# ASTR 160: Frontiers and Controversies in Astrophysics

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# Frontiers and Controversies in Astrophysics

## ASTR 160 - Lecture 10 - Tests of Relativity

### Chapter 1. Velocity and Mass in Special Relativity [00:00:00]

Professor Charles Bailyn: Okay. The subject is special relativity. And right at the end of last class, I had written down this factor, γ. And γ is the key thing, which tells you how relativistic you are. γ = 1 over the square root of [1 - (V2 / c2)]. And we talked about this factor a little bit. If V over c is equal to zero or approaches zero, then γ, obviously, is 1. And when γ is 1, that’s the Newtonian case–then, everything is just like Newton’s law said.

Okay. On the other hand, as V over c goes to 1–that is to say, as the velocity approaches the speed of light, this γ factor goes to infinity, because 1 minus 1 in the denominator–that’s zero in the denominator, so the thing has to go to infinity. And then, all these bizarre relativistic effects start taking place. And the one we talked about in particular came about from an example of how this γ is used–namely, that the relativistic mass is equal to γ times the rest mass, which is the Newtonian mass. And, obviously, if γ = 1, then the Newtonian mass is equal to the–then, the mass is equal to the Newtonian mass, and you’re in Newton’s laws, and everything is fine. When the velocity approaches the speed of light, then this total relativistic mass goes to infinity–the consequence of which is that you can no longer accelerate, regardless of how much–so, no more acceleration–regardless of how much force is applied, because force equals mass times acceleration. And if the mass is infinite, then any amount of force will not give you an acceleration. An acceleration is a change in velocity, and so, the consequence of this is that you can’t go faster than the speed of light. It’s also another side consequence of this–sorry–there was? Oh, excellent, yes ask it.

Student: [Inaudible.]

Professor Charles Bailyn: V–okay. V is the velocity that something is traveling. There’s no escape velocity here, at the moment. There are all kinds of different Vs floating around, so it’s important to keep them straight. Yeah, can’t–you can’t go faster than the speed of light.

A side comment from this is that photons, particles of light, which obviously, by definition, do go at the speed of light, have to have zero rest mass–because otherwise they’d have–they’d end up having infinite mass and infinite energy which isn’t–which isn’t physical. So, photons which go at the speed of light, for which γ is therefore infinite, have to haveM–this little M0 here, equal to zero. So, you have zero times infinity, and that can equal a finite number, otherwise they’d have infinite energy. Yes?

Student: I was just wondering if we’re talking about velocity as a factor or, like, the speed of light?

Professor Charles Bailyn: At the moment what I’m talking about is velocity as a speed. So, I’m talking about the magnitude of the velocity. And you can tell, actually, that that’s the case, because it comes in as the velocity squared. So, even if it’s a vector, when you square it, that gives you a scalar quantity.

Okay. So, let me go on from here and talk about an intermediate case. We’ve talked about V equals zero, we’ve talked about V equals c. Let me talk about an intermediate case. And the particular intermediate case–and this will show up on your problem set ‒ this is the thing we didn’t get to on Thursday. This is what’s called the post-Newtonian approximation. This is when you’re just a tiny little bit relativistic. So, V over c, or more properly, V2 / c2 is small, but not zero.

And here, a little mathematical concept that may have flashed before your eyes when you were in eleventh grade, or something, comes in, which is the series expansion. We won’t do the mathematics of this, but here it is. If you take 1 plus epsilon [ε]–and I should say, epsilon is a–used by mathematicians to mean something small. Famous mathematician named Erdos used to refer to children as “epsilons.” That seems to be carrying it a little bit too far. But, this is how mathematicians think. So, epsilon, in this case, is something that’s much, much less than 1.

And if you take 1 plus epsilon–so that’s a number that’s slightly greater than 1–and you take it to the nth power, you can then expand this as a series. And the series goes like this: 1 + n ε, plus a bunch of other terms, each of which is multiplied by a higher power of epsilon. So, there’s a term in ε2 and there’s a term in ε3, and so forth. And this is an infinite series.

But the thing is, if epsilon’s small, then ε2 is even smaller. So, if epsilon’s already small and nε is pretty small, then ε2 is far smaller than that, ε3 yet smaller, and these are negligible. And so, the approximation is, (1 + ε)n is approximately equal to 1 + n ε, if epsilon is much, much less than 1.

So, here’s what we’re going to do. We’re going to use this series expansion, and we’re going to generate expressions for some things–in particular, the mass, but some other things, too–where the 1 is the Newtonian term. And the thing represented by this n ε, which is much smaller, is a correction to the Newtonian term, and is the first sign that things are becoming relativistic. So this is, in this context, going to be a Newtonian term, and this is the post-Newtonian approximation. Okay.

Oh, so, let’s just do an example. Let’s think about the motion of the Earth–the Earth’s orbit around the Sun. Earth’s orbit, you may recall, from the last section of the class, how fast the Earth moves. It’s about 3 x 104 meters per second. Thirty kilometers a second. The speed of light is 3 x 108 meters per second. And so, V2 / c2 = [(3 x 104) / (3 x 108)]2 = 10-8. And so, that quantity, which is going to turn out to be the post-Newtonian effect, is one part in 108 for the Earth’s orbit. And so, pretty small. Considering that we think that 9 = 10, something in the eighth decimal place isn’t going to do a lot of damage. So, it’s not a particularly relativistic situation.

### Chapter 2. Gamma and Post-newtonian Mass [00:08:34]

All right, so, now I want to go back and I want–that’s just an aside. I want to go back and apply this approximation to &γ;. γ is equal to 1 / (1 - V2 / c2), the whole thing to the ½ power. That’s (1 - V2 / c2). I got to get a better pen, sorry about that. And now, this can be expanded. This can be expanded in just this kind of way. The first term is 1. Epsilon is - V2 / c2.

n is - ½. Minus times a minus is a positive, so this is plus ½ V2 / c2–plus additional terms, the first of which is (V2 / c22, which in the case of the Earth would be 1016. So, it would be vastly smaller still, so, we ignore that. And so that–and so, you can substitute this for γ in situations where V is kind of small. For example, let’s go back to this mass equation. 1 + ½V2 / c2. This is the total mass. Now–oh, let’s separate these terms out. M0 –that’s the Newtonian term. And now, here’s a post-Newtonian approximation.

½ M0 V2 divided by c2. Okay, high school physics experts, do you recognize this term? Kinetic energy. Absolutely right. This is from high school physics. The kinetic energy–that is to say the energy in the motion of this object. So what is this whole term? This whole term is the kinetic energy. So, M is equal to the Newtonian rest mass, plus the kinetic energy divided by c2.

Now, you’ll recall that E / c2 = M. So, this is the mass equivalent of the kinetic energy. So, given this nice relativistic idea that mass and energy are interchangeable, well, here it is. Here is the kinetic energy expressed in terms of mass. And so, this little equation M = M0 γ has a variety of consequences. When V / cV2 / C2, really, goes to zero, then it’s just Newton. It just says, mass equals mass, because γ is equal to 1. And that expresses an important Newtonian concept, that the mass of something is an intrinsic property of that object, which doesn’t change–which is how things are in Newton, but not how things are relativistically.

In the other extreme, where V2 / c2 approaches 1, this equation expresses the fact that light–the speed of light is a speed limit. And in the post-Newtonian case where V2 / c2 is small, but not zero, this same expression expresses the fact that Emc2.

So, all three of these concepts come out of the same equation. It’s this kind of thing that makes physicists say things like, the theory of relativity is incredibly beautiful. It’s hard to know–I realize, it’s hard to know what that means, you know. What makes a piece of mathematics beautiful? It’s when you get a situation like this, where a simple mathematical concept sort of spawns all kinds of new and unexpected ideas, depending on how you look at it–and that they all kind of come together as one.

I have this vivid memory in graduate school, I was sitting in a class, you know, and the professor was doing the thing that professors do. He was writing down all kinds of miscellaneous information really, really fast, from relativity theory and nuclear physics. All sorts of stuff was going up on the board. I was doing the thing that students do, where you desperately try and write it all down, so that you can then go back and figure out what the hell he was talking about later. And suddenly, in the midst of this rather typical class, I realized what was happening: that in about twenty minutes, he was going to put all this stuff together and prove Chandrasekhar’s limit. We talked about Chandrasekhar’s limit a week or so ago. That’s the limit whereby a white dwarf can’t be bigger then 1.4 times the mass of the Sun or it continues to collapse. And, I suddenly saw where all this was going and I sort of wrote down in my notebook, “Chandra,” and underlined it, put my pencil down. And then, I just watched for twenty minutes. It was great. It was–the only thing I can compare it to is listening to a great piece of music, because it kind of unfolds in time, and you see where it’s going, and it’s just a great feeling.

So, don’t be condescending to the physics majors when they’re working like hell on those problem sets late on a Thursday night, because they have access to realms of aesthetic experience that you can only imagine.

### Chapter 3. What Is Mass? [00:15:34]

All right. I’ve now heard the same question twice, so let me answer it, and then we’ll ask for other things. The same question, which has been asked in a couple of different ways, is, what is mass? That’s an excellent question, because it’s awfully hard to parse what’s going on with this M0 and this other kind of M, and stuff, if you don’t know what the concept means. So, that’s a really good question.

What’s mass? Mass is defined–you can think of it this way. What is mass? Mass is defined as from this equation, in a certain sense. F = ma. It tells you, for a given object, how hard it is to move–or, more precisely, how hard it is to accelerate. So, higher mass requires greater force to accelerate by a certain amount. It’s a property of an object, and it tells you how much it resists being pushed around. It’s sometimes referred to as “inertial mass” for this reason. It’s an expression of the object’s inertia–how much it resists being accelerated.

And so, in Newtonian physics, this is purely a property of the object itself. You can say, if you’ve got a basketball, or if you’ve got a car, or something, that the car has more mass than the basketball. Why? Because if I go–do I have a basketball? I don’t. All right, car, a car has–the lectern has more mass than the little piece of cardboard here, because if I apply force to the cardboard I can move it. And if I apply the same amount of force to the lectern, I move it a whole lot less. And so, what is the difference between those two things since the force applied is the same? The difference is that this one has a whole lot more mass.

And in Newtonian physics, the mass is a property of the object. In Einsteinian physics, there is something that’s the property of the object–that’s its rest mass. But the inertial mass, the amount by which it resists being pushed, is also a property–not just of the rest mass, but also of its motion. So, if this were moving at a large fraction of the speed of light and I applied some kind of force to it, it wouldn’t move nearly as much as it does–or, it wouldn’t change its motion nearly as much as it does when it’s standing still. And so mass–the inertial mass of the thing varies depending on how fast it’s moving. So, that’s an important question for understanding this. Let’s see–yes?

Student: Let’s say that you’re on Earth. If you were to push an object that’s standing still versus an object in motion, you’re going to have the same affect on it?

Professor Charles Bailyn: No, you’re not going to have the same effect on it, because if the object in motion is going to have a little extra piece to its mass.

Student: Okay, so it has a very small effect [inaudible]

Professor Charles Bailyn: A very, very small effect. In fact, if you try this on the Earth itself, which is moving at 30 kilometers a second, that effect is going to be 1 part in 108.

So, that’s what we calculated before, where we–where was it? Yeah, here it is. That’s this purple part here. You calculateV2 / c2, that’s 10-8. So, it’s a difference if something moving even as fast as 30 kilometers a second, which is pretty fast–this difference is only going to be–actually, ½ of 1 part in 108. So, it’s a hard thing to see. Now, the place you can see this stuff is in particle accelerators, because there, you can take sub-atomic particles and accelerate them to very close to the speed of light. And then you see these very dramatic effects. Yes?

Student: So, if something’s going at the speed of light, so that the force you would need to stop it is infinite [inaudible]?

Professor Charles Bailyn: Yes, yes, that’s correct. You cannot stop something that’s going at the speed of light, because the force you’d need to accelerate it, which could either be a–making it faster or slower, would be infinite. But, that’s why photons don’t have any rest mass at all. That statement that you made only applies to something which has non-zero–greater than zero rest mass.

Student: And they never go?

Professor Charles Bailyn: And they never go to the speed of light, because you can’t get them that fast. You’d need an infinite amount of force to get it to go that fast. Photons are a different story. And, of course, if you stop a photon, it disappears. Because if it’s going less than the speed of light, then it’s got no energy, because it’s a finite γ times zero rest mass. Yes?

Student: What about all the weird, like, time stuff you always hear about?

Professor Charles Bailyn: The weird time stuff you always hear about. Excellent. Yeah, okay. A couple of other Lorentz transformations. Time is equal to γ times time zero. This is time dilation. The faster you go, the slower your clock works. This is the origin of all that nice science fiction, where you get on a rocket ship. You go close to the speed of light. You go somewhere. You turn around. You come back. Your clocks are running slow, so, it only takes a year in your time. And you come back and everybody–you know, a hundred years have passed. All your friends are dead, and so forth. So, that’s a whole science fiction thing.

There’s also length contraction. That looks like this. Take the pen. Put it in motion. Make that motion at some substantial fraction, at the speed of light, and it gets shorter. Amazing. We’ll talk more about this in a little while.

Yeah, these are–these things with the γs in them, they’re collectively known as the Lorentz transformations. That’s just a name. And these are the ways in which basic properties–space, time, mass–change with velocity. Yeah?

Student: So with the equation, M = M0 times γ. M0 is the intrinsic mass?

Professor Charles Bailyn: Yes. No, no, no, M0–yes. Let me think. M0 is the intrinsic mass. M is the inertial mass.

Student: Okay.

Professor Charles Bailyn: That’s a way of thinking about it.

Student: So, if something was traveling at a very small speed, wouldn’t it mean that V2 / c2 was close to zero?

Professor Charles Bailyn: Yes.

Student: Which means the inertial mass would be close to zero?

Professor Charles Bailyn: No. If it goes–so, you were right up to the very end. Remember how this works.

γ = 1 / [(1 - V2 / c2)½]

If V / c goes to zero, γ goes to 1. And in that case, the inertial mass is equal to the intrinsic mass. And, if it’s going faster, then the inertial mass is bigger than the intrinsic mass. The intrinsic mass is usually referred to as the rest mass.

Student: Well, then how did you get V2 / c2?

Professor Charles Bailyn: Ah. That’s the second term of the series expansion.

1 / [(1 - V2 / c2)½] expands to be 1 + ½ V2 / c2 plus other terms. Here’s the first term. That’s the rest mass, because we’re going to multiply this by M. And this is then the kinetic energy divided by c2.

Student: So the [inaudible]

Professor Charles Bailyn: It’s the second term.

Student: [inaudible] inertial mass can never be zero. It’ll just go closer to the front?

Professor Charles Bailyn: Yeah, the rest mass doesn’t change. If V2 / c2 goes to zero, what happens is, the kinetic energy goes to zero, which is exactly what you want it to do. Yes?

Student: How does–how is momentum conserved when something’s relativistic?

Professor Charles Bailyn: How is momentum conserved when something’s relativistic? Excellent. There is an equation of relativistic momentum. And it’s that thing that’s conserved, which is different from M times V, because V behaves differently and M behaves differently. And you can work out exactly where all the γs go in that, and I don’t remember it off the top of my head. But there is a quantity that is conserved, but it doesn’t look quite the same as the Newtonian quantity. In the limit where the velocity is small, it reduces down. The first term of that series is M times V. Yes sir?

Student: When you define mass by the equation force equals mass times the acceleration, how, then, do you define force?

Professor Charles Bailyn: Yeah, okay. So, this gets a little bit circular, right? Okay. Force–you can define it either way. Force is the thing that–force is the ability to do work. That’s the technical definition. Force: ability to do work. And then, work also has a technical definition. The problem with these things is you kind of do–if you just define them all around the circle, they do tend to come back and bite themselves in the tail. I agree with that.

But force is also related to energy. How much energy is required to exert a certain force? So, all these things, interrelate–but it is definitely true that there’s a circularity to the definitions, if you follow them all the way around. So, if you keep asking that kind of question, I’m going to go around until I get back where I started from. Yes?

Student: Why is the speed of light constant in all frames?

Professor Charles Bailyn: Why is the speed of light constant in all frames of reference? That’s problem two of your problem set, and I’ll come–I’ll say a few things–that’s an insufficient answer, but I’ll come back and say a little bit more about that in a minute. Yes?

### Chapter 4. Lorentz Transformations and Relativistic Effects [00:26:25]

Student: We were just a little bit confused with the original Lorentz transformation–how, in most cases, that transforms toE = mc2. Could you show that to us?

Professor Charles Bailyn: Yeah. So, let’s go back and take another look at that. All right, so, here’s what I’m going to do. First of all, I’m going to take γ–this quantity–and I’m going to expand it. This is equal to γ, and now I’m going to expand it. That gives me 1 + ½ V2 / c2. Let’s see what have I done with the–my–the pieces of paper are getting out of order, here. Here it is.

So, inertial mass is equal to the rest mass, times γ. So, now, I’m going to substitute in this approximation for γ. And then, I’m going to multiply M0 through both terms. So this is–so, the inertial mass is equal to the Newtonian mass, plus a term that looks like this. One half M V squared is, in Newtonian terms, the kinetic energy. And it appears here, divided by c2. And so, a way you can write this is that the inertial mass is equal to the rest mass, plus the kinetic energy, divided by c2. And that is an example of the fact that energy divided by c2 is another way of expressing mass.

Student: Then, how is it not saying that energy over c2 equals–why is it not from the energy of c2, literally, algebraically, whether it be M minus M0 [inaudible]

Professor Charles Bailyn: Oh, which M are you talking about?

Student: Yes.

Professor Charles Bailyn: Yeah, its M0 here, and it’s M over here. And so, somebody asked–actually you’re now the second person who’s asked this question, so I better answer it explicitly. I answered the question, “What is mass?”

What’s energy? E = mc2 squared. So, what do I mean by this? M–this is the inertial mass. So, that’s M, which is equal to, in this case, M0 plus kinetic energy over c2. And notice the difference here. This is one kind of energy. This is kinetic energy. That’s the energy in–that’s the energy contained in the motion.

Another way you could write this is the rest energy over c2. So E, here, is the total energy. And there are two kinds of energies being expressed here. There’s a rest energy, divided by c2, and there’s a kinetic energy, divided by c2. But in the Newtonian case, you think of them as two completely different things–whereas, in the relativistic case, they’re two manifestations of the same thing, and they add to make the inertial mass. Yes?

Student: At what velocity are a fractions of c– would something need to be traveling on Earth for us to–for there to be a perceptible relativistic effect?

Professor Charles Bailyn: Okay. So, how fast do you have to go to have a perceptible relativistic effect? That depends how good your instruments are. If you’ve got something that can perceive–that can measure a velocity or an energy or a mass to eight decimal places, then it doesn’t have to go so fast. If you have something that’s fairly crude, then it has to go much faster.

But I should say–special relativity, these Lorentz transformations, have been measured upside down and backwards in the laboratory. This work to many, many decimal places.

Here’s an example. If you take sub-atomic particles that are unstable, that decay–they have a half life, so, half of them will decay in ten seconds, or something. And you take those particles, and you accelerate them to a large fraction of the speed of light. It takes much longer for them to decay, because they’re going so fast and their time clocks slow down.

One manifestation of this is, there’s a kind of unstable particle that’s created at the top of the atmosphere. These things are called “muons.” What happens is, cosmic rays come in. They blast the top of the atmosphere. They make these muon things. The muons propagate close to the speed of light, and they arrive at Earth. And you can measure, if you have a little muon detector–never mind what the heck a muon is–but imagine that you have something that could detect it on Earth, and you see a whole bunch of these things caused by cosmic rays. But, if you ask, how long did it take them to get from the top of the atmosphere down to where your muon detector is, it’s much, much longer than their decay time. So, you really shouldn’t see any of them at all. But you do see them. And the reason is because they’re going at close to the speed of light. And so this is–these effects are observed and measured to great accuracy in the laboratory and in astrophysical situations. Yes?

Student: I know it’s practically impossible, but what would happen here if we–if you were to go faster than the speed of light?

Professor Charles Bailyn: Faster than the speed of light? Well, you’d have super infinite mass. This would not be good. Well, okay. So, let us imagine what the equations would do. What would happen is, as you go faster than the speed of light, if you try and slow down toward the speed of light, you’d have the same problem. That is to say, it would take you more and more effort to slow down to close to the speed of light. And the consequence of this is that if you’re faster than the speed of light, you can’t slow down to the speed of light, or slower than that.

There are hypothetical particles–and this doesn’t exist in the real world–but, there are hypothetical particles that do this. These are called tachyons. They have the odd effect that they go backwards in time–because again, if you believe the equations, right? Time dilation. T = &γ; T0. Now, look at γ. Square root of 1 - V2 / c2. So, this is awkward. If V is bigger than c, this is a square root of a negative number. That becomes imaginary. Right? Square root of a negative number–hard to do. So, this becomes imaginary. So, your time axis becomes imaginary, and all sorts of very bad things start to happen to you. Yeah, yes?

Student: So, as your mass increases with velocity, do you just–does the particle or the entity have a greater gravitational effect?

Professor Charles Bailyn: Does it have a greater gravitational effect? Excellent question. That’s the next lecture, because that’s general relativity. And what you’re doing is you’re assuming that Newtonian gravity is correct. Newtonian gravity–gravity is a force proportional to the mass.

Turns out–and this is one of the things that led Einstein to his theory of general relativity–turns out that relativistically, it’s–that gravity should not be thought of as a force, at all. And there’s a whole different way of thinking about it, and your question has to be, in a kind of Zen sense, unasked. We’ll get to that. We will get to that. But the point is that, in relativistic theory, gravity is not a force, anymore. And so, the question doesn’t arise in quite the same way. Yes?

Student: [Inaudible.]

Professor Charles Bailyn: Sorry?

Student: [Inaudible]

Professor Charles Bailyn: E = M0 / c2 + K.E. Yeah. You have to put–you have to get the c squareds in the right place, and that’s basically a unit conversion.

Student: What’s K.E. / c2?

Professor Charles Bailyn: No, E -

Student: Oh, E.

Professor Charles Bailyn: - is equal to c2 M0 + K.E. Basically, what I’ve done is I’ve multiplied this equation by c2 on both sides. Yeah. Yes?

### Chapter 5. Time, Light, and the World [00:35:02]

Student: I don’t know if this is really a worthwhile question, but, what is–what are the implications for, like, what time is?

Professor Charles Bailyn: Well, this is the–

Student: If you have one meaning of what time is?

Professor Charles Bailyn: Right. So, what are the implications for what time is? This is the big stumbling block, from a philosophical point of view, to all of this stuff.

Turns out, time isn’t absolute. That is to say, one has the feeling, as we go through our everyday life, that, you know, my watch and your watch are kind of measuring more or less the same thing, to the extent that they’re accurate to do so. So, if we have identical, perfectly accurate watches, and we go about our daily life and come back, they’re going to read the same time at the end if they read the same time at the beginning–because time moves the same way for me as it does for you. This turns out to be false.

And so, time is not an absolute quantity. Neither is space, for that matter, because length changes also. This is quite disturbing. And the reason it’s so disturbing to us is our brains have evolved in a situation where we’re always moving really, really slowly compared to the speed of light. And therefore, you don’t have to worry about this stuff in everyday life, down to some factor of 10-12. But, nevertheless, it turns out to be true that time does not tick off in an absolute way. And this is why people get so freaked out by this stuff–because it seems counter to everyday experience. But, it can be measured. And it turns out to be true.

So, the Newtonian case, what we’re used to and the way our brains work, turns out to be only an approximation of the way things really are. There are other things that are conserved, no matter how fast you go. And we can get to that at some point–but time isn’t one of them. Yes?

Student: Why is the speed of light this magic number?

Professor Charles Bailyn: Why is the speed of light this magic number? Okay. So, the fact that there is a magic number comes out of this kind of equation. Because, when this quantity is equal to 1, bad things start to happen, and all the equations blow up. The fact that it is the speed of light that happens to be that, in that equation, is because light consists of particles with zero rest mass. And a particle at zero rest mass has to have this thing go to infinity or it doesn’t exist at all.

Student: [Inaudible]

Professor Charles Bailyn: Why is the world this way?

Student: Well [inaudible]

Professor Charles Bailyn: That I–no, seriously, that’s what you’re asking, and it’s a good question, and I can’t answer it. You know, because this is where physics turns into–seriously, this is where physics turns into theology. You can’t–all you can say from science is that this is the way it works. You can’t answer the “why” question. You got to talk to my colleagues in some other department about that one. I’m sorry, because it’s the question one would really like to know the answer to, right? But that one I can’t cope with. Yes?

Student: Thinking about black holes, when the light enters the black hole across the event horizon, I’ve heard that it often times orbits or may orbit Jupiter? How is that possible if the light doesn’t have mass?

Professor Charles Bailyn: Okay, so, the question is, can light go into orbit around a black hole? Basically. And the answer is, yes it can. And then, the next part of the question was, how does that work if it doesn’t have any mass? You’re asking the same question he did over there, because what keeps it in orbit is gravity. And if you reject the view that gravity is a force, then that question doesn’t become important. And we’ll–as I say, we’ll talk about that later. So, that, again, is a question that pops up because you’re thinking of gravity as a force.

Student: How does–with that equation, not to say “why,” if it is, why it is, but where–how is it derived?

Professor Charles Bailyn: How is it derived? Okay. So, let me go on with the presentation in the last couple of minutes of the class, here. Here’s–so, let me ask your question in a different way. Here’s what–why did Einstein think this up? What a crazy-ass thing to do, right? Why not–there’s little discrepancies when things are moving at the–at high speeds, but who the heck cares? It’s hard to measure things at high speeds. Why not just stick with Newton, which has done very well for two-and-a-half centuries?

Here was the problem. In the late nineteenth century, there were a whole series of experiments, all of which, one way or another, had the same consequence–namely, that light–the speed of light is the same in all–for all observers. Okay? That’s extraordinarily weird. It’s not obvious immediately why that’s so weird. But let me try and explain it.

Here’s a guy, and let’s say he’s a baseball pitcher, and he’s throwing a ball. So, here’s the ball and it’s moving this way at some velocity, which we’ll call V1. And you’re down here, or over here somewhere, it doesn’t really matter. You’re down here, and you’ve got a radar gun, and you measure the velocity of that baseball. How fast is it going? Well, if he throws it at V1, and you’re at rest with respect to him, then you measure V1.

Now, supposing we put our pitcher on top of a moving train, all right? And the train is moving at some other velocity, V2. So, the guy’s on top of the train. He throws it 100 miles an hour. The train’s moving at 100 miles an hour. You’re standing next to the track with your radar gun. How fast do you measure that baseball to move? Well, you measure a total velocity, which is the sum of the velocity of the baseball with respect to the train, and the train with respect to you. Okay? That seems pretty clear.

But now, supposing this guy–instead of throwing a baseball, he’s got a flashlight. So he’s got some light source, and the light is moving at the speed of light, because that’s the speed that light moves. And you’ve got some device, down here, which measures how fast that light moves along. And he’s standing at rest with you. And you measure the speed of light, and it comes out to be c–not surprisingly, the same speed of light.

Now, let’s put the guy on the train. Here he is, moving at V2, or any V. You would expect that the speed of light that this guy measures down here would be c + V2. Right? Because, it would the speed of light with respect–from the flashlight, with respect to the train, plus the speed of the train, with respect to you. You would expect it to be c plus V2.

But it isn’t. It’s cVtotal = c, not c + V2. Very weird. The speed of light is the same no matter how fast you’re moving relative to the source. You could be moving at 99% of the speed of light, toward a light source, and it would not be coming at you at 1.99 times the speed of light. It would be coming towards you at the speed of light. You could have a friend who’s standing still, and you would both measure the same speed of that light, even though you’re moving really fast compared to your friend. Very strange. And there are a whole series of experiments, the most famous of which is something called the Michelson-Morley experiment, which, one way or another demonstrated that this was the case.

What to do? And, what Einstein did was, he said, hmmm. Okay, various things were tried. The experiments were wrong. The equations were slightly screwy, who knows. But Einstein took–Einstein’s great genius was to take this seriously, and to say, okay something is really screwed up with velocities. What is velocity? Velocity is space over time–miles per hour. Therefore–so, space and time are messed up when you get close to the speed of light. Now, of course, it also has to be true that when you’re going slowly, the original Newtonian result is recovered.

So, this is your problem set. I have written down the relativistic formula for the addition of velocities. And your problem set is going to be to use that series expansion to demonstrate that in the Newtonian limits, you get this. And in the limit where one or the other of those velocities is the speed of light, you get this. So, you’ll see. The algebra’s actually not so bad. And that’s what prompted Einstein to go on this whole rampage, generating all of these γs, and so forth. It was the experimental evidence that there’s a serious problem with velocities. Okay, we got to stop. More next time.

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