again by Einstein. And this is the equivalent of a law of gravity. You’ll remember that the way one has to calculate orbits in the Newtonian Theory is, you take Newton’s Law of Gravity and you plug it into Newton’s Laws of Motion. So, this is a change, or an enhancement, in the laws of motion. This is a change or an enhancement in the law of gravity. In both of these cases ‒ and this is important to realize ‒ it doesn’t overturn what Newton did. This is a misconception about how scientific revolutions work. This is as revolutionary a set of ideas as you could possibly imagine. And yet, Newton’s laws are–still hold. They don’t overturn them. What they do is, they say, Newton’s laws are not general laws, but laws that only operate in certain specific situations. These are now the general laws that operate always, and Newton’s calculations are a subset of these things that only work under certain conditions. And just to jump ahead a little bit–the condition is, there’s a quantity–so, now I’m going to talk, first, about special relativity. We’ll leave gravity until later. But now, let’s just talk about how things move. And this is important, because this gives rise to the limit on–to the limit on velocity. All velocities must be left less than or equal to the speed of light. You can’t go faster than the speed of light. And just to jump ahead, the way Newton’s Laws of Motion relate to special relativity is through a parameter, a quantity called gamma [γ]. And this is equal to this quantity. 1  V^{2} / c^{2}, the square root of that, in the denominator. Okay. 1 / (1  V^{2} / c^{2})^{½} And when V is much, much less than the speed of light, then gamma is equal to 1. You can see. If this–if V is much less than the speed of light, this quantity is more or less zero. You got 1 / 1. Gamma = 1. Newtonian physics assumes that gamma is equal to 1. So, this leads directly to all of Newtonian physics. But, as velocities start to get higher, gamma starts to be different from 1. And if–as V approaches the speed of light, what’s going to happen? This is going–if V is the speed of light, V^{2} / c^{2} = 1. You’ve got 1  1 in the denominator, so you’ve got zero in the denominator. So, as V approaches the speed of light, gamma approaches infinity. And then you get–so, you start to get these relativistic effects. So, it turns out, for example, in relativistic physics, that mass isn’t exactly what Newton thought mass was. Mass is what’s called the rest mass; this is the Newtonian mass–times gamma. So, if everything’s moving slowly, and gamma = 1, then the mass is exactly what Newton thought it was going to be. It’s a property of objects. It’s constant. It doesn’t change. Mass is conserved and you can do all the wonderful calculations that you can do in Newtonian gravity. Now, the properties of gamma are such–remember this goes to infinity–that, therefore, as the velocity goes to the speed of light, the mass approaches infinity, because gamma approaches infinity. And you’re multiplying the Newtonian mass by some number that’s getting larger and larger and larger and larger as you get closer to the speed of light. Now, what does this mean? Remember Newton’s Laws of Motion. Let’s go all the way back to Newton. This is–force is equal to mass times acceleration. Yeah, go ahead. Student: What does gamma stand for? Professor Charles Bailyn: Gamma is a parameter that tells you how relativistic you are. It is defined by this. So, it’s a function of your velocity. Yeah. And whether–and how–how much bigger than 1 it is tells you how far away from the Newtonian Theory relativity is. And, the faster you go–the closer you get to the speed of light, the more relativistic you become and the less Newtonian you are. So, it’s a parameter that tells you how close you are to Newton. Other questions, yes? Student: How exactly is gamma derived? [inaudible] Professor Charles Bailyn: How was gamma derived? I’ll talk about the history of this next time, but basically it was–Einstein revised the Laws of Motion in response to some experiments, which weren’t giving the right Newtonian answer. That’s the short version, and we’ll do the long version next time. And, it was the genius of Einstein that he recognized how it–how the Laws of Motion should be modified, and indeed that they should be modified at all, which was not most other people’s initial reaction to these experiments. Okay, so now, what happens if your mass becomes infinite? Right? That means for a–remember this is acceleration, and this is force. If your mass becomes infinite then F / M, which is equal to the acceleration, this goes to zero, which means you cannot have any acceleration, no matter how much force you apply. And that’s why you can’t go faster than the speed of light. Right? Because as you get closer and closer to the speed of light, this gamma factor becomes larger and larger. That makes your mass larger and larger. And then, to get a given amount of acceleration, since this is huge, you have to have an increasing amount of force. By the time you get to the speed of light the mass is–sorry, it’s this whole thing that’s zero, not the mass. By the time you get to the speed of light, you need an infinite amount of force to accelerate yourself. So, you need an infinite amount of force in order to go faster. And so, by the time you get out to the speed of light, you can’t go faster, because it would take an infinite amount of force to do so. Yes? Student: Does that mean that when you’re traveling the speed of light, not only can you not speed up, but you also can’t slow down, since it would take force to slow you down? Professor Charles Bailyn: That’s right, and what it really means, though, is that nothing with a nonzero mass can actually ever get to the speed of light. And therefore, what about light? Light has zero mass. It has to, otherwise it can’t go at the speed of light. So if M_{zero} is zero, then this whole thing applies in a very different way. And so anything that goes at the speed of light has to be massless, for just this kind of reason. Yeah? Student: If light is energy then doesn’t it have a mass equivalent? Professor Charles Bailyn: It has a mass equivalent, but its rest mass is zero. So, the reason it has energy is because you’ve got zero times infinity. And so, if you slow light down, then it disappears altogether. Right? Because–if you slow light in a vacuum down, I should specify, it disappears altogether, because if this isn’t infinite and this is zero, then you’ve got no mass energy at all. Yes? Student: If you can’t accelerate or decelerate, how do you get to your sort of intrinsic velocity? Professor Charles Bailyn: Well, what is an intrinsic velocity? This is where relativity becomes relative. Depends on how fast the person who’s observing you is going. But remember, this only applies if you’re already moving at the speed of light. If you’re not moving at the speed of light, then this isn’t infinite. You have some finite mass and you can change your velocity. Remember, acceleration is a change in velocity by applying force. And as long as the mass isn’t infinite, it just–you know, you calculate how much force it takes. And that’s how much it takes to change your velocity. Other questions? Excellent questions all around. This is not at all straightforward and we will continue next time. [end of transcript] Back to Top 
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