PHYS 200: Fundamentals of Physics I
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Fundamentals of Physics I
PHYS 200 - Lecture 12 - Introduction to Relativity
Chapter 1. The Meaning of Relativity [00:00:00]
Professor Ramamurti Shankar: So, let’s begin now. First of all, I’m assuming all of you have some idea what special relativity means. There are two theories of relativity, one is the special theory and one is the general theory. The general theory is something that we won’t do in any detail. Special theory is something we will do in reasonable detail. So, it’s good to begin by asking some of you what is your present understanding of what the subject is all about. Yes, sir? The Yale cap, what do you think it’s about?
Student: It’s about relative speed in two reference systems.
Professor Ramamurti Shankar: Okay, it’s about relative speed in two reference systems. Yes? I’ll come to you; then I’ll come to you.
Student: It’s based on the postulate that the laws of physics are the same in any two references moving in uniform motion relative to one another and the speed of light is constant in all references.
Professor Ramamurti Shankar: Okay, I will take the last row there.
Professor Ramamurti Shankar: Okay, so what I’ve heard so far is that the laws of physics are the same for two people who are both in inertial frames of reference and the velocity of light’s a constant. Right. That’s certainly the way we understand the special relativity theory. But it’s a very old one. It’s been going on long before Einstein came. There was a theory of relativity at the time of Newton and that’s where I want to begin. Relativity is not a new idea at all, it’s an old one. And the old idea can be illustrated in this way and it will agree with your own experience.
So, the standard technique for all of relativity is to get these high speed trains. I’m going to have our own high speed train; this is the top view of the train. And like in everything I do, we’ll get away with the lowest number of dimensions, which happens to be this one spatial dimension and of course there is time. So, the train is moving along the x axis. You are in this train. You board the train and all the blinds are closed because you don’t want to look outside. That’s not because you’re traveling through some parts of New Jersey; you don’t want to look outside for this particular experiment. You get into the train, you settle down and you explore the world around you. You pour yourself a drink, you play pool, you juggle some ping pong balls, tennis balls, and you have a certain awareness of what’s happening, namely, your understanding of the mechanical world, and then you go to sleep. When you are sleeping, some unseen hand gives to this train a large velocity, 200 miles an hour. The question is, “When you wake up, can you tell if you’re moving or not?” That’s the whole question. Will this speed, whatever I gave you, 200 miles per hour–will it do anything to you in this train that will betray that velocity? So, when you wake up will you say I’m moving or not? Now, you might say, I’m not moving because I’m on Amtrak and I know this train is not going anywhere.
That kind of sociological reason, by the way, there are many of them, you cannot invoke. You can only say, “I’m on this train. Is anything different?” And the claim is that nothing will be different. You just will not know you are moving. Now, if the train picks up speed, or slows down, you will know right away. If it picks up speed or accelerates, you find yourself pushed against the back of the seat or if the driver slams on the brake, you will slam into the front of the seat in front of you. No one is saying that when the motion is accelerated, you will not know. Accelerated motion can be detected in a closed train without looking outside. The question is uniform velocity, no matter how high, can that be perceived? Can that be detected?
So, at the time of Galileo and Newton everybody agreed that you cannot detect it. Remember that if you started out and Newton’s laws worked for you, you are called an inertial observer. One of the laws you want is, if you leave something, it should stay where it is. When the train is accelerating, that won’t be true. You leave things on the floor when it’s accelerating, things will slide backwards. So, with no apparent force acting on it, things will begin to accelerate; that’s a non-inertial frame. We are not interested in that. You started out as an observer for whom the laws of Newton work, the laws of inertia work, F = ma, then you go to sleep and you wake up. So, when I said everything looks the same, I really meant that the laws of Newton continue to be the same because if the laws of Newton are the same, everything will look the same. That’s what it means to say “everything looks the same.” Our expectations of what happens when I throw it up or what happens when two billiard balls collide, everything is connected to the laws of Newton. So, the claim is, the laws of Newton will be unchanged when this velocity is added on to you.
Now, we should be clear about one thing. If there is a train next to you in the beginning – let me just put it on this side for convenience – and you got in and you boarded this train but you looked at this train and it was not moving. If you lift the blind and look through, you’ll see the other train and there’s another passenger in the other train and you look at each other, you’re not moving. When you wake up after this brief nap, you find when you look outside the other train is moving at 200 miles an hour. The question is, “Can you tell if it’s you who’s responsible for this relative motion, or maybe nothing happened to you and the other train is moving the opposite way?” And the claim of relativity is that you really cannot tell. You can tell there is motion between the two trains that wasn’t there before. That’s very clear if you look outside but there is no way to tell what actually happened when you were sleeping. Whether you were given the velocity of 200 to the right or the other train was given a velocity of 200 to the left or maybe a combination of the two, you just cannot tell. That’s the word “relative.”
So far – I didn’t tell you – if you have only one train, what I told you earlier, is that uniform velocity does not leave its imprint on anything you can measure. If you look outside, of course you can see the motion of the other train, but you still cannot tell who is moving. You cannot distinguish between different possibilities. So, you have every right to insist that you are not moving and the other train is moving the opposite way. Once again, you can make this argument only for uniform relative motion. If your train is accelerating, now I’m saying it as if it is an absolute thing, and it is. You cannot say, “I’m not accelerating, the other train is accelerating in the other direction.” You cannot say that because you’re the one who is barfing up and throwing up and slamming your head on the wall; nothing is happening to the other person. You cannot say “I’m still in the same frame, you are going the opposite way.” If you are going the opposite way, why am I throwing up? Or if you are in a rocket and the rocket’s taking off and the G forces are enormous, many times your weight, it is the astronauts who are going through the discomfort. At that time they cannot say we are at rest and everyone is going the opposite way because no one else is in danger, but they are.
So, accelerated motion will produce effects. You cannot talk your way out of that. But uniform velocity will produce no effects on you and no effects on the other person. You can detect relative motion but you cannot in any sense maintain that you are moving and he’s not or that he’s moving and you are not. You can say, “I am at rest, things are the same as before, the train is moving the opposite way.” Now, if you go in the Amtrak and you look outside and you don’t see another train, but you see the landscape, you see trees and cows and everything, going at 200 miles an hour in the opposite direction, you have some reason to believe that probably the ground is not moving and you are moving. But that’s just based on what I called earlier some sociological factors. In other words, it’s completely possible to devise an experiment in which somebody puts the whole landscape on wheels and when you go to sleep the landscapes, cows and trees are made to move the opposite way. Not very likely, but that’s because we know in practice no one is going to bother to do that just to fool you. But if that did happen, you won’t know the difference. So, the reason we rule that out is we know some extraneous things not connected to the laws of physics. That’s why we don’t like to open the window and look at the landscape because then we have a bias. Open the window and look at another train and you just won’t know. That is the principle of relativity, that uniform motion between two observers, both of whom are inertial, is relative. Each one can insist that he or she is not moving; the other person is the one who is moving.
Of course, now, if the two, in reality, if the two trains were at rest–Let’s imagine my train got accelerated. So, during the time it was accelerated, I would know, but if I was sleeping at the time, I don’t know and when I wake up and the acceleration is gone and the velocity is constant, that’s when I say, “I just cannot tell.”
All right. Now, let’s show once and for all that the laws of Newton are not going to be modified. So, you find the laws of Newton before you go to sleep, you wake up, you find them again, you’ll get the same laws; that’s the claim. I hope you understand that all the mechanical things you see in the world around you come just from F = ma. We have seen projectiles and collision of billiard balls and rockets; they’re all Newtonian mechanics. So, to say that things will look the same is to say the laws of Newton that you will deduce before and after waking up will be the same. So, let’s show that. When you show that, you’re really done with it once and for all.
So, let’s do the following. Here is the x axis and here is my frame of reference. This is my x axis. Let’s call this the origin. The frame goes to negative and positive x values. Pick some object sitting at the point x. Now, we are going to first define the notion of an event. An event is something that happens at a certain place at a certain time; that’s called an event. For example, if there’s a little firecracker going off somewhere at some time, the x is where it happened and the t is when it happened. So, this is space-time. Once again, space-time does not require Einstein coming in at all. We have known for thousands of years that if you want to set up a meeting with somebody, you’ve got to say where and you’ve got to say when and things do happen in space-time. The fact that you need x and t, or, if you’re living in three spatial dimensions, the fact that you need x, y, z and t is not new. That is not the revolution Einstein created. The fact that you need four coordinates to label an event is nothing new. What he did that is new will be clear later. So, does everyone understand what an event means? Okay? An event is something that happens and to say exactly where and when it happened, in our world of one dimension, we give it an x and we give it a t.
Now, that’s me, and I’m going to give my frame of reference the name S. It turns out S is not just based on just my name. This is the canonical name for two observers, one is called S and one is called S′. So, S′, let’s say, is you. So, your frame of reference is going to be taken to be sliding relative to mine. So, let’s draw a y axis here. We don’t really deal with the y coordinate but just to give you a feeling, this is my y axis; that’s my x axis; this is my origin. y is not going to play a big role. Now, you are sliding this way, to the right, and your speed or velocity is always denoted by u. Some number of meters per second, you are zooming to the right at some speed u. So, imagine now you are going past me. At some instant–I’m sitting here at the origin of coordinates, you cross me and then a little later you are somewhere there. So, that’s your y frame axis; that’s your origin. And the same event you say has a coordinate x′. We arrange it so that when you zoom past me, you set your clock to zero and I set my clock to zero. When you want to set the clock to zero is completely arbitrary. So, we will decide right when you pass me. I’m at the origin of my coordinates, you’re at the origin of your coordinates. When you pass me, we’ll click our stop watches and we will set the time to zero. So, here’s an event. You and I crossed. What are the coordinates for that event? According to me, that event occurred at x = 0 and the time was chosen to be zero. According to you, your origin was also on top of my origin, so x′ was zero and the time is just the time. Everybody has a single time, and that time is called zero. That is one event. We made the coordinate of the event zero zero for both you and me. It is zero in space because my origin crossed your origin. That crossing took place at my origin; that’s why my x is zero and took place at your origin, that’s where your x′ is zero and the common time we chose to be zero by convention.
Then, we want a second event. So, let’s say the second event is some firecracker going off here. Here is something I should explain that I used to forget in the previous years. When I say I am moving, I imagine I am part of a huge team of people who are all moving with me. So, I’ve got agents all over the x axis who are my eyes and ears; they are looking out for me. So, even though I am here, if there’s a firecracker exploding here, my guys will tell me. And you are carrying your own agents. Let’s say at every point x you have a reporter, x = 1, 2, 3, 4; there are people sitting and watching. So, when I say I see something, I really mean me and my buddies, all traveling the same train at the same speed, all over space taking notes on what’s happening. We’ll pool our information later but we know this explosion took place here. I’ll simplify it by saying I know an explosion took place here at location x at time t.
So, this is our crossing. This event is when we crossed. Then, there is a firecracker. For the firecracker, I have to give some events [should have said “coordinates”]. I say it took place at location x at time t. What do you say? You measure the distance from your origin, you call this some x′, the time is still t. In Newtonian mechanics, the time is just the time. How many seconds have passed is the same for everybody. The question is, “What is the relation between x′ and x?” That’s what we want to think about. So, you guys should think before I write down the answer. What’s the relation of x′ to x? Well, this event took place at time t, so I know that your origin is to the right by an amount u times t. So, the distance from your origin for this event, I maintain is x′ is x - ut. Again, I want all of you to follow everything. These are all simple notions. Our origins coincided at zero time for the event that occurs at time t. Therefore, in a sense, you are rushing toward the event. You’ve gone a distance ut. Therefore, the distance from your origin to the event will be less than mine by this amount ut. This is the law of transformation of coordinates in Newtonian mechanics.
Chapter 2. The Galilean Transformation and Its Consequences [00:18:10]
If you have an event–if you want, formally, you can define it time, t′ for the primed observer. It goes without saying that t′ and t are the same. There’s no notion of time for me and time for you. There’s universal time in Newtonian mechanics. It just runs. We can call some time a zero. Once we have agreed, if you say you and I met at t = 0 and an explosion took place 5 seconds after our meeting, it’s going to be 5 seconds after our meeting for me and it’s going to be 5 seconds after the meeting for you. The time difference between two events is the same for all people. This is called the Galilean transformation. What are the consequences of Galilean transformation? Well, let’s look at the fact that x′ is x - ut. Remember, everything is varying with time. So x′ is a function of time and x is a function of time, if you are watching a moving particle.
Suppose this firecracker is not just one event, but it’s a moving object. Let’s give the object some speed; it’s moving to the right. Then the velocity, according to me–I’m going to call v as dx/dt is the velocity. Let’s just call it a bullet, according to S. Then w – it’s the standard name – is the velocity of the bullet according to S′. So, what I’ve done is, I first took one event and I gave it some coordinates and I told you how to transform the coordinate from one person to the other person. But now, take that point x not to be a fixed location but a moving object, so that as a function of time that body is moving. Then its velocity at any time is dx′ dt according to you; that is the dx/dt according to me, minus the derivative of this, which is u. Now, does that make sense? This should agree with common sense.
For example, if that bullet is going at 600 miles per hour to the right, that is 600 for me, and you are going to the right in your train at 200 miles per hour, you should measure the bullet speed to be reduced by 200 and you should get 400. That’s all it means. The two people will disagree on the velocity of the bullet because they are moving relative to each other. This is the way you will add velocities. But let’s look at the acceleration. dw/dt is going to be dv/dt - 0 because u is a constant. That means you and I agree on the acceleration of the body. We disagree on where it is. We disagree on how fast the bullet’s moving. But we agree on the acceleration of the body because all I’ve done is add a constant velocity to everything you see. Therefore, if according to you the velocity of the body is not changing, according to me the velocity of the body is not changing, because the constant added will drop out of the difference. Or, if the body has an acceleration, we’ll both get the same answer for the acceleration. So, that is the common acceleration a. So, if you like, a′ is the same as a. So, the acceleration of bodies doesn’t change when you go from one frame of reference to another one going at a constant speed.
All right, so let’s look at F = ma, which is md2x/dt2 is equal to some force on the body. And you look at the body and you say d2x′ over dt2 is the force on your body. First, I want to convince you we wanted to see that the left-hand sides are equal because the acceleration’s the same. Then, I want to convince you that the right-hand sides are also going to be equal. I can take many examples but eventually you will get the point. Let us not consider one body, but let’s consider two bodies. Two bodies are feeling a certain force due to, say, gravitation. And gravitation is, of course, a force in three dimensions but let’s write the force in just one dimension. And let’s say the force of gravity is equal to 1 over x1 - x2. Force on 1 due to 2 and the force on 2 due to 1 will be minus 1 over x1 - x2. The real forces are separation in three dimensions but this is a fictitious force. I want to call it gravity. It is any force that depends on the coordinates of the two particles.
So, I will say m1d2x1 over dt2 is 1 over x1 - x2. And m2d2x2 over dt2 is minus 1 over x1 minus–I have forgotten constants like g and m1 and m2. They don’t matter for this purpose. So, here are two bodies. They feel a force for each other and I’ve discovered what the force is. It’s 1 over x1 - x2. I don’t care if it’s 1 over x1 - x2 or (x1 - x2)2; that’s not important. What’s important is, it depends on x1 - x2. You come along and you study the same two masses. What will you say is happening? You will say, m1d2x1′ over dt2 is equal to 1 over x1′ minus x2′. Maybe I will–I’m sorry. Let me do it a little better. I can tell you what you will see. Given this is what I see, I can tell you what you will see. Let’s do that in our head. We know that the acceleration is the same for any mass so I’m going to write this thing as m of dx over prime dt2.
In other words, the acceleration according to me is the same as the acceleration according to you. Then, I’m also going to write the right-hand side as x1′ minus x2′. Do you understand that? If there are two bodies feeling a force, if you see it from a moving train, the distance between the two bodies is the same for you and me, because x1′ is x1 - ut and x2′ is x2 - ut. Take the difference; the difference between the location of the particles is the same for you and me. Acceleration is the same, mass is postulated to be the same, so I know that you will get the same law that I get. You will get F = ma; your acceleration will be the same as mine; the force you attribute between the two bodies will also be the same. That is why I know that you will also deduce the same Newtonian laws that I will.
You can also say it differently. If I woke up from my nap and I am now in a moving train and I examine the world around me, I’m going to get the same F = ma. Because as seen by a person on the ground, the masses obey F = ma. I am in this moving train now but I have the same acceleration for each mass and I have the same force. So, if you want, I’ll complete the second equation. m2d2 x2′ over dt2 will be minus 1 over x1′ minus x2′. If this is a little difficult, we should talk about this. I’m telling you that if I deduce F = ma and the F depends on the separation between the particles, then I’m sure that you will find the same laws of motion because the acceleration is the same that I get because we have seen a′ is the same as a. And the force will also be the same because the force depends on the separation between particles. And that doesn’t depend on which train you’re in or it’s not affected by adding a constant velocity to the frame of reference.
So, if you like, this is the way you prove in Newtonian mechanics the principle of relativity. So, not only is it something you observe by going on trains and what not, you can actually show that this is the reason everything looks the same. In other words, if the train was at rest on the platform and you and I were comparing notes and we both find F = ma, I go to sleep and I’m waking up and the train is going at a constant speed, if you can look through the window and look at the objects in my train, you will say they obey F = ma because nothing has happened to you. But you will predict I will also say F = ma because if you see an acceleration, I will see the same acceleration. If you see a distance between two masses to be one meter, I’ll also think it’s one meter. If the force is 1 over the square of the distance, we’ll agree on the force, we’ll agree on the acceleration, we’ll agree on everything. And once you’ve proven F = ma is valid, it follows that every mechanical phenomenon will behave the same way. That’s the reason things behave the same way. Yes?
Student: If for some weird reason, suppose different frames of reference, the rule F = ma was to fail, what would happen?
Professor Ramamurti Shankar: You mean if the rule failed in the other frame?
Professor Ramamurti Shankar: Yes. Suppose, hypothetically, that happened. Then, it would mean that when you wake up in the train, you will look at the world around you, it will look different because F ≠ ma. You will conclude, when I went to sleep it was F = ma; when I got up, F ≠ ma, the train is moving. So, you will have to conclude that uniform velocity makes detectable changes. And if you look outside the train, to the other train, the other train’s going backwards. You can now no longer say, “You’re going the other way, I’m not moving”, because the other person will say, “Hey, F = ma works for me.” It doesn’t work for you. So, you’re the guy who’s moving.” So, you’ve lost the equal status with other inertial observers because those for whom F = ma worked will say they’re not moving and for you, it doesn’t work, so you will have to concede you are moving. So, uniform velocity, if it makes perceptible changes, can no longer be considered as relative. It’s absolute and if you and I find each other moving, there may be a real sense in which I am at rest and you are moving, because for me F = ma works and for you it doesn’t. Well, that’s not what happens.
In real life, you find it works for both of these and either of us can maintain we are not moving. So now, you’ve got to fast forward to about 300 years. This goes on, no problem with this principle of relativity and 300 years later, people have discovered electricity and magnetism and electromagnetism and electromagnetic waves, which they identify as light. And then, it was discovered that what you and I call light is just electric and magnetic fields traveling in space. You don’t have to know what electro-magnetic fields are right now. They are some measurable phenomenon. They are like waves. And the waves have a certain velocity that Maxwell calculated and that velocity is this famous number 3 times 10 to the eight meters per second. And the question was, “For whom is this the velocity?”
For example, you can do a calculation of waves on a string, something we’ll be able to do in our course. Waves on a string will be some answer that depends on the tension on the string and the mass density of the string and that’s the velocity as seen by a person for whom the string is at rest. But if you calculate the waves of sound in this room – I talk to you, you hear me slightly later – the time it takes to travel is the velocity of sound in this room. That is calculated with respect to the air in this room because the waves travel in the air. In fact, the fact that all of us are sitting on the planet, which itself is moving at whatever, 1,100 miles per second, doesn’t matter, because the air is being carried along so even if the Earth came to a sudden halt, as far as the velocity of sound in this room is concerned, it won’t matter, because we are carrying the medium with us.
Chapter 3. The Medium of Light [00:31:35]
So, people wanted to know what is the medium which carries the waves of light, electromagnetic waves? First of all, the medium is everywhere because–How do we know it’s everywhere? Can anybody tell me? Yes?
Student: It travels through the vacuum of space.
Professor Ramamurti Shankar: Right. It travels in the vacuum of space. We can see the Sun; we can see the stars, so we know the medium is everywhere. Then, you can sort of ask, “How dense is the medium?” It turns out that the denser the medium, the more rapidly signals travel, in most of the things that we know. For example, when we look at waves in sound and when we look at sound waves in a solid, or in iron, you find in a very dense material, that the velocity is very high. So, this medium, which is called “ether,” would have to be very, very dense to support waves of this incredible velocity. But then, planets have been moving through this medium for years and years and not slowing down. It’s a very peculiar medium. But it has to be everywhere so we are all immersed in this medium because we are able to send light signals to different parts of the universe. And the question is, “How fast is the Earth moving relative to this medium?” You understand? This medium is all pervasive. We know that we can see the stars so it’s going all the way up to the stars and beyond. And we are immersed in this and we are drifting around in space. What is our speed relative to the medium? That’s the question that was asked.
Well, to find the speed relative to the medium, you calculate the velocity in the medium by Maxwell’s theory. So, here’s the medium in which waves travel at a certain speed. This is planet Earth going around the Sun. At some instant, you will have a certain velocity with respect to ether. And therefore, the velocity of light as seen by you will be modified from c to c - V. In particular, suppose the waves are traveling to the right in ether. Let me draw it this way. The Earth is going at this instant at the speed V. We expect the speed to be c - V because part of the speed is neutralized because you are going along with the waves. You’ll see a slower velocity. So, Mr. Michelson and his assistant Morley–they did the experiment. And they got the answer equal to c. What does that mean?
Student: The speed of light [inaudible]
Professor Ramamurti Shankar: No, no, but you cannot jump to that right now. If you are following Newtonian physics, your expectation is, it should be c - V. Yes?
Student: It means that there is no ether.
Professor Ramamurti Shankar: Well, that’s–not so fast, but it certainly means the following. Well, there’s a simpler answer than that. Yes?
Student: It means that the Earth is moving with respect to the ether.
Professor Ramamurti Shankar: At what speed?
Professor Ramamurti Shankar: Zero. Because you don’t have to–Look, you guys are ready to overthrow everything because you know the answer. But you’ve got to put yourself in the place of somebody in the early 1900s. There’s no reason to overthrow anything. The answer is, you’re going at the speed zero. Of course, you realize, that it is incredibly fortunate that on the one day Michelson wants to do the experiment, we happen to be at rest with respect to the ether. Fine. But we know that luck is not going to last forever because you are going around the Sun. On a particular day may be. But that velocity was such that on that day the Earth was at rest with respect to the ether. It’s clear that six months later, when we are going the other direction, you cannot also be at rest with respect to the ether. But that’s what you find when you do the experiment. You find every day you get the same answer and you jolly well know you are not at rest. You are moving around the Sun for sure. Yes?
Student: Did they postulate a drag for the Earth turning [inaudible]
Professor Ramamurti Shankar: Yes. So, people tried other solutions. But it is simply a fact that when you move one way or six months later in the opposite way, you get the same answer c. So, one possibility is, you don’t want–Look, don’t be ready to do revolutions, try to avoid it. So, one answer is, look at the speed of sound. You and I talked to each other then and we talked to each other six months from now; we get the same speed of sound. The speed of sound is published in textbooks, right? Seven hundred and something miles per hour. How come that doesn’t change from day to day? Anybody here on this side can tell me why the speed of sound doesn’t change from day to day even though we are moving? No one here can guess?
Professor Ramamurti Shankar: No, we are moving. Even six months from now, we get the same speed of sound in this room. When I talk to you, does it matter what time of year it is?
Student: The medium is moving along with us?
Professor Ramamurti Shankar: Yes, we are carrying air. As the Earth moves through space, it carries the air with you and the speed of the wave is with respect to the medium. If you can carry the medium with you, then it doesn’t matter how fast you’re moving. So, they tried that. They tried to argue that the Earth carries ether with it the way it carries air with it. Then, it’s not an accident you are at rest with respect to the ether because you’re taking it with you. But it’s very easy to show by looking at distant stars that you cannot be doing that. I don’t have time to tell you why that is true. So, you cannot take the ether with you and you cannot leave it behind, and that’s the impasse people were in. So, it’s as if there’s a car that’s going to the right at a certain speed c. You move to the right at some speed, maybe c/2. I expect you to get a speed c/2. But you keep getting c. You go three fourths of the velocity of light; you still get the velocity of light. That is very contrary to what we believe. In fact, that’s in violent opposition to this law here. If this V were not a bullet but a light beam, suppose for me traveling at a speed c and you’re traveling to the right at speed u, you should get c - u. That’s the inevitable consequence of Newtonian physics. And you don’t get that. And that was a big problem.
So, people tried to fix it up by doing different models of ether, none of which worked. And nobody knew why light is behaving in this peculiar fashion, so that’s when Einstein came in and said, “I know why light is behaving in this peculiar fashion. It is behaving in this way because if it didn’t behave in this way, if the speed of light depended on how fast I’m moving, then when I wake up in this train, all I have to do is measure the speed of light. Originally, I got some number; now I get to get a different number and the difference will give me the speed of the train. So, it would have been possible to detect the velocity of the train without looking outside, just by doing an experiment with light. So, even though mechanical laws involving F = ma are the same, laws of electricity and magnetism would be such that somehow they would betray your velocity. And that would mean uniform velocity does make an observable change because it changes the velocity of light that you would measure. “
But conversely, the fact that you keep getting the same answer means that electric and magnetic phenomena are part of the conspiracy to hide your velocity. Just like mechanical phenomena won’t tell you how fast you’re moving, neither will electromagnetic phenomena. Because to Einstein, it was very obvious that nature would not design a system in which mechanical laws are the same but laws of electricity are different. So, he postulated that all phenomena, whatever be their nature, will be unaffected by going to a frame at constant velocity relative to the initial one. That’s a very brave postulate because it even applies to biological phenomena about which I’m sure Einstein knew very little. But he believed that natural phenomena will just follow either the principle of relativity or they won’t. And that is something you should think about. Because that was the only reason he had. He just said, “I don’t believe chapters 1 through 10 in our book obey relativity and chapters 20 through 30 where we do E&M doesn’t.” These are all natural phenomena that will obey the same principle, which says all observers that are uniform relative to motion are equivalent.
Now, that’s really based on a lot of faith and even though scientists generally are opposed to intelligent design, we all have some bias about the way natural laws were designed; there’s no question about it. You can talk to any practicing physicist. We have a faith that underlying laws of nature will have a certain elegance and a certain beauty and a certain uniformity across all of natural phenomena. That is a faith that we have. It’s not a religious issue; otherwise, I wouldn’t bring it up in the classroom, but it is certainly the credo of all scientists, at least all physicists, that there is some elegance in the laws of nature and we put a lot of money on that faith, that the laws of nature will do this and will not do this. Who are we to say that? Who are we to say nature wouldn’t have a system in which mechanics obeys the laws but electricity and magnetism doesn’t? We haven’t run into somebody called nature. We don’t worship a certain deity called nature but we believe the laws of nature obey that. So, even though scientists or physicists in particular may not believe in design by any personal God, they do believe in this underlying, rational system that we are trying to uncover. You could be disproven, you could be wrong in making the assumption, but here it was right. It was really driven by this notion that all laws of physics should obey the same principle of relativity. So, Einstein’s postulates are that light behaves in this way because if it didn’t behave in this way, it would violate the principle of relativity, whereas we know mechanical phenomena do and electrical phenomena would not and that cannot be the case. You have a question?
Student: Why would that not apply to the speed of sound?
Professor Ramamurti Shankar: Yes. Because in the case of the speed of sound, you can take the medium with you; there is no such experiment you could perform. See, in the train, if you could carry the ether with you, there’s no surprise you would get the same answer. But we know we cannot carry the medium with us, that comes from extra-terrestrial experiments. That’s why the velocity of sound is not elevated to a fundamental velocity on which everybody will agree. So, the two great postulates–You’ve got to know where the postulates came from.
Chapter 4. The Two Postulates of Relativity [00:43:22]
Postulate Number 1 is simply a restatement of the relativity principle. I’ll just say it in one sentence. Exact wording is not important. All inertial observers are equivalent. “Equivalent” means each one of them is as privileged as any other one to discover the laws of nature. The laws of nature, we found, are not an accident related to our state of motion. If I find some laws, and you’re moving relative to me, you’ll find the same laws. And if you and I find each other in relative motion, you have as much right to claim you are at rest and I am moving and I have as much right to claim that I am at rest and you are moving. There is complete symmetry between observers in uniform relative motion. There is no symmetry between people in non-uniform [relative] motion. As I said, non-uniform motion creates effects which can destroy me and not destroy you. So, no one’s trying to talk their way out of acceleration, whereas you can talk your way out of uniform velocity. That’s the first principle. So this was there even from the time of Newton. What is true here is that all inertial observers are equivalent with respect to all natural phenomena, meaning all natural laws. And that is a generalization, when we say “all” instead of just mechanical. And the Second Postulate, you call it a postulate because there is just no way to deduce this, is that the velocity of light is independent of the state of motion of the source, of the observer, of everything. If a light beam is emitted by a moving rocket, it doesn’t matter. If a light beam is seen in a moving rocket, it doesn’t matter. All people will get the same answer for the velocity of light.
Student: Is there a reason why the speed of light is constant?
Professor Ramamurti Shankar: No. That’s why it is a postulate. You can show a few things later on. You can show that if there is any other speed, which is the same for everybody, that would have to be the speed of light. In the final theory of relativity, there are not two or three velocities that come out the same for everybody. There is only one velocity that can have the same answer for all people. That velocity is the velocity of light. By that I don’t mean it has to be light itself. For example, gravitational waves travel at the speed of light. It’s not just the light. It has to do with the velocity of light being a single number which has to have the same value for everybody.
Okay, so it looks like he has solved a big problem because he has said why light behaves this way; light behaves this way because it is part of the big conspiracy to hide uniform motion. But you will see that you have made a terrible bargain because once you take these two postulates, you have restored the relativity principle to all phenomena. Okay. You’ve gone beyond mechanical phenomena to electro-magnetic phenomena.
Chapter 5. Length Contraction and Time Dilation [00:46:48]
But you will find that you have to give up all the other cherished notions of Newtonian physics. Think about why. We are saying, here is a car going at 200 miles an hour, according to me. You get into your own car and follow that car at 50 miles an hour. You should get 150 but you keep getting 200. This may not be true for cars going at the speeds I mentioned but when finally you are talking about a pulse of light, that is true. And you’ve got to agree that is really not compatible with our daily notions or with the formula I wrote down, w = v - u. When you put v = c, w has got to come out to be c. And that’s not a property of the Newtonian transformation. So, what we are looking for is a new rule for connecting x and t and x′ and t′, such that when the velocities are computed and applied to the velocity of light, you get the same answer. That’s what we want to do now.
So, here is how we are going to do this. Now, let’s think about it. Let me send a pulse to the right at speed c. You are going to the right at three-fourths of c. My Newtonian expectation is, you should get the speed of the pulse to be one-fourth of c. But you insist it is c. So, what will I say to you? What will I accuse you of doing? I say you should get c/4 and you’re getting c. And you’re finding velocity by finding the distance it travels and dividing by the time so you are jacking up a number like one-fourth c to c itself. So, what could make you do that? Yep?
Student: If I perceive that you’re moving forward, or you’re contracting your length, then you’re going to measure velocity keeping in mind you should have had a greater length to begin with.
Professor Ramamurti Shankar: Yes. So, one option–Let me repeat what he said. I will say your meter sticks are being somehow shorter. When you and I were buddies and were on the same train, we agreed on the meter stick. But now we have gone on the moving train. I will say there is something wrong with your meter stick. Not only something wrong. Specifically, I would say your meter sticks have shrunk. For example, if they have shrunk to one-fourth their size, it is very clear that you would get a velocity of four times what I expect. But there’s another possibility.
Student: Time may be running slow.
Professor Ramamurti Shankar: Your clocks may be running slow. So, you let the light travel for four seconds and you thought it was only one second. That’s why you got four times the answer; or it could be both. But something has to give. And that is why it is an amazing theory. That’s why it is also amazing to me that somebody who was 26 years old would simply follow the consequence of this theory and take it wherever it takes you but it is at the very foundations of space and time that you have to modify. So, even though you restored the relativity principle and brought it back to the front, the price you have to pay is to give up your notions that length is length and time is time. We used to think a meter stick is a meter stick and a clock is a clock. If I have a clock that ticks out one second, you take it on a train, I expect it to be ticking one second, but we’re saying it’s not. So, something has to give in length measurement or time measurement or both. And that’s what we’re going to find out.
So, here’s how you find that out. Let us say that–Maybe I’ll do it all on one blackboard because this is the key to the whole calculation. You remember now, if there is an event here, you call it x′ and I call it x. And according to me, you have traveled the distance ut. So, x′ equals x - ut is what we used to say in the old days. And the converse of that is x = x ′ + ut. But now, we’ll admit the fact that maybe t and t′ are no longer the same. But that’s not all we will do. I will say, whenever you give me a length x′, I just don’t buy it. I take any length you give me and I jack it up by a factor of γ [correction: should have said 1/γ] to get the length according to me. And you will take any answer I give you and you will multiply it by the same factor.
[Note: The preceding discussion and a similar one later on are poorly worded and unclear. The point is simply this: In the old Newtonian days, I would have said you should get for the coordinate of the event x’ = x - ut but now I will multiply this expectation by a factor γ because your meter sticks could differ from mine. It is not known yet if your meter sticks are shorter or longer according to me. The fact that ultimately γ >1, means your meter sticks are shortened according to me. (A person with a shortened meter stick will assign a numerically larger value to a length.) Strangely, it will also mean my meter sticks are shortened according to you. Don’t worry about this yet!]
In other words, we don’t buy our units of length, so if you say it should be x′ + ut′, that’s your formula backwards for me. The coordinate of the event, according to you, in the old days, was x′ + ut. Now, we admit that t and t′ may not be the same. Then we say, but I will not take your lengths, I will multiply them by γ to get the lengths according to me. And you will not take my expectations, but you will multiply it by the same γ; that is the symmetry between the two observers. In other words, if I think your meter sticks are short and γ [correction: should have said 1/γ] is a number less than 1, I’m allowing you to accuse me of having meter sticks which are short. It is very interesting. If I said your meter sticks are short and you say my meter sticks are longer than average, that’s an absolute difference. But we both accuse each other of using shortened meter sticks, and so we use the same factor of γ. We are going to find this γ now.
Student: How do you know that both the distance and time are different?
Professor Ramamurti Shankar: We’ll give it the possibility that they are different and then we will see that they are different. We know something’s wrong with space and something’s wrong with time. So, we’ll not assume that t′ is equal to t. You have to open up the possibility. In the end, it may be that the nature will say t′ is t and something happens to length alone. But we’ll find the answer is more symmetric.
Student: Why do we say that the symmetry is the same?
Professor Ramamurti Shankar: Because that is the symmetry between the two observers. If I want to say your meter sticks are short, why should you concede that? You should be able to accuse me of saying–The only difference between you and me is you are moving to the right and I’m moving to the left. Other than the sign of the velocity, each person says the other person is moving and so we will say that any length you give I’ll discount by a factor of γ. By symmetry, anything I call a length, you will discount by the same factor of γ.
Now, let’s apply this. (x,t) was a certain event, right? Let’s apply it to the following event. You have to follow this very carefully. When you and I crossed, remember that was at the origin of coordinates, x equals to t equals to zero and x ′ equals to t′ equals to zero. At that instant when our origins touched, let us emit a flash of light. Okay, maybe when the origins touched, there’s a spark, the light signal goes out and the light signal is detected here, by a light detector. That second event, detection of the pulse, it has a coordinates (x,t) for me; it has a coordinates (x′, t′) for you. The same event is given different coordinates. We’ve already used the fact that coordinates will be different but we are saying not only will x′ not be equal to x; t′ may not be equal to t either. Okay, but now let’s write down one important condition. What is the relation between x′ and t ′ in this particular pair of events? What is x′ ? It is the location of the light pulse after certain time? So, what’s the relation of the location of the detection of the light pulse and the time t′ ? Yes?
Chapter 6. Deriving the Lorentz Transformation [00:55:34]
Professor Ramamurti Shankar: He is saying x′–Do you guys understand that? Do you agree with that statement? This is not a random event. The second event was the detection of a light pulse. Light pulse left the origin t′ seconds earlier and has come to this point, according to this guy, and the ratio of the distance to the time is the velocity of light. But it is also true that for me, the light went to distance x in a time t. So, that’s the relation of x to t for me also. I’m going to use those two results and combine it with this to find this factor γ and we will do that now. I want you to multiply the left-hand side by the left-hand side and the right-hand side by the right-hand side of that equation.
I hope you understand that in the Galilean days, in the old days–Let’s see what you will say. I will say, x′ is x - ut because the origins have shifted by an amount ut. And you will say x is x′ plus ut with the same time. Now, I’m saying time is different. Not only that, I don’t buy your length. If you expect me to have this length, I say “no.” You exaggerated everything. I’ll scale it down by γ and vice versa. So, if you multiply the left-hand side by the left-hand side, you get xx′, the right-hand side you get γ2 times xx′ plus uxt′ minus ux′ t minus u2tt′. Now, divide everything by xx′. Then, you get 1 = γ2 times 1 + u. If you divide this by xx′, you’ll get t′ over x′. If you divide this by xx′, you’ll get t/x. And here you get u2 times t/x, times t′ over x′.
So, what does that mean? Well, t′ over x′ is 1/c and t/x is also 1/c because of what I wrote here. If you want, let me write this as t′ over x′ equals the c and t over x equals the c. So, they will cancel. And I’ll get 1 = γ2 times 1 - u2 over c2 because this t/x is a 1/c and t′ over x′ is another 1/c. So, that gives us the result that γ is 1 over square root of 1 - u2 over c2. If you plug the γ back in, you will find x′ is x - ut divided by [square root of] 1 - u2 over c2. Now, once you do that, once you got the relation between x′ and x, you can go to the lower equation and solve for t′. I don’t feel like doing that. It’s a simple algebraic equation once you’ve got x′ how to solve for t′. Take the second of the two equations and solve for t′. That detail I won’t fill out, but you will get t′ is t - ux over c2 divided by this. So, I’ve not done every step but I’ve given you all the things you need to do the one step. There are equations up there that relate x and x ′ to t and t′ so if you can get x′ in terms of x and t, the other equation that you solve will give you t′ in terms of x and t. And this is the result. Okay. This guy deserves two boxes because it is the greatest result from relativity; it’s called the Lorentz transformation. And we’ve been able to derive the Lorentz transformation with what little we know. And you can see, you can be a kid in high school and you can do this. There’s no calculus or anything else involved, other than being open to the fact that the velocity of light behaves in this strange way. Yes?
Student: How do you get t/x = c?
Professor Ramamurti Shankar: Why is t/x = c? Oh, of course, you’re quite right. So, you caught the mistake here. t/x is 1 over c and 1/c. I really meant to write x = ct. Yes?
Professor Ramamurti Shankar: No. If you define γ to be the absolute value by which you transfer lengths from you to me, then you can take the positive root.
Professor Ramamurti Shankar: Well, I can also tell you other reasons. Let’s take this formula here. Let’s take the case where the velocity is very small compared to the velocity of light. That u/c is a very small number. This number is almost 1 and I get x′ because x - ut, which I know to be the correct answer at lower velocities. If you pick the minus sign, I’ll get minus of x - ut, that’s not the right answer, not even close to the right answer. At low velocities, if you go to velocity u over c much less than 1, you have got to get back to Galilean transformation. You can see if u over c goes to zero. You can forget all about this factor here. You get x′ is x - ut and here, u over c can be neglected, forget all that, you get t′ equals t. So, this coordinate transformation would reduce to the Galilean transformation if the velocity between me and you is much smaller than the velocity of light. So, the formula really kicks in for velocities comparable to velocity of light. Yes?
Student: [inaudible] What happens if u > c?
Professor Ramamurti Shankar: Well, you start getting crazy answers, right? You can already see that the theory will not admit velocities bigger than the velocity of light. You can already see it in this formula. That tells you that the one single velocity that you wanted to be the same for everybody is also the greatest possible velocity; that no observer can move at this speed with respect to another observer that is equal to what is in excess of the speed of light. So, the speed of light, which came out to be a constant in the beginning of the theory, is also turning out to be the upper limit on possible velocities. That’s the origin of the statement that no observer can travel at a speed bigger than light, but we’ll discuss it more and more. But you have to understand what it is that is being derived, what is the meaning of this formula here. What is it telling you? If I say, this is called the Lorenz transformations, what do they tell you? What are these numbers and what’s their significance? Would you like to try?
Student: Well, one thing that they tell you is if u happens to be greater than c [inaudible]
Professor Ramamurti Shankar: No, no, I don’t mean what happens in the formula in special cases. What is it relating? What is xt and what is x′ t′?
Student: Okay, so x′ would be the distance that the person who’s traveling at the higher speed experiences. And [inaudible] and so for that person, distance is going to seem shorter?
Professor Ramamurti Shankar: No. See, I’m not even telling you to get consequences of the equation. What are the numbers x and t in this equation? And what are the numbers x′ and t′ in this equation?
Student: x and t are distance and time for a person who is in an inertial frame of reference, who is not moving.
Professor Ramamurti Shankar: Right.
Student: And x′ and t′ are distance and time for the person who is moving at the speed of [inaudible]
Professor Ramamurti Shankar: And when you say distance and time, what do you mean, distance and time?
Student: The way that the length of the distance will seem to them that they travel.
Professor Ramamurti Shankar: But what is happening at xt; it’s the coordinates of what?
Student: Of their location [inaudible]
Professor Ramamurti Shankar: They are located at zero, zero. Right? What’s happening at x and t? Yes?
Student: It’s observing the event.
Professor Ramamurti Shankar: It’s the event. The key I was looking for is an event. You’ve got to understand what the formula is connecting. Things are happening in space and in time, right? Something happens here. That something has a spatial coordinate and a time coordinate, according to two observers. The observers originally had their origins and their clocks coincide when they passed; that’s how they’re related. And one is moving to the right at speed u. Then, the claim is that if one event had a coordinate x and t for one person, for the other person moving to the right at speed u, the same event would have coordinates x′ and t′ and the relation between x and t and x′ and t′ is this. Yes?
Student: The two observers [inaudible] don’t they observe different laws and [inaudible]
Professor Ramamurti Shankar: No. The fact that an event has different coordinates doesn’t mean that you are observing different laws. For example, let’s take that fire extinguisher. We look at it, it’s obeying F = ma, right? The coordinates of the fire extinguisher with me as the origin is quite different from you as the origin.
Professor Ramamurti Shankar: You mean in these new equations? Yes, in these new equations, F = ma will not work; that’s correct.
Student: They are inertial references, even though they are moving?
Professor Ramamurti Shankar: Ah. Yes. So, the point is the laws of Newton themselves have to be modified. F = ma will be modified in a certain way but the new modified laws will reduce to F = ma at low velocities, which is why in the old days it looked like F = ma. But there will be new laws, but they will also have the property that when I measure them I’ll get the laws that will agree with what you measure. Yes?
Student: Does individual values for time or distance still have to agree?
Professor Ramamurti Shankar: The coordinates of an event will differ from person to person. That’s not the same as saying the laws as deduced will be different. For example, there are two stars which are attracted to each other by gravitation and they are orbiting around their common center of mass. If I see them, I will find that they obey the law of gravitation with m1m2 over r2 where r is the distance between the points and the acceleration is whatever I think it is. You can go on a rocket and look at the same two stars. They will be in a more complicated motion, maybe the whole system will be drifting a little to you, but their acceleration will be the same as what I get and the force between them will also be the same as what I get and the laws that you would deduce by looking at that star would be the same laws that I would deduce. So, that’s a difference between the laws being the same and the coordinates being the same.
No one said x′ and x are the same in that equation there. They are different. We are looking at it from different vantage points. But the fact is that force is equal to mass times acceleration is the same for the two people. Okay. The laws will be the same but things won’t look the same. For example, you can stand on your head, don’t even have to go to another frame of reference; you can stand on your head, your z coordinate is the minus of my z coordinate; to every point I give a z, you will give a minus z. But the world, even though you are a little messed up and want to stand on your head, you have every right to do that and you will find that F = ma. So the point is, the way we see events may depend on their origin or coordinates, but the laws we deduce are to be distinguished from the perception that we have. Okay? For example, if I’m on the ground, I send a piece of chalk, it goes up and it goes down. If you see me from a moving train, you would think it went on a parabola. So, no one says the chalk will go up and down for you. For you, it will go like this but its motion will still obey F = ma, is what I’m saying. That’s all you really mean by saying things look the same.
So, what you have to understand is that Lorentz transformations are the way to relate a pair of events, given events. Here’s a simple example. If you live in the xy plane, there’s a point here. It’s not an event, it’s simply a point in the xy plane. You measure it this way and that way and you call it the coordinates. If somebody else picks a different coordinate system with an angle θ, that person will say that’s x′ and that is y′ and x′ and y′ are not the same. I remind you, x′ is x cos> θ [delete “minus y”] plus y sin θ, etc., and y′ is something else. θ is the angle between the two observers. So, the point is the point. It certainly looks different to the two people, but the same point has two coordinates. Similarly, the same event, like the collision of two cars, will have different events for different people. That’s not the new part. The new part is that the rules for connecting xt to x’t’, is quite different from the Galilean rules, new rules. It’s what you guys have to understand.
And finally, why did Einstein get the credit for turning the world into four dimensions instead of three? After all, x and t were present there, too. The point is, t′ is always equal to t no matter how you move, whereas in the Einstein theory, x and t get scrambled into x′ and t ′ just the way x and y get scrambled into each other when you rotate your axis. So, to have time as another variable that doesn’t transform at all is not the same as making it into a coordinate. The four-dimensional world of Einstein is four-dimensional because space and time mix with each other when you change your frame of reference. That’s what makes t now a coordinate as previously it was something the same for all people. All right, I’ll see your guys on Wednesday.
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