12. Introduction to Relativity
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00:00
Ring strainWatercraft rowingLightTrainFahrgeschwindigkeitInitiator <Steuerungstechnik>ForceInertiaPlane (tool)Combined cycleYearReference workBallpoint penAutumnShip breakingScrew capRocketGalileo (satellite navigation)AccelerationHot isostatic pressingSensorHot workingSpantFictitious forceGround (electricity)Direct currentSchwimmbadreaktorCar seatEngine knockingParking meterWeather frontHourScrewdriverSpare partAstronautBulk modulusCardboard (paper product)Effects unitKopfstützeNewton's laws of motionSpeed of lightGround stationCylinder headLecture/Conference
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Transcript: English(auto-generated)
00:00
So, let's begin now. First of all, I'm assuming all of you have some idea of what special relativity means. There are two theories of relativity. One is special theory and one is a general theory. General theory is something we won't do in any detail. Special theory is something we will do in reasonable detail.
00:20
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? What do you think it's about? Okay, it's about relative speed and two reference systems.
00:43
I'll come to you, then I'll come to you. Yes? Okay, I will take the last row
01:01
there. Yes. Okay, so what I've heard so far is that the laws of physics are the same for two people who are both inertial frames of reference and the velocity of light is a constant. Right. That's certainly the way we
01:22
understand the relativity theory, but it's a very old one. It's been going on long before Einstein came. There is a theory of relativity at the time of Newton, and that's where I want to begin. So, relativity is not a new idea at all. It's an old one. And the old idea can be
01:43
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.
02:02
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
02:24
outside. That's not because you're traveling through some parts of New Jersey. You don't want to look outside. You don't want to look outside for this particular experiment. You get into the train, you settle down, and you explore the world
02:40
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.
03:00
When you're 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 those speed,
03:22
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 an Amtrak and I know this train is not going anywhere.
03:40
That kind of sociological reasons, by the way, there are many of them, you cannot invoke. You can only say, I'm in this train, is anything different? And the claim is that nothing will be different. You just will not know you're moving. Now, if the train picks up speed or slows down, you will know right away.
04:03
If it picks up speed or accelerates, you will find yourself pushed against the back of the seat. If the driver slams on the brake, you will slam in the front of the seat in front of you. No one is saying that when a motion is accelerated, you will not know. Accelerated motion can be detected in a closed train without looking outside.
04:23
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're called an
04:44
observer. One of the laws you want is, if you leave something, you 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.
05:03
We're not interested in that. You started out as an observer for whom the laws of Newton work, the law of inertia works, F is equal to 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,
05:22
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.
05:43
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 looked up blind and looked through, you would see the other train, and there's another passenger
06:02
on the other train, and you look at each other, and 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
06:21
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
06:42
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
07:00
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.
07:24
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's 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
07:43
you're the one who is barfing up and throwing up and slamming your head on the wall. Nothing's happened to the other person. You cannot say, I'm still in the same frame, you're going the opposite way. If you're going the opposite way, why am I throwing up? So, but if you're in a rocket, the rocket's taking off.
08:00
The G forces are enormous. Many times you wait. There's the astronauts who are going through the discomfort. At that time, they cannot say, we are at rest and everyone's going the opposite way because no one else is in danger, but you are. So, accelerated motion will produce effects. You cannot talk your way out of that, but uniform velocity
08:21
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 he's moving and you're 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
08:44
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 as some sociological factors.
09:04
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 landscape, cows and trees are made to move the opposite way. Not very likely, but that's because we know in practice no one's going to
09:21
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 take, 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.
09:40
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, in reality, if the two trains were at rest,
10:00
let's imagine my train got accelerated. So, during the time it was accelerated, I would know. But if I was sleeping at that time, I don't know. And when I wake up and the acceleration is gone, the velocity is constant, that's when I say I just cannot tell. All right, now let's show once
10:21
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. And I hope you understand that all the mechanical things you see in the world around you just come from F equals m a. We have seen projectiles and
10:41
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.
11:00
Here is the x-axis and here is my frame of reference. This is my x-axis. Let's call this the origin. So, the frame goes to negative and positive x-values. Pick some object sitting at the point x.
11:22
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,
11:41
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
12:01
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 that 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, everyone understands what
12:21
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'll give it an x and we'll give it a t. Now, that's me and I'm going to give my frame of reference the name S. Turns out S is not just based on my name.
12:43
This is a canonical name for two observers. One is called S and one is called S prime. So, S prime, let's say, is you. So, your frame of reference is going to be taken to be sliding relative to mine. So, I don't know how to draw
13:01
this. 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.
13:28
So, 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.
13:43
You cross me, and then a little later, you are somewhere there. So, that's your y prime axis. That's your origin. In the same event, you, say, has a coordinate x prime. We arrange it so that when you zoom past me,
14:03
you set your clock to zero, and I set my clock to zero. When you want to set the clock to zero, it's 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, right when you pass me, we'll click our stopwatches, and we will set that time to
14:20
zero. So, here's an event, u and I crossed. What are the coordinates for that event? According to me, that event occurred at x equal to zero, and the time was chosen to be zero. According to you, your origin was also on top
14:41
of my origin, so x prime was zero, and the time is just the time. Everybody has a single time. The second time, that time is called zero. That is one event. We made a coordinate of that 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
15:01
zero, took place at your origin, that's why your x prime 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
15:22
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 got agents all over the x axis, who are my eyes and ears. They're looking out for me. So, even though I am here, there's a firecracker
15:40
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 equal to 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,
16:03
taking notes on what's happening. We'll pull our information later, but we know this explosion took place here. I will simplify it by saying, I know an explosion took place here at location x at time t. So, this is our crossing.
16:21
This event is when we crossed. Then, there is the firecracker. For firecracker, I have to give some events. I say it took place at location x at time t. What do you say?
16:41
You measure the distance from your origin, you call it some x prime, 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 prime and x?
17:02
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 prime 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.
17:24
So, the distance from your origin for this event, I maintain, is x prime, is x – ut. Again, I want all of you to follow everything. These are all simple notions. Our origins coincided at zero
17:45
time, but the event occurs at time t. Therefore, in a sense, you're rushing towards the event, you've gone a distance ut. Therefore, the distance from your origin to that event will be less than mine by this amount ut.
18:02
This is the law of transformation of coordinates in Newtonian mechanics. If you have an event which exists, if you want formally, you can define a time t prime for the primed observer. It goes without saying that t
18:22
prime and t are the same. There is no notion of time for me and time for you. There's universal time in Newtonian mechanics. It just runs. We can call something a zero. Once we have agreed, if you say you and I met at t equal to zero, an explosion took place five seconds after our meeting. So, the time difference between two events is the same for all
18:40
people. It's going to be five seconds after a meeting for me, and it's going to be five seconds after the meeting for you. The time difference between two events is the same for all people. This is called a Galilean transformation. What are the consequences of
19:02
Galilean transformation? Well, let's look at the fact that x prime is x – ut. Remember, everything is varying with time. So, x prime is a function of time, and x is a function of time, if you're watching a moving
19:22
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
19:40
dx dt is the velocity. Let's just call it a bullet. So, I'm going to call it a bullet according to s. Then, w is a standard name, is the velocity of the bullet according to s prime. So, what I've done is I
20:05
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.
20:23
Then, its velocity, or any time, is dx prime dt according to u. That is the dx dt according to me minus the derivative of this, which is u. Now, does that make sense? They should agree with common
20:42
sense. For example, if that bullet's 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.
21:04
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 minus 0 because u is a
21:23
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
21:40
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 I added will drop out of the difference. Or if the body has an acceleration, we'll both get the same answer for the acceleration.
22:00
So, that is the common acceleration, a. So, if you like, a prime 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 constant speed. All right, so let's look at F equals ma,
22:21
which is m d2x over dt square is equal to some force on the body. And you look at the body and you say d2x prime over dt square is the force on your body. First, I want to convince you,
22:41
we've already seen that the left-hand sides are equal because the acceleration is 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.
23:02
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 minus x2, force on 1 due to 2.
23:23
And the force on 2 due to 1 will be minus of 1 over x1 minus x2. The real forces are R1, the 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 m1 d2x1 over
23:47
dt2 is 1 over x1 minus x2. And m2 d2x2 over dt2 is minus 1 over x1 minus. I have forgotten constants like
24:02
g and m1 and m2. They don't matter for this purpose. So, here are two bodies. Two bodies, they feel a force for each other and I've discovered what the force is. It's 1 over x1 minus x2. I don't care if it's 1 over x1 minus x2 or x1 minus x2 squared. That's not important. What's important is it depends on x1 minus x2.
24:23
You come along and you study the same two masses. What will you say is happening? You will say m1 d2x1 prime over dt squared is equal to 1 over x1 prime minus x2
24:40
prime. 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 d x1 prime over
25:01
dt squared. 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 prime minus x2 prime. Now, 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
25:22
you and me, because x1 prime is x minus ut, x2 prime is x2 minus ut. Take the difference, the difference between the location of the particle 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 equals ma,
25:43
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 see it differently.
26:00
If I woke up from my nap and I'm now in the moving train and I examine the world around me, I'm going to get the same F equals ma, because as seen by a person on the ground, the masses obey F equals ma. I'm in this moving train now, but I have the same acceleration for each mass and I have the same force.
26:21
So, if you want, I'll complete the second equation, m2 d2x2 prime over dt squared will be minus 1 over x1 prime minus x2 prime. Now, if this is a little difficult, we should talk about this. I'm telling you that if I
26:41
deduce F equals 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 prime is same as a, and the force will also be the same because the force depends
27:01
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
27:23
you observe by going on trains and whatnot, you can actually show that this is the reason everything looks the same. In other words, if the train was at rest in the platform and you and I were comparing notes and we both find F equals ma, I go to sleep and I'm waking up in the train going at a
27:40
constant speed. If you can look through the window and look at the objects in my train, you will say they obey F equals ma because nothing has happened to you, but you will predict I will also say F equals 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 will also think it's one meter.
28:00
If the force is one 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 prove an F equals ma is valid, it follows that every mechanical phenomenon will behave the same way. That's the reason things behave the same way. Yes?
28:22
Pardon me? You mean if the rule failed in the other frame? Yeah, suppose hypothetically that happened. Then it would mean that when you wake up in the train, you will look at the world
28:41
around you, it will look different. Because F is no longer ma, you will conclude when I went to sleep it was F equals ma, when I got up F is not equal to 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, other trains going backwards,
29:03
you cannot no longer say, you're going the other way, I'm not moving, because the other person will say, hey, F equals 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 equals ma worked will say they are not moving,
29:23
and for you it doesn't work, so you will have to concede you're 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 addressed and you are moving, because for me,
29:40
F equals ma works and for you it doesn't. Well, that's not what happens. In real life, you find it works for both and we say either of us can maintain we're 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,
30:04
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.
30:21
You don't have to know what electric and magnetic fields are right now. There are some measurable phenomena. They are like waves and the waves had a certain velocity that Maxwell calculated. That velocity is this famous number, 3 times 10 to the 8 meters per second. And the question was, for whom is this the velocity?
30:44
For example, you can do a calculation of waves on a string, something we'll be able to do in our course. The 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 a velocity as seen by a person for whom the string is at rest.
31:01
Or if I calculate the waves of sound in this room, and 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
31:22
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're carrying the medium with us. So, people wanted to know, what is the medium which carries the waves of light,
31:43
electromagnetic waves? First of all, the medium is everywhere because, well, how do we know it's everywhere? Anybody tell me? Yes? Right, it travels through the vacuum of space.
32:02
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 denser the medium, more rapidly signals travel in most of the things that we know. For example, when we look at waves in sound, or when we look at sound waves in a solid or an
32:22
iron, you find if it's a very dense material, 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 are moving through this medium for years and years and years, and not slowing down.
32:41
It's a very peculiar medium. But it has to be everywhere, so we are all immersed in this medium because we're 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?
33:00
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. 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
33:21
the medium by Maxwell's theory. So, here is 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
33:41
from c to c minus v. In particular, suppose the waves are traveling to the right. We expect in ether, and let me draw it this way, the Earth is going at this instant at a speed v. We expect the speed to be c minus v, because part of the
34:00
speed is neutralized because you're going along with the waves. You will see a slower velocity. So, Mr. Michelson and his assistant morally, they did the experiment. And they got the answer equal to this. What does that mean?
34:24
What does that mean? Yes? No, no, but you cannot jump to that right now. If you are following Newtonian physics, your expectation is it should be c minus v.
34:41
Yes? Yes? Well, that's not so fast, but it certainly means the following. Well, that's a simpler answer than that. Yes? With what speed? Student 1 and Student 2. Zero. Zero. Because you don't have to look, you guys are ready to overthrow everything because you
35:02
know the answer. Right? Well, you've got to put yourself in the place of somebody in early 1900. There's no reason to overthrow anything. The answer is you're going at a speed zero. Of course, you realize it's incredibly fortunate that on the one day Michelson wants to
35:22
do the experiment, we happen to be at rest with respect to the aether. Fine. But we know that luck is not going to last forever because you're going around the Sun. On a particular day, maybe, that velocity was such that on that day the Earth was at rest with respect to the aether,
35:42
it's clear that six months later when you're going the other direction, you cannot also be at rest with respect to the aether. 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're moving around the Sun for sure. Yes?
36:03
So, people try other solutions, but it's simply a fact that when you move one way or six months later the opposite way, you get the same answer C. So, one possibility is you don't want to, look, don't be ready to do
36:20
revolutions. You try to avoid it. So, one answer is look at the speed of sound. You and I talk to each other then, and we talk to each other six months from now, we get the same speed of sound. The speed of sound is published in textbooks, right? 700 and something miles per hour, how come that doesn't change from day to day?
36:41
Anybody here this side tell me why the speed of sound doesn't change from day to day, even though we are moving? No one here can guess? No? We are moving even six months from now. We get the same speed of sound. In this room, when I talk to you,
37:01
does it matter what time of year it is? 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
37:23
way it carries air with it. Then, it's not an accident you're addressed 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.
37:41
So, you cannot take the ether with you. You cannot leave it behind. 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 over 2.
38:00
I expect you to get a speed c over 2, but you keep getting c. You go at three-fourth 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
38:21
but a light beam, suppose for me it's traveling at a speed c, and you're traveling to the right at speed u, you should get c minus 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
38:40
doing different models of ether, but none of which worked. And nobody knew why light is behaving in this peculiar fashion. So, that's when Einstein came and said, I know why light is behaving in this peculiar fashion. Behaving in this way, because if it didn't behave in this way,
39:00
the speed of light depended on how fast I'm moving, the speed of light. Then, when I wake up in this train, all I have to do is measure the speed of light. Originally, I got some number, and now I get a different number. The difference will give me the speed of the train. So, it would have been possible to detect the velocity of the train without
39:21
looking outside just by doing an experiment with light. So, even though mechanical laws involving F equals ma are the same, laws of electricity and magnetism would be such that somehow they will betray your velocity. So, that would mean uniform velocity does make an observable change,
39:42
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.
40:02
Because to Einstein, it was very obvious that nature will 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
40:22
one. That's a very brave postulate, because it even applies to biological phenomena, both of 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,
40:44
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. He believes all natural phenomena will obey the same principle, which says all observers are uniform relative
41:01
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 is 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
41:22
certain beauty and a certain uniformity across all of natural phenomena. That is a faith we have. It's not a religious issue, otherwise I won't bring it up in the classroom, but it's certainly the credo of all scientists, or at least all physicists, that there is some elegance in the laws of nature.
41:41
And we put a lot of money on that faith, that the laws of nature will do this or 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.
42:02
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
42:22
notion that all laws of physics should obey the same principle of relativity. So, the 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. Electrical phenomena would not.
42:43
That cannot be the case. You have a question? Yes? Yes, because in the case of speed of sound, you can't take the medium with you, so there is no such experiment you can perform.
43:02
See, in the train, if you could carry the ether with you, it's no surprise you get the same answer. But we know we cannot carry the medium with us. That comes from 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. So, you've got to know where
43:20
the postulates came from. Now, postulate number one 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. By equivalent means each one
43:45
of them is as privileged as any other one to discover the relative laws of nature. So, 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
44:03
other in relative motion, you have as much right to claim you're at rest and I'm moving, and I have as much right to claim that I'm 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 motion. As I said, non-uniform motion creates effects which can destroy
44:22
me and not destroy you. So, no one's trying to talk your way out of acceleration, but 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,
44:42
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,
45:01
which 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
45:24
a moving rocket, it doesn't matter. If the light beam is seen in a moving rocket, it doesn't matter. All people will get the same answer for the velocity of light. Yep, no, that's why it's a postulate.
45:40
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's only one velocity that can have the same answer for all people, that velocity is the velocity of light.
46:02
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 the big problem because he has said why light
46:23
behaves this way. Light behaves this way because it's part of the big conspiracy to hide uniform motion. But you will see that you have made a terrible bargain now, because once you take these two postulates, you have restored the relativity principle to all phenomena.
46:42
Okay, you've gone beyond mechanical phenomena to electromagnetic phenomena. 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's a car going 200 miles an hour, according to me. You get into your own car and
47:03
follow that car at 50 miles an hour. You should get 150, but you keep getting 200. At least it may not be true for cars going at the speeds I mentioned, but if finally you're 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
47:22
formula I wrote down, w equals v minus u. When you put v equal to c, w has got to come out to be c, and that's not 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
47:45
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.
48:01
You are going to the right at 3 fourths of c. My Newtonian expectation, you should get the speed of the pulse to be 1 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 over 4 and you're getting c. And you're finding velocity by
48:22
finding the distance it travels and dividing by the time, so you're jacking up a number like 1 fourth to c itself. So, what could make you do that? Yep, so one option, and let me repeat what he
48:51
said. I will say your meter sticks are being somehow shorter. When you and I were buddies and were in the same train, we agreed on the meter stick.
49:01
Now, we have gone into the moving train. I will say that there's something wrong with your meter stick. Not only something wrong, specifically I would say meter sticks have shrunk. For example, if they are shrunk to 1 fourth their size, it's very clear that you will get the velocity before times what I expect. But there's another possibility. Yes?
49:21
Your clocks may be running slow, so you let the light travel for 4 seconds and you thought it was only 1 second. That's why you got 4 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's also amazing to me that somebody who is 26
49:41
years old would simply follow the consequence of this theory and take it wherever it takes you, but 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
50:01
stick is a meter stick and a clock is a clock. If I have a clock that ticks out 1 second, and you take it on a train, I expect it to be ticking 1 second. But we're saying it's not. Or something has to give in length, measurement, or time measurement, or both. And that's what we're going to find out. So, here is how you find that out.
50:20
So, let us say that maybe I'll do it all in one black board because this is the key to the whole calculation. You remember now, if there is an event here, you call it x prime and I call it x.
50:44
And according to me, you have traveled a distance ut. So, x prime equals x minus ut is what we used to say in the old days.
51:04
And the converse of that is x equals x prime plus ut. But now, we'll admit the fact that maybe a t and t prime are no longer the same. But that's not all we do. I will say, whenever you give me
51:24
a length x prime, I just don't buy it. I take any length you give me and I jack it up by a factor γ to get the length according to me, the coordinate according to me. And you will take any answer I give you and you multiply it by
51:43
the same factor. In other words, we don't buy our units of length. So, if you say it should be x prime plus ut prime, right, that's your formula backwards for me. The coordinate of an event according to you, in the old days, was x prime plus ut.
52:02
Now, we admit that t and t prime 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. Now, that is the symmetry between the two observers.
52:21
In other words, if I think your meter sticks are short and γ is some number less than one, I'm allowing you to accuse me of having meter sticks which are short, which 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 short and meter
52:41
sticks, and so we use the same factor γ. We're going to find this γ now. Yes, we'll give it the possibility they are different. Then we will see that they are different.
53:00
We know something is wrong with space and something is wrong with time, so we'll not assume that t prime equal to t. We have to open up the possibility. In the end, it may be that nature will say t prime is t and something happens to length alone. But we'll find the answer is more symmetric. Yes, because that is the
53:25
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're moving to the right and I'm moving to the left. Other than the sign of the velocity, each person says the
53:43
other person's moving. And so, we will say that any length you give, I'll discount by a factor γ. By symmetry, anything that I call a length, you will discount by the same factor γ. Now, let's apply this. xt was a certain event, right? Let's apply it to the following event. You've got to follow this very
54:00
carefully. When you and I crossed, remember that was at the origin of coordinates, x equal to t equal to 0 and x prime equal to t prime equal to 0. At that instant, when our origin starts, let us emit a flash of light. Okay, maybe the origin starts, there's a spark,
54:21
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 coordinate xt for me, it has a coordinate
54:43
x prime t prime for you. The same event is given different coordinates. We are already used to the fact that coordinates will be different, but we are saying not only will x prime not be equal to x, but t prime may not be equal to t either.
55:04
Okay, but now let's write down one important condition. What is the relation between x prime and t prime in this particular pair of events? What is x prime? It's the location of the light pulse after a certain
55:21
time. So, what's the relation of the location of the detection of the light pulse and the time t prime? Yep, he's saying x prime, do you guys understand that? Do you agree with that
55:42
statement? This is not a random event. The second event was the detection of a light pulse. The light pulse left the origin t prime 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's also true that for
56:01
me, the light went a distance x and 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.
56:22
So, 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 prime is x minus ut because the origins are shifted by an amount
56:40
of time. So, x is x prime plus ut, and you will say x is x prime 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,
57:00
you get x x prime. The right-hand side, you get γ square times x x prime plus u x t prime minus u x prime t minus u squared t t prime.
57:23
Now, divide everything by x x prime. Then, you get 1 equals γ square times 1 plus u. If you divide this by x x prime, you'll get t prime over x prime. If you divide this by x x prime, you will get t over x. Then here, you get u squared
57:45
times t over x times t prime over x prime. So, what does that mean? Well, t prime over x prime is
58:03
1 over c, and t over x is also 1 over c, because of what I wrote here. If you want, let me write this as t prime over x prime equal to c, and t over x equal to c. So, they will cancel. And I will get 1 equals
58:23
γ squared times 1 minus u squared over c squared, because this t over x is a 1 over c and t prime over x prime is another 1 over c. So, that gives us the result that γ is 1 over square root of 1 minus u squared over
58:47
c squared. If you plug that γ back in, you will find x prime is x minus ut divided by 1 minus u squared over c squared.
59:15
Now, once you do that, once you got the relation between x prime and x, you can go to the lower
59:20
equation and solve for t prime. I don't feel like doing that. It's a simple algebraic equation. Once you got x prime, how to solve for t prime? Take the second of the two equations and solve for t prime. That detail I won't fill out, but you will get t prime is t minus ux over c squared divided by this.
59:44
So, I've not done every step, but I've given you all the things you need to do the one step. I mean, there are equations up there that relate x and x prime to t and t prime. So, if you can get x prime in terms of x and t, the other equation, if you solve it, will give you t prime in terms of x and t.
01:00:00
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.
01:00:23
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 velocity of light behaves in a strange way. Yes? Student
01:00:41
x equal to c. Oh, of course, you're quite right. So, you caught the mistake here. T over x is 1 over c and 1 over c. So, I really meant to write x equals ct. Yes?
01:01:02
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. 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,
01:01:23
so that u over c is a very small number. This number is almost 1 and I get x prime equal to x minus ut, which I know to be the correct answer at low velocities. If you pick the minus sign, I'll get x prime is minus of x minus ut. That's not the right answer, not even close to the right
01:01:40
answer. At low velocities, if you go to velocity u over c much less than 1, you've got to get back the Galilean transformation. You can see if u over c goes to 0, you can forget all about this factor here. You get x prime is x minus ut, and here, u over c can be neglected.
01:02:03
Forget all that, you get t prime equals t. So, this coordinate transformation will reduce to a Galilean transformation if the velocity between me and you is much smaller than the velocity of light. So, the formula really kicks in for velocity is comparable
01:02:22
to the velocity of light. Yep, well, you start getting crazy answers, right? So, you don't, 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
01:02:43
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 a speed with respect to another observer that is equal to or in excess of the speed of light. So, the speed of light, which came out to be a
01:03:01
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. So, you have to understand what it is that is being derived. Now, what is the meaning of this formula here?
01:03:22
What is it telling you? If I say, okay, these are called the Lorentz transformations, what do they tell you? What are these numbers and what's the significance? Would you like to try? Yes? Yeah?
01:03:43
No, no, I don't mean what happens in the formula in special cases. What is it relating? What is xt and what is xt'? I don't even want to know.
01:04:14
See, I'm not even telling you to get the consequence of the equation. What are the numbers x and t in this equation? And what are the numbers
01:04:23
x' and t' in this equation? That's all I'm saying.
01:04:41
And when you say distance and time, what do you mean distance and time? But what is happening at xt? It's the coordinates of what? They are located at 0,0, right?
01:05:01
What's happening at x and t? Yes, it's the event. The key I was looking for is an event. So, here is, you've got to understand what the formula is connecting. Things are happening in space and in time, right? Something happens here.
01:05:21
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 were 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,
01:05:43
for the other person moving to the right at speed u, the same event will have coordinates x' and t'. And the relation between x and t and x' and t' is this. Yeah? No, the fact that an event has
01:06:10
different coordinates doesn't mean that we are observing different laws. For example, let's take that fire extinguisher. We look at it, it's obeying F equals ma,
01:06:21
right? The coordinate of the fire extinguisher with me as the origin is quite different from you as the origin. You mean in these new
01:06:44
equations? Yeah, the new equations, F equals ma will not work. That's correct. Yes, so the point is the laws of Newton themselves have to be modified. F equals ma will be modified
01:07:02
in a certain way. But the new modified laws will reduce to F equals ma at low velocities, which is why in the old days it looked like F was equal to 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?
01:07:28
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're
01:07:41
orbiting around the common center of mass. If I see them, I will find they obey the law of gravitation with m1m2 over r squared, 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 relative to you.
01:08:00
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 will deduce by looking at that star will be the same laws that I will deduce. So, that's the 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.
01:08:21
They are different. We are looking at it from different vantage points. But the fact that force is equal to mass times acceleration is the same for the two people. The laws will be the same, but things won't look the same. For example, you can stand on your head. You don't even have to go to another frame of reference.
01:08:40
You stand on your head. Your z-coordinate is a minus of my z-coordinate. For every event I give a z, you will give minus a 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 because you will find F equals ma. So, the point is the way we see events may depend on the origin of coordinates, but the laws we deduce have
01:09:02
to be distinguished from the perception that we have. For example, if I am on the ground, I send a piece of chalk, it goes up and goes down. If you see me from a moving train, you will 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 equals ma,
01:09:23
is what I'm saying. That's all you really mean by saying things look the same. So, what you have to understand is the Lorentz transformations are the way to relate a pair of events, a given event. So, here's a simpler example. If you live in the xy plane,
01:09:40
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 coordinate system at an angle θ, that person will say, that's x prime and that is y prime. And x prime and y prime are not the same. I remind you x prime is
01:10:02
x cosine θ minus y plus y sine θ, etc., and y prime 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
01:10:23
for different people. That's not the new part. The new part is that the rules for connecting xt to x prime, t prime, 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
01:10:41
world into four dimensions instead of three? After all, x and t were present there too. The point is, t prime is always equal to t, no matter how you move. Whereas in the Einstein theory, x and t get scrambled into x prime and t prime, just the way x and y get scrambled into each other when you rotate your axis.
01:11:01
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, whereas previously it was something the same for all people.