WSU: Space, Time, and Einstein with Brian Greene

In the spring of 1905 a storm broke out in Albert Einstein's mind, at least that is how Einstein himself described it later in his life, and the result of that storm of thought is the special theory of relativity, a theory that completely transforms our understanding of

space

. and

time

, matter and energy, I mean, Einstein discovered that, contrary to experience, moving clocks keep

time

at a slower rate, he discovered that moving objects contract along their direction of movement, he discovered that clocks that a group of individuals say are in motion. Synchronization with each other, he discovered that if someone moves with respect to those clocks, they would say that they are not synchronized and of course he also found the most famous equation in all of physics, e equals m c squared, establishing this deep hidden connection between mass and energy. you should tell yourself if there is such an unexpected nature in reality that we have overlooked in our everyday experience, why we have missed it, that is, why we are not aware of the special relativity in our bones and the answer is when Look at the universe, we recognize that there are a huge variety of scales that make up reality and we humans only have access to a very small part of that whole, so to give you an idea, let's look at an axis, an axis of length, and if you look at the scales that are there in length atoms 10 to minus 10 meters viruses 10 to minus 8 meters red blood cells 10 to minus 6. single-celled organisms there we humans are 2 meters away the earth 10 to 5 meters from the solar system 10 to the 13 galaxy and the observable universe itself, a huge range of scales when it comes to length and we humans really only have direct access through experience to a small part of that and that's just the axis in length, imagine that we look. on the mass axis there we will also find a wide variety of scales, so if we go back and look at atoms, they weigh from 10 to minus 26 kilograms and go down to red blood cells from 10 to minus 15 humans, well, it depends on who you are.

We're talking, but you know, about 100 kilograms of the solar system in the entire universe, the observable part of 10 to 52 kilograms, a huge variety of mass scales, we humans only have direct access to a small part of it, an axis More to look at is the speed axis, so humans walk around the world at certain ordinary everyday speeds, sometimes you get on airplanes, but there is a wide variety of speeds, the growth of human hair, which is quite small, the human speed, typically the

space

shuttle 10 at 3 meters per second speed of light, that's a number that will come up a lot in our discussion.

From 10 to 8 meters per second, approximately, the point is in this spectrum of all possibilities in length, mass and also in speed, humans occupy a small part, so our experience, the experience that has given us our intuition, it's built from a very limited sense of what's really out there, so our intuition, which actually comes in a sense from evolution, so we evolved in the jungle and Our intuition developed so that we can survive. The survival value of understanding the environment is what matters. We humans only have access to a small part of the totality of what is out there and therefore it would be surprising if what we have experienced actually does. tell us about physics at all possible scales of length, speed and mass and it turns out that, in fact, when you look at the extremes of mass, length or speed, the world operates, the universe operates in ways that we are not used to if you are looking at extremes let's say a very small size the new physics that comes into play is called quantum mechanics if you are looking at extremes of enormous mass the new physics that comes into play is the general theory of relativity if you are looking at extreme speeds, how the universe behaves at very, very high speeds, then it is the special theory of relativity that comes into play, that is what we are going to discuss, so, in a nutshell, the whole discussion we are going to have here is focused on how the universe at very high speeds, that is the special theory of relativity, since speed is the fundamental core of what drives the special theory of relativity.

Let's start at the beginning and ask the most basic question of all, which is what is speed? We all know the answer to that, but let's all be on the same page, so if that car has a speed of sixty miles per hour, we all know what that means if we look at how far it has gone divided by how long it takes. to get there, let's say it's a one hour trip, if you are going 100 kilometers per hour, we know that you will have traveled 100 kilometers in that hour, that is what speed is, in essence, speed is nothing more than distance that runs through an object divided by the duration. expressed in that language speed may seem like I don't know a kind of boring concept a pedestrian concept why worry about speed the answer is clear if we recognize that distance is a measure that has to do with space duration is a measure that It has to do with time, so if we discover, as we will find in the future, that velocity has unusual characteristics when the velocities involved become very large, close to the speed of light, what we will really learn is that the Space and time have strange and strange characteristics. characteristics that is what we are looking for, well, that is where we are heading, let's start by thinking first about the non-strange characteristics of speed, the characteristics that we all have in our intuition, so what are the basic characteristics of speed?

Well, first of all, speed is a concept that even before Einstein was met is a relative concept, what do I mean by that? Well, imagine we look at this car, we say it's going 100 kilometers per hour, what we have to say is 100 kilometers per hour relative to the road, why else? that road itself is moving, let's say you are on a boat, the boat is moving, then the speed of the car relative to the water is not a hundred kilometers per hour and if we zoom out and look at that car on the surface of the earth and we realize that the Earth itself is orbiting the Sun with respect to the frame of reference.

Our perspective at the moment is that the car is executing quite a complicated movement, not just at 100 kilometers per hour, so it is always vital when we talk. about speed to recognize that you can only frame the idea of ​​speed for ordinary objects that we encounter, since the object has this or that speed relative to this or that object, it is necessary to specify the reference so that the speed that you are specifying has any meaning okay, so that is a very basic characteristic of speed, which is relative, another basic characteristic of speed is that it is additive and subtractive, so what do I mean by that?

Well, let's imagine that we have two characters who are playing a game of catch, say, a soccer ball, George and Gracie, two characters that we will often encounter in our discussions and imagine that they are throwing that soccer ball from a back and forth at, say, five meters per second, that's very well understood, but let's imagine that they go out to play ball another day and Gracie is surprised to see that George has a hand grenade, now she doesn't like hand grenade and that's why she runs away when he throws it because she knows that by running away from the hand grenade, she can change the speed at which he approaches it, she can subtract her own speed from the speed of the approaching grenade and from that way make it approach her, not to mention at the initial speed at which it was thrown, which for example could be like the soccer ball at five meters per second, on the other hand, if she flees at three meters per second, she knows that now the grenade will approach her more slowly at two meters per second and that is good when it comes to hand grenades, similarly it is the case that if an observer like Gracie should not run away but say run towards an object that is being thrown at her, the speed at which she approaches the object will be added to the speed at which she is approaching her, so if it was thrown at five and she is running at three, well, we all know that means which will approach it at eight meters per second.

Speed ​​is additive and subtractive. You can change the speed at which an object comes towards you, either by running towards it or away from it. Now let me. quickly mention as a little footnote these basic calculations we've done here; Surprisingly we will find that they are only approximate when we take into account some of the strange features of relativity, but that is something we will find out later, especially if we are doing the mathematics version of this course, but in terms of the basic idea that comes from From our intuition, you would certainly anticipate that if you run towards an object, its speed will approach you more quickly.

If you run away from an object, it will come closer to you. More slowly, the third basic characteristic of speed that we are all familiar with is that when you execute a very special type of movement, what we call constant velocity movement, movement that has a fixed speed, a fixed magnitude and a fixed direction, then you can't feel it. that movement correctly, you cannot feel that movement for a very good reason, if you are executing a movement at constant speed, you are completely justified in claiming that you are at rest and that the rest of the world is moving next to you, in that sense, the Movement at constant speed is very special because it is a movement that is completely subjective, there is no absolute notion of being in motion when the speed is constant, so let me give you a quick example of that, let's imagine that we have the same physical situation described from two different perspectives. so what I'm going to imagine is to have george and gracie floating in space, okay, floating in space, now here is george's vision of the events, he looks out and sees a character, gracie, coming towards him and she waves as she passes, as does he, his perspective. is that he is stationary and she is running alongside him.

Well, now I'm going to show you the exact same situation, but from Gracie's perspective, so what is Gracie saying from her perspective? She says she is stationary in space, she looks out. distance and she sees George running alongside him, she does too and he follows her merry way. One perspective is right and the other wrong at all. You are completely justified if you are not accelerating, if you are not changing your speed or shifting. the direction of motion to say that you are stationary now the reason why I am emphasizing motion at constant speed is something that you have all experienced well if you are in a car and you make a sharp turn and you feel your body being pushed in this direction you know that you are moving if you are on a plane and it takes off while it is accelerating as it accelerates you feel like you are being pushed back in your seat you know you are moving but if you are not accelerating you don't feel the movement and in fact there is no way to that you detect motion at all, so for example, if you imagine that George and Gracie were in two floating laboratories in space and they do experiments to solve the laws of physics, they will solve it. exactly the same laws because there will be absolutely no remaining experimental implication of its relative motion if that motion has a constant velocity in a fixed direction there is no way to determine its state of motion because you are justified in saying that it is at rest now that you ideate that idea that you can say that you are at rest that there are no implications of motion at constant speed it doesn't start with special relativity it doesn't start with albert

einstein

this is an idea that actually goes back a long time it goes back to galileo and galileo wrote a description wonderfully poetic of this idea.

Let me show you a little visual representation of what he said and I'll also read his words while he plays. Then he said: "Lock yourself in a big boat and there you get mosquitoes, flies and other small winged creatures. He said to hang a bottle that lets its water come out drop by drop in another narrow-necked bottle placed underneath and then with the boat still, watch how the winged animals fly with the same speed he told all parts of the room how all the drops of distillation fall into the bottle below and then he says that, having observed all these details, make the ship move with the speed that want, so here is the ship that sets in motion and says that as long as the motion is uniform by which he means constant speed, he will not be able to discern the slightest alteration in the four named effects nor will he be able to deduce from any of them whether the ship is moving or is still, that is the same idea that you cannot detect movement at constant speed there is no.impact on your observations, old idea with Galileo, so where does Einstein come into this story?

Einstein's new contribution is to say that among the four named effects that Galileo was talking about, he was only talking about the mosquitoes, the flies, and the water that fell into the bottle. He adds that those things would not change if you always kept moving. when it is uniform

einstein

added something to the list einstein added to the list of things that would not change he added the speed of light this is Einstein's new and surprising idea let's see what it means the speed of light is constant right, that's one of the most famous phrases in all of science the speed of light is constant now what does it mean and why should you care about getting there, let's think about how einstein came to the idea that the speed of light is constant, it's a story interesting in that over many centuries many people struggle to understand light and perhaps a good place to pick it up is between the years 1600 and 1800, when a whole group of physicists spent a lot of time and a lot of effort trying to understand light. measure the speed of light and they did a pretty good job. speed of light measurements with modern updates, we now know that the speed of light is 671 million miles per hour, if you like those units, that's 300 million meters per second or a little more precisely, 299 million 792 458 meters per second, but I'll round that up to 300 million meters per second for the most part, so it was good that people understood the speed of light, but still the physicists did not understand what light really was and that was when two physicists, Michael Faraday and James Clark Maxwell, who through experiments mainly Faraday through theorizing mainly Maxwell realized something quite surprising because they studied electromagnetic waves they studied waves in a electromagnetic field and came to a surprising conclusion, so Maxwell did this mathematically based on Faraday's experimental results and generally speaking, what Maxwell finally concluded from the mathematical equations was that an electromagnetic disturbance always travels at a speed particular regardless of the wavelength which is the distance between one crest and another and notably the speed that he found for electromagnetic waves again regardless of whether they have a very long wavelength or if, for example, they have a much shorter wavelength , like this guy who came here and found in the equations that the speed of those electromagnetic disturbances would always be equal to a particular number and that number turned out to be 671 million miles per hour or 300 million meters. per second, so this was an amazing idea and again, if you haven't studied electromagnetism, it doesn't matter, the only thing that matters here is that Maxwell had these equations and from the equations arose from a calculation a velocity that was equal to the velocity of light, which Maxwell concluded well, naturally, he said that if the speed of electromagnetic disturbances is equal to the speed of light, then light itself must be an electromagnetic disturbance, it must be an electromagnetic wave, which was a great It was a step forward now that we understood what light really is, but even with the progress that that represented, it still posed a deep mystery and that mystery is this, as we described it before, whenever you talk about speed, you have to say that an object has this speed in relation to that object, you must express things.

In that way, speed even has some meaning, but when it comes to the equations Maxwell was studying, they didn't specify what speed was relative to 671 million miles per hour, so if you have sound waves, the speed of sound is relative. to still air if you have water waves the speed of the water wave is relative to still water what was the thing relative to the speed of light that was being calculated nothing seemed obvious so physicists tackled this mystery by inventing an answer They said that maybe there is something called aether that fills space and when you talk about the speed of light you are talking about the speed of this electromagnetic wave relative to the ether.

Experiments were done to try to find the ether and tell a long story. In short, there is no evidence of ether, so the enigma remained. This is where Einstein's genius comes into the story because Einstein had this uncanny ability to look at something that everyone else had been looking at and see it in a new way and Einstein said, "Look." If the equations say the speed of light is 671 million miles per hour but the equations don't specify what the relative speed is, perhaps it's because nothing needs to be specified. Einstein said the speed of light is 671 million miles per hour. time relative to anything, as long as it travels through empty space.

This is a strange idea. It is a non-conformist idea. You could say that it is a crazy idea because we are not used to any speed that is not relative. We are not used to any speed that is not relative. for example, you can't change it by running towards it or away from it, but that's what Einstein was saying, so let me give you just a little visual example of what the constant speed of light is and the fact that Einstein was saying it can't be changed. I need to specify the reference for the speed of light, it's just a number, a law of physics, here's what that would entail, so imagine we have George and Gracie out there again.

Gracie has a meter that can measure the speed of light when stopped. George fires the laser beam at her and she reaches 300 million meters per second, but now let's change things up a bit and imagine that Gracie runs away. You'd think the speed should be less because she's fleeing, but no Einstein would say it's still 300 million meters. per second is a constant it is a law of nature that the speed of light is that number relative to anything similar if gracie ran towards george you would think the speed would increase because she is running towards the approaching laser beam not 300 million meters per second again and the same would be true if it's not Gracie that's running but George, then the fountain, if the fountain is running, do you think the speed should be a little bit faster 300 million meters per second not one bit more? large and similarly if If George were running away, you would think that the speed he should measure would be less than 300 million meters per second, but according to Einstein it is still the same fixed number, a constant 300 million meters per second Now if this is true, right?

It's from Einstein. One idea, if true, is telling us, as we mentioned before, that speed has some very unusual properties when you're talking about very fast speeds close to the speed of light. 300 million meters per second or 671 million miles per hour is very fast. circle the earth seven times in a single second, but einstein says that at those speeds you begin to reveal a feature of nature that you wouldn't anticipate based on soccer balls or hand grenades or any of the ordinary objects of everyday experience , so if it is true If speed has these strange characteristics when you talk about speeds close to the speed of light, then that would mean that space and time because speed, distance over duration, space over time, would mean that space and time have strange characteristics, that is why this is such a critical idea but of course the essential question is whether it is correct, if the speed of light is really constant, we care that the speed of light is constant because, again, velocity is a measure of space times time, so if velocity is doing something strange, then that must mean that I have already emphasized that space and time must also be doing something strange and in this section we are going to describe one of the most surprising implications of the constant nature of the speed of light, which is that there is no universal agreement on what things happen at At the same time, that's where we're going to get to.

Let's start with the basic intuitive understanding of time. Over many centuries, humans have learned to measure time quite well. We have developed all types of clocks that over time have become increasingly better at measuring the time interval between one event and another with absolutely astonishing precision. Now that we've said that we've still struggled for years to really understand what time itself is, we still don't have an answer. to that question, but we have some basic understanding of the properties of time, for example, we all agree that clocks that are working properly and set correctly, all of those clocks will keep time at the same rate, so they will all be synchronized with one. another, everyone will agree with each other, we also generally agree that people who measure the duration of an event with properly working clocks will get the same answer, we all agree on how long it takes for something to happen and we all agree.

I agree in general terms what things happen at the same moment, those are the basic characteristics of time as we experience it in everyday life, this is what the constant nature of the speed of light tells us that all of that is wrong, it tells us that clocks that work properly and are set correctly do not. we generally agree with each other tells us that we generally don't all agree about what happens at the same moment and generally we won't all agree about how long it takes for something to happen now those are some pretty surprising statements, I'm not going Let's describe them all now, but let's address one of them.

I want to describe how the constant nature of the speed of light ensures that the different perspectives of individuals moving relative to each other do not coincide. what events happen at the same time and to do that I'm going to frame it in the context of a little story that goes like this imagine there are two nations at war, an advanced land and a backward land, and they have just reached an agreement. are ready to sign a treaty, except that each president stipulates that he does not want to sign the treaty before the other, so the secretary general of the united nations must devise a plan that will convince them that in the procedure they are going to follow to use each president will sign at the same time here is the procedure that the secretary general comes up with says look we are going to have both of you sitting at opposite ends of a table we are going to put a light bulb in the middle we will turn on the light bulb when the light reaches your eye you sign the treaty you are equidistant from the light bulb and therefore the light should take the same time to reach you you must sign simultaneously so here is the setup the light goes off the flash goes to each of the two presidents hits them and they sign the treaty and everyone is very very happy well now everyone is very happy that this agreement was reached and a few months later they reach another agreement that again they want to sign at the same time, except this time both presidents want to do it in a slightly different Although they have many differences, each of the presidents of the advanced lands and the backward lands, both have a deep love for trains, so they want to do the treaty signing ceremony on a train that crosses the border between the advanced lands and the backward lands, so they set up the same scenario and here they are on the train, the train is going, there is a table again in one of the cars the presidents are equidistant on the train from the light bulb the light bulb will light up Same as in the previous case and when each president sees the light he will sign the treaty ok so here we go the light bulb goes off the light flashes it goes to each of the presidents and they sign the treaty and everyone on the train is again very happy with the treaty. result, but here's the thing, right after signing the treaty, it is said that the people on the platform are fighting, they are fighting because those people in front The land claims that they have been deceived, they claim that their president of the land in front signed the treaty first, how could they come to that conclusion?

Well, that's how the people of the land ahead go, they are on the platform watching this happen and from their perspective they watch. What happens when the flash goes off? The president of the backward lands is moving away from the flash from his perspective, so it takes him longer to reach it than the president of the advanced lands who is moving towards the flash, so let me show you that again, watch what happens when the flash goes off the president of the earth in front moves towards the flash the president of the earth behind moves away from the flash the light has to travel further to reach the president of the earth behind than the president of the earth in front has the same speed as the speed of light is constant, so if you have to travel further, it will take longer to get there from the perspective of the people watching on the platform and therefore claiming that the two presidents do not signed at the same time, they claim that the president of theadvanced earth I signed the treaty first let me show you one more variation of that so you can really see the detail of what's happening this time I'm going to draw a line where the flash takes place that's where the flash took place look how far away the light is has to travel to reach the president of the advanced earth versus the backward earth he only has to travel this distance to reach the advanced earth he has to travel this distance to reach the president of the backward earth light travels at the same speed if he has to Traveling further is going to take longer to get there, so now we have an interesting situation: those people on the train are absolutely convinced that the two presidents signed at the same time, those people on the platform are equally convinced that they did not sign at the same time.

Same time. So the big question is who is right and who is wrong and the answer is both are right the reasoning of each group of individuals is absolutely impeccable for those who are on the train the presidents are equidistant from the light bulb the light bulb turns on the light travels the same distance to each so they sign at the same time perfect reasoning those on the platform say that the flash goes off and does not have to travel as far to land a forward land as it does with the president to land backwards the speed of the light is constant and therefore they do not receive the flash at the same time, that reasoning is absolutely impeccable, so what this is telling us is that the constant nature of the speed of light means that events taking place at the same time from the perspective of one group of individuals will not take place at the same time from the perspective of another group of individuals moving relative to them now, this depends of course on the constant speed of light because what Newton would have said is that I would say that the projectile would say that the light would receive an additional kick from the train moving in this direction and that additional speed would allow it to cover this distance in the same amount of time that the light going in this direction would have its speed decreased and Therefore it will travel a shorter distance and when you take those two effects into account Newton would say that both presidents are hit at the same time regardless of their point of view, but due to the constancy of the speed of light we come to a very different conclusion: This is what is known as the relativity of simultaneity and is one of the most surprising implications of the constant nature of the speed of light.

The relativity of simultaneity strongly hints at The fact almost necessarily requires that the speed of motion affects time itself, the speed at which time passes must be affected by motion, that is the only way we can really conclude, as We have already done it, that simultaneity depends on your perspective, that is, at the treaty signing ceremony, if the clocks on board the train coincided with the clocks on the platform, then everyone would agree about what happens in a At any given time, everyone would agree about whether the president signed at the same time, but they don't agree, so the clocks move relative to each other.

Now we need to mark time differently so we can refine that idea to really understand how movement affects the passage of time. We need a way to measure the passage of time. Of course, we need a watch right now that you can wear for everything we're talking about here in relativity, you should feel free to wear any watch you like, your favorite rolex, your favorite grandfather watch, any watch you want to wear, without However, I am going to use a special type of clock. That's pretty unknown, but as you'll see, it's a very powerful type of clock for evaluating the effect of movement on time, so I should take a moment to address the question of what a clock is.

What is a watch? What is a watch? system that experiences repetitive cyclical motion and performs that cyclical motion, experiences those cycles uniformly, so if you're talking about using the Earth as a clock, the Earth spins around its axis uniformly and we use that to say every time that rotates once, that's one day, we can talk about the earth in revolution around the sun, right, it does so in a fairly uniform cyclical manner and we call each revolution a year and on a more standard wristwatch, if you have one , should we say that one of the old ones that has a seconds hand that moves around it makes that sweeping motion cyclically sweep after sweep after sweep and we call each of those sweeps a minute, so that's what conceptually a clock is , the new type of watch. which I am going to present has the same type of characteristic cyclic movement a cyclic process happens over and over again but the process itself is a bit unknown because the type of clock I am talking about is called a light clock which is a light clock A light clock is a gadget where we have two mirrors facing each other and a ball of light bounces between them and every time the ball goes up and down, you can think of it as tick tock, right tick tock.

The ball just goes up and down, so let me show you a quick picture of this type of clock, this light clock there it is, so every time it goes up and down, let's make it tick tock, tick tock, it's a cyclical motion. regular that could be used to measure how much time elapses between one event or another, the reading is at the top of this light clock and what I want to emphasize from the beginning is that this light clock, as unknown as it is, is correct, It's not familiar to you, you can't go. Go to Walmart and buy one of these light clocks, but conceptually a light clock is no different from any other type of clock, which means that any conclusion we come to about the nature of time makes use of this light clock. as an intermediate part of the reasoning.

That conclusion applies to any clock, it would simply be harder to come to that conclusion with a clock that had a more complicated internal mechanism because, as I will show you in a moment, the beauty of the light clock is because the mechanism is the ticking mechanism. tac. It is so simple that we can very easily determine the effect of movement over time. Well, to do that I'm going to want to introduce a second of these light clocks because I'm going to want to compare the speed at which time passes in one compared to the other, not when they're stationary like they are here, but I'm going to want to put one of them in motion before you do it, let me tell you what you're going to see.

Just to prepare you because this is a wonderful result that we are going to find and I want you to be completely prepared for when it arrives, imagine in your mind that I have one of these light watches and I am going to walk. with it is now moving from your perspective, think about the path the light will travel from your perspective, the light will start here, it hits the top of the mirror here and then it hits the bottom mirror here, so from your perspective the light He passed through a diagonal path up and down as I walk with the light clock, so I'll show you this in a moment, but let me answer a quick question first.

You might think well if you are walking in this light. the clock with the two mirrors will not miss the ball of light the top mirror because you are moving as you go answers absolutely no what is the argument the argument is simply this from my perspective okay I am experiencing a constant speed moving at the same speed in a fixed direction, which means that from my point of view I can say that I am stationary and it is you and the rest of the world that are running towards me and therefore, from my perspective, it has to be that the ball of light It just goes up and down and up and down because my view I'm not moving so the ball of light has to hit the top mirror if it hits the top mirror from my view it has to hit the top mirror from your view also the ball So light can't fly into space, so what would happen is we put these two clocks aside and look at that ball of light moving in the diagonal path of the clock moving up and down.

Now notice something, the amount of time that passes on the two clocks is different. Because? Well, think about it. Look at the path of light on the moving clock because it is a double diagonal going up and down the diagonal. The path it follows to make it work. talk and let me show you this in slow motion the trajectory it takes to tick and talk is longer so let's take a look at what got these guys moving this guy has already gone up and down record one this guy because he's going in a longer trajectory from your perspective and yet the speed of light is constant, a longer trajectory will take longer to get there, which means this guy has ticked, this guy still has to get to the top of the mirror, so if we let this guy continue and then notice that this guy has reached two, this guy is just bouncing around in the background, he's too far from reaching two to allow him to continue, and so on, and so on, you will see the speed at which time passes. on the moving clock is slower than the speed at which time passes on your stationary clock and it all comes down to the speed of light being constant, the perspective that is from the perspective of laboratory observers, those people who are observing the moving clock, they see that the ball of light in the moving clock keeps bouncing up and down between the two mirrors, but from the perspective of those of us in the laboratory, the path that the light must follow to make ticking on the moving clock, the path is longer because it goes along this double diagonal path from our perspective and if the path is longer but the speed of light is the same, that means the Tick ​​tocks occur at a slower rate on the moving clock, our clock goes tick tock tick tock. the moving clock is ticking ticking ticking time itself is passing slower in the moving clock, so this is this wonderfully amazing idea that we have now established with this light clock that, from the perspective of those in the laboratory who If you look at a moving clock, you will conclude that time runs slowly on that moving clock and again I have used the light clock as a tool as an intermediate step because as we have just seen, I can easily see the effect of movement on the passage of time.

The same would be true if you were to wear any other watch a rolex a grandfather clock because what we are talking about is how movement affects time itself and the bottom line is that from our stationary perspective a watch that is moving will keep time at a slower pace Now we know that time passes more slowly on a clock that moves relative to you, and we're going to calculate shortly the speed at which that moving clock keeps time relative to your clock, but first I want to address two vital questions about this topic of time slowing down and The first is if you are moving with that moving watch and someone on the platform looking at you says that your watch is ticking slower.

Do you feel like time is passing more slowly? The answer is absolutely no, you don't, because Once again, it comes back to the same point that I emphasized at the beginning, right when we said that George and Gracie were there in space, so they were there in space and they passed each other, and I emphasized that each could claim to be at rest and that the other is moving next to them, so that same idea here we are just talking about movement at constant speed, fixed speed and a fixed direction that tells us that the person you see moving with that clock In motion that person can claim that they are at rest and it is you who is moving, so from their perspective time is passing as it always does, if you wish, the light clock in relation to them goes up and down and up and up and down.

It goes down just as it always does with you and the rest of the The world is running along, so the bottom line is that no one internally feels that time is running slower due to the fact that everyone can claim to be the person who has the clock. Resting. Well, that being said, if you're really looking at such a relative watch. for you is moving, you should see time passing more slowly if you are not moving with that clock, then the question is why do we never notice that time passes slowly on a clock that is moving relative to us, because?

Did it take Einstein's genius to discover this? Why don't we know this in our bones? Why don't we experience this in everyday life? And the answer is the same answer we have arrived at in analogous questions that As we have seen before, it all has to do with the fact that everyday experience only takes advantage of a small part of how the world is configured and, in this particular case , everyday experience does not imply that we observe objects that reach speeds close to the speed of light. which is where these effects kick in the most, so let me give you a little demo where you can see that idea in action.

Here you have your own light clock that you can play with on your own and what you do with it. Inthis light clock, you set the clock speed to whatever value you want and it's all in fractions of the speed of light. Okay, now if you set the clock speed to be relatively small, non-zero, but relatively small, let's look at the ticking speed of that clock compared to what it would be doing if the clock wasn't moving at all and notice that the diagonal path here has almost no diagonal because the speed of the clock is very slow compared to the speed of light that light goes up and down and has practically no capacity for the clock to move to the right during any of the ticks, so that the movement of the clock has very little effect on the passage of time when the speeds are slow but when the speeds increase, let's do another version of this, let's put the speed, I don't know, about 60 of the speed of light, now the clock can move significantly between its ticking because it's going fast, it's going at a speed on par with the speed of light it's half the speed of light a little bit more and now the diagonals are actually longer than the straight ones up and down and just to emphasize that point as much as possible in this little demo, play with this again here we are at 99.9 percent speed. of light, so if I now turn this guy on here, look at that look, how far he didn't even make his first tick, that's how fast this clock was moving relative to the speed of light, so, When is? stationary would tick tock this was it didn't even get the tick k this is how far this clock moved to the left because of its very high speed so this clearly shows us that again it is the speed of the clock that determines the speed at which time will pass more slowly than on a stationary clock, that's the key point, but we want to go further because ultimately what we really want to do is derive a formula for how much slower time passes. a moving clock compared to a stationary one and I will derive it mathematically for those of you who are taking the mathematical version of this course in the next section, but let me give you the essential idea here and what the essential idea is, well, the essential idea.

The idea can be obtained by analyzing one of these moving light clocks a little bit, so here is our little schematic version of a light clock that is moving and if you think about the process of ticking according to what we have described . That little demonstration we just did, the key thing we need to think about to know how fast this clock will tick is to look at the length of the path for the little photon of light, if you will, to start here and to continue . tick has to take that trip and to go backwards it has to take that trip and since the speed of light is constant, the most relevant thing here is how long that trip is in relation to how long it would be for the stationary clock and for the stationary clock it just goes up and down, so if I mark it just in case, let's give that guy a different color, let's call it blue, so if this length here is, let's say, equal to l and this length here is equal to d, then this will also be d, so this guy ticks on the stationary, goes up and down, so goes l plus l, goes to l to tick on the stationary and moving clock to tick is d plus d is equal to 2-d now again, since the speed of light is constant, this tells us the duration of each tick on the moving clock, let's compare that to the duration of this guy ticking on the stationary clock , so that's the duration of the tick tock in the motion compared to the duration of the tick tock in the stationary clock, well that relationship is just the relationship of the distances because the speed of light is the same. so that's 2d over 2l, which is d over l, so that's the essence of the problem.

The ticking of the moving clock compared to the ticking of the stationary clock is the ratio of the path length on the moving clock from our perspective. looking at it compared to the path length on the stationary clock, so that's the key formula that describes the rate at which the ticking occurs on the two clocks, but now let me take this a little further and notice the next point that can easily be confusing, it is a very simple point, so if you ever find yourself confused on this, calm down, take a deep breath, think about it carefully and you will be able to figure it out, what is this if you are considering the amount of time that passes. between two events Correct, if you are measuring the time between two events on any clock, but particularly a light clock, you want to know the number of ticks that tells you how much time has passed so far if the duration of each tick tick tick is longer, then less time will pass on that clock longer tick tocks mean less time will pass, so what that means is that if we are not looking at the duration of the ticks but at the elapsed time, then the elapsed time and I Complete what clock in a minute divided by the elapsed time here and I will complete the clocks right now, so if the duration of ticking on a moving clock is compared to the duration of ticking on a stationary clock, let's say this is a ratio of five to one, meaning you will spend five times as much time on the stationary clock, where the ticking is faster compared to the amount of time spent on the moving clock, so this translates to the elapsed time on the stationary clock. compared to the time spent in motion, these are inversely related to the duration of the ticking, this is equal to d over l, so again, if this length here is five times the length for the ball to go up and down, the ball of light must go. up and down on the stationary clock, that means the ticking is happening five times slower on the moving clock, which means you will spend five times as much time on the stationary clock compared to the moving clock because the ticking they're happening faster here, so that means that now we can take the little formula that we've stated here and translate it to the elapsed time on the stationary clock to the elapsed time and the moving clock is the ratio of the length of that diagonal that we have here to the length of the straight line up and down, so now we've basically reduced the calculation of how much time the moving clock slows down compared to a stationary clock to really a little bit of geometry and trigonometry and I'll show you how it works if You're taking the math version in the next section, but let me give you the answer.

This is the answer that we will establish if this clock that we are looking at here let's say this guy has a velocity that is going this way and this way. velocity is equal to v, so that formula takes a velocity v as input and tells us how much slower the moving clock is compared to the stationary clock. Now, for those not taking the math version of this course, this is one of only two equations. I'm going to show you the other one, of course, which is equal to m c squared, but this formula is just as important as e equals m c squared, it's not as famous, but it tells us that the time that passes in a moving clock is slow in relation to that of a stationary clock by a factor of 1 over the square root of 1 minus v over c squared this expression this little formula is so important that we give it its own name we call it gamma again we will derive it in the next section but only I want to give you an idea of ​​this result before we do it, we'll look at two clocks and a clock that you can imagine is here on Earth, which we'll call stationary, a clock is on a rocket, so now we've gone from a train to a rocket because we want some of these speeds to be able to be really fast and what this demonstration will do is just take the formula that I've told you and that we'll derive in a moment. so remember what that formula is so that this is equal to we claim 1 over the square root of 1 minus v over c squared the proof will take the v that you entered in the proof calculate that and show us the amount of time the rocket elapses compared with the amount of time that passes on Earth, okay, let's do that, so let's be conservative at first.

I have chosen that the speed of the rocket is 12 of the speed of light. Get this guy moving and you can. We start to see a little time difference between them, it's hard to see, but there is a little difference, but now let's turn this up and go to 66 67 the speed of light and now you can really start to see that the elapsed time in the rocket is minor from the perspective of those of us here on earth looking at our stationary clock and then let's get inspired and you should do this on your own again to feel the formula, this formula for this object called gamma in your bones, now I'm at 95 , I don't know, let me push it all the way, let's go to 98 and a half percent of the speed of light and now you really start to see the difference between these two, it's dramatic, here we are. the Earth and hour after hour passes in the usual way, but from our point of view, the time of the moving clock moves very, very slowly, so this factor that this guy called gamma time dilation, as we call it, it activates substantially for speeds close to the speed of light at everyday speeds time dilation is still there, so this guy here where speed is in the denominator that activates at any speed, right, but that number is so close to one for everyday pedestrian speeds we don't notice it, so clocks we move around the world all the time we don't realize they are ticking time at a different rate just because that formula is such that gamma is very close to one, so the relationship between the time on the moving clock and the time on the stationary one is practically indistinguishable from being equal to each other, but the effect is there, anyway, having said that, let me emphasize a small The loophole that we will come back to is a curious loophole, which is that if there are individuals that are moving at relatively slow speeds but are very, very, very far apart, that can amplify this effect, so there can be big differences in the time even at slow speeds.

If we're talking about observers that are far apart in space, we'll come back to that and it's curious implications as we go, but putting that aside, if you're talking about observers that are at reasonable distances from each other, distances that could be planetary scales or even galactic scales, this time only when they move relative to each other very quickly The dilation effect is activated substantially, but it is there all the time, so in a sense we all keep our own time, this completely breaks the unity of time that Newton imagined. Newton imagined that there is a clock in the cosmos ticking second after second. after a second the same for all of us this directly shows that that is not true, each one wears his own watch and our watch keeps time at a rate compared to others that depends on the relative speed between us.

I have been talking as if time dilation is an established fact about how time itself behaves and there are good reasons for it, we arrived at this idea of ​​time dilation based on an experimental fact that the speed of light is constant and then, following in Einstein's footsteps, we have turned it into an understanding. that time on a moving clock runs more slowly than on a stationary clock, well, okay, that's all good, but you know, when you talk about an idea that is as strange and as contrary to experience as time dilation , you are simply happier. we're just more convinced if we have some direct experimental support for that idea, so the question is whether there is any direct experimental support for time dilation, and there is plenty of experimental support.

I'm just going to give you two small examples that really help solidify the idea that this is really tapping into the true nature of time. Well, the first example is to look at the simplest and most direct way to verify that time on a moving clock slows down. It is an experiment that was carried out in the 1970s and What happened in this experiment is very simple, the scientists took two atomic clocks, they put one of those clocks on a plane and the other atomic clock was left on the runway. , they flew this plane around the world and finally they landed the plane and they compared the amount of time on the moving clock with the amount of time on the stationary clock and lo and behold, when they compared the two clocks, they discovered that an amount of time had elapsed different in each one, in fact, the time difference between them is exactly what Einstein's ideas predict.

It's a little bit more complicated to figure it out relative to the formula that we've derived here, the formula that we derived, this gamma factor plays a role in the analysis, but because this plane is flying, it's not at a constant speed, gravity comes into play. the history.a little more complicated but it states absolutely very directly that time in moving clocks runs at a different rate and when the detailed analysis is done taking into account all the complexities that we are not going to talk about, all the ideas are confirmed . that we have described, so look once you see that these two clocks show a different amount of elapsed time, I think you should be convinced that these ideas are correct, but anyway let me give you another piece of experimental evidence that is kind of fun . and we'll come back to this in a bit.

It has to do with a species of particles called muons. You don't need to know what muons are, but they are very similar to electrons, they are a little heavier. but the essential characteristic of muons is that they are unstable, which means that they disintegrate, they fall apart in a fraction of a second, which means that when these particles are produced as they are in the upper atmosphere, they can fall towards the Earth, but at some point in the journey, the particle disintegrates, it disintegrates into other particles, it essentially disintegrates and the question is how far can the particle travel before disintegration occurs and it is worth studying that for a moment to Let the particle start here and fall to the end. up there and the problem is how far it can travel before breaking.

Now everyone knows what the answer is. It must be the case that the distance it travels is equal to the speed multiplied by time and this time here is its lifespan how long does it live before it disintegrates before disintegrating into other particles now in the laboratory scientists have measured the lifetime of these muons and it turns out that the answer is 2.2 times 10 to the power of minus 6 seconds and since scientists also know the speed that these muons have in the upper atmosphere as they descend, they know how far they should be able to travel and here's the remarkable thing: when you do that calculation you discover that the muon that you would think should only have enough time before explodes to go that far but observations show that the muon goes much further what's the explanation well let's think about time for a moment because this is the time measured in the laboratory the muon is moving, that means its clock is ticking time slows down, which means from our perspective the muon, its clock is ticking slowly, so it should be gamma multiplied by delta t, as measured in the laboratory, when at rest.

The way to think about this is as if the muon, if I don't mind putting it in a bit violent language, the muon has a gun to his head and when the clock that the muon is wearing reads 2.2 times 10 minus 6 seconds, you pull the trigger and it falls apart, but if the muon is moving from our view, your watch is ticking more slowly, so our clock will have advanced 2.2 times 10 to minus 6 seconds and the mu1 you still won't have pulled the trigger because, from your view, that amount of time. hasn't elapsed yet, so what this means is that the distance d that the muon should be able to travel given this time dilation is now v times 2.2 times 10 to the power of minus 6 seconds of its lifetime in rest times gamma, so the formula then is 2.2 times 10 to the power of negative 6 seconds times v divided by the square root of 1 minus v over c squared and it's that formula, this number is greater than just the product Of the two things in the numerator, this guy here makes it bigger which explains that the muon can travel from here to here without decaying, so let's get an idea of ​​this result: muons travel farther than I would have thought based on Newtonian reasoning because of this time dilation factor, so this little demonstration of what it does here You can choose the speed of the muon there in the upper atmosphere and this will show how far the muon travels before it decays, so again at slow speeds there is not much difference with Newton, but then it really comes into its own at high speeds. and in fact this one allows you to show the Newtonian answer.

That dotted line that you have at the bottom, if you can see it, it will be easier for you to see it on your own when you play with this, so this dotted line is the distance that Newton would say that muons will be able to travel. before they decay, so that's only 2.2 times 10 to the minus 6 times their speed, but then if you take time dilation into account, you see that the muons can travel much further and they do How can we explain how they can reach from the upper atmosphere downwards. to the surface of the earth where Newton would have thought they would have disintegrated long before touching the surface of the earth, giving us two solid experimental proofs that time dilation is really simple direct experiments that can really only be explained by This idea that movement clocks keep time slowly there is a wonderfully surprising implication of time dilation that is often not emphasized as much as it should be and I would like to briefly describe it to you now as it has to do with the following fact, so we know By the formula for gamma, the effects of time dilation only kick in significantly when the relative velocity being studied in a given situation approaches the speed of light, that's all true, but there is another way In fact, the relativity effects of simultaneity can be amplified over very large distances, these effects can become significant even when the speeds involved are ordinary speeds of every day, so how does this work to set it up?

Let's first think about time for a moment from the perspective of experience, so generally we all think of time as a kind of continuous development, a continuous flow, but for our purposes it is useful to also think about the time of In a different way, it is a kind of series of moments, a series of snapshots of one moment after another, moment after moment and any physical process, of course, can be described in this way: a flower, a wild animal running moment after moment after another, a horse running, etc., are just a series of snapshots that capture each subsequent moment in time.

In fact, if you want, you can even go out into space and think about the Earth in its orbit around the Sun again, moment by moment. Well, then what I'd like to do is start with that way of thinking about things. I would like to compare my set of snapshots, my sequence of events unfolding over time and I want to compare my snapshots with the snapshots of another person moving relative to me and to do that there is a related idea that I want to present , which is the concept of a portion of now and by a portion of now what I mean is that I consider the world and think about all the things that are happening right now, like the strike of 12 on a clock or that moment my cat jumping or maybe other events like a bird flying. flight at this very moment let's say in Venice or we can go cosmic in this too so that we can imagine at that very moment a meteor hitting the surface of the moon or go even further, we can imagine a supernova explosion in the farthest reaches of our galaxy now a portion now is a portion in this image here where I write down all those events that I say happen at one moment in time and if I look at one portion now after another this is the unfolding of one moment after another after another So each one of those events that are in a certain portion constitutes those things that I say were real, that happened at a given moment, one now after another, now after another, now there are two points that I want to emphasize about this, we give a name to this picture.

That makes a lot of sense. We call this space-time because we have all the space in each of these portions. Let's imagine that the portion continues forever and includes everything in the cosmos at any given time. In this direction, of course, we have the development of time, so we have space and time, that's where the name comes from. The second point is common sense and everyday experience would tell us that every observer in the universe, regardless of its motion, must agree on what is in a given portion of the now, which is the Newtonian. vision of how the world is put together, but when Einstein enters history that changes radically because with Einstein we have now learned that the constant speed of light means that observers who are in relative motion do not have the same sense of simultaneity that they do not have. . agree on what is happening at any given moment and that has a surprising implication that I would like to describe and to do that, let me use a little metaphor here, it is one that I actually used in my nova program tissue of the cosmos if you have seen, but if not, it's a simple metaphor, think of this whole expanse, this whole expanse of space-time as if it were some kind of big cosmic loaf of bread and what these slices are.

Basically, I'm cutting off this space bar. time into pieces that represent all of space at a single moment in time from my perspective, if someone moves relative to me they have a different perspective than what is now simultaneous and that means they cut the bread at a different angle to mine. so let me show you that schematically so let me imagine that I consider the bird's eye view of that image just because it's easier for me to draw and let me write my cuts now there so from the bird's eye view I will draw the space in a moment of time space in let's say the next moment in time the next moment in time and so on, so these are all my cuts from now and just so I have them labeled in a way that we all understand, put them here going right on This image is what I consider the future and going in this direction is what I consider the past.

Now someone is moving relative to me and let's say he's also interested in drawing space-time slices, so let's draw his and why they're moving relative. For me, they will cut this region of spacetime at, say, a different angle relative to me, they cut the bread with a knife that is at an angle relative to my slice because their notion of simultaneity (what is happening at one moment given) differs from mine now if we are trying and this is the point if we are dealing with low speeds, the person who is far away has a low speed, so we are not talking about speeds close to the speed of light, which what that translates to in this image is that the angle that we have here this angle is relatively small, so in the vicinity of where that person is sitting, the low speed movement has practically no impact, but the point and I'll show you an animation of this in a moment is that at increasingly greater distances, let's say I'm here and let's say someone else who is doing the movement is very far away at very large distances a small angle can be amplified into a very large time difference a very large difference in our conception of what is happening at any given moment so let's take a look.

With that idea in animated form, let's imagine that we are looking at a great expanse of space and time and we have a character, an alien far, far away in space, and we have a more familiar looking character, a human being sitting still on a bench above. here now, if initially these two individuals do not move relative to each other, they share the same idea of ​​simultaneity if there is no movement, so they cut spacetime in the same way, they both agree on what is happening at a moment given in time. okay, but now let's change things up a bit, let's let our alien friend get on an alien bike, say, and say the alien starts moving away from me due to the relative motion between the alien and me or the guy on the bench that the alien has. a different conception of simultaneity a different notion of what is happening now and what that means is that when the alien cuts the bread of space-time across all of space at any given time, the cut will now go through it at a different angle and again The point is that a small speed means a small angle, but consider a small angle at increasingly larger distances between us and that small angle becomes a large change over time, so in fact the aliens Now they move into the past and it can be a significant sweep into the past when you put in some numbers, as we will do later in this course, you find that the sweep goes further back than when that guy was a baby, it goes further back in time than that and in terms of events on earth that the alien would claim are happening right now from their perspective, it could be hundreds of years ago, say beethoven putting the finishing touches on the fifth symphony, now what's not completely obvious about this and requires some math and if you are taking the math version of this courseWe'll do the math if you're not taking the math version.

I hope this is exciting enough that you can take the math version of the course, but putting that aside, why was it a thing of the past and not? tell the future here's a quick way to think about it remember the treaty signing ceremony president of the backward lands, right, the backward lands were running away that president and he signed the treaty late, right, he wasn't the one who did it first , he did it secondly if you remember, in essence, if you are moving away, you are sweeping into the past, you are old news from that perspective, but that also means thinking now from the perspective of the treaty, the president of the advanced earth , if you are getting closer, if you are moving forward, your notion. of simultaneity should extend into the future and in fact that is the case, so if the alien gets back on the bike but turns around, say, and does not move away from Earth but travels towards Earth, then, of course, In fact, the alien's notion of what's happening right now on Earth extends from what we consider the present to what we consider the future and could include things strange from our perspective, like this guy's great-great-granddaughter, perhaps teleporting from one place in the universe to another, so the point is the whole.

The notion of what you consider real, what you consider to be happening right now depends entirely on your emotion, so initially, when the alien was not moving, say in relation to us, let's put ourselves in the position of the guy on the bench. our From our point of view, we agree with whatever the alien says is happening right now, whatever is real, we totally agree at this moment, when the alien gets on a bike, We don't suddenly dismiss the alien's perspective because he's on a bike, so if the alien then says that other events are happening. considered as real in its portion now at any given time, we must give that statement the same status and the same credibility as when the alien was not moving relative to us, so that if the alien tells us that things in our distant past are real, they are in their portion now at a given time, we must take that into our perspective on what is real, if the alien tells us that things in our future are in their portion now at a given time, we must also keep that in mind, so what this tells us collectively is that the traditional way we think about reality the present is real the past is gone the future is yet to be that has no real basis in physics what we're really about learning from these ideas is that the past, present and future are all equally real.

Time dilation is one of the strangest counterintuitive concepts you can come across and it's good to have a sort of mental mnemonic, a sort of shorthand way of thinking about this strange idea that maybe makes it a little more intuitive. I would like to give you such an intuitive way of thinking about time dilation now and let me tell you that you can justify the explanation that I am about to give you mathematically and if you are taking the mathematical method. In this course version we'll justify it a little later, but if not, don't worry, this gives you a good way to think about why time passes more slowly when a clock is moving. here's the idea let's forget about time for a moment, let's think about space and imagine that we have a car heading north at 100 kilometers per hour now let's imagine that the car swerves and drives east without changing its speed, now Your movement in the northward direction will not be as fast as it was before because some of the northward movement has been diverted towards the northeastward movement, so what that means is that movement can be shared between dimensions and when the Movement is shared that way, the movement was completely dedicated to one direction.

The direction is diverted to another direction, so the movement in the initial direction slows down. Let me show you a little picture of that. Here we have our car. I am going to show three versions of the car, one goes north and the others go northeast. at various angles and there you see the point where this car has traveled much further north than these cars because these cars have diverted some of the northward movement to the east, so the idea is that when you go in a different direction through space you deviate somewhat from your initial movement in that new movement in the new direction okay, now let's take that idea and apply it not to space but to space and time, okay, right now here I am and you would say that I'm not moving in relation to you say, but of course I am moving, look at my watch, my watch ticks second after second after second, taking me forward in time, forward in the dimension of time, if you are good, now imagine that I get up and I start walking, basically, Einstein. told us that as I walk I deflect part of my previous motion through time into this motion through space, which means that I move through time less quickly, like here this car is going less fast heading north because it has deflected part of the initial movement towards the north. towards this movement when I start to move I deflect my initial movement through time towards this movement through space so that my passage through time slows down.

That idea to me is the simplest intuitive way to understand time dilation. You can do it mathematically.precise, but putting that aside, if you want to think about why a clock slows down when it's moving, just think about when it's still, all its movements through time, when I see it move through space, has deflected some of that motion through time passes through space, so it passes through time more slowly, that's why time moves slower in a moving clock, the constancy of the speed of light that we have seen in the context of all the ideas of special relativity has a dramatic effect. impact on our understanding of time It is also surprising that the constant speed of light has a dramatic effect on space and also on mass and we are going to talk about both, but for now let's move on to the first, the implications of the constant speed of light light for our understanding of space, so first, why would we expect motion to affect space?

Well, it's pretty simple, because speed, as we emphasized, is distance divided by duration, which of course is space divided by time, so if we learn as we have that the speed of light is constant. Well, we have also learned that time is not constant, meaning that space must somehow compensate for the non-constant aspects of time so that its ratio remains the same, allowing for the speed of light. In order not to change, in schematic language, to ensure that the speed of light is constant, space must adjust in conjunction with time so that the proportion of light remains fixed, so the image you must have in mind is something like this if we consider that time is not constant, therefore space must also change in relation to movement, so that the space-time relationship is such that the speed of light can remain unchanged and what we What I would like to do is take that rough idea and make it explicit.

Now we want to determine what the effect of movement in space is, how are we going to do it? Well, let's work in the context of a concrete example. Let's imagine that we have a train and we want to ask ourselves how we would measure the length of a train. train well, that's a pretty simple thing to do when the train is stopped, because if the train is stopped, you take out a tape measure and measure the length of the train correctly, so let's get going, let's imagine that's the situation and Let's Consider the length of a train from the perspective of someone on the train, so that would be our intrepid train passenger, George.

From his point of view, the train is at rest relative to him, so he takes out his tape measure and simply stretches it. from one end of the train to the other and that way you measure the length of the train and let's say you find that the train is 210 meters long, well, everything is perfectly simple, let's now imagine that our second character, Gracie, is on the platform. So from his point of view, the train is moving, so he has to use another approach to measure the length of the train. Actually, he can't take out his tape measure and measure the length of the train because the train passes quickly, so that's not the case. a way in which you are going to measure the length of the train, instead you do something smarter, let's assume you know the speed of the train, let's also assume you have a stopwatch, what you can do is this: you will start the Stopwatch just when the front of the train is passing her, she will stop the clock when the back of the train passes her to find out how long it took for the train to pass by her.

She knows the speed of the train and she simply multiplies them, multiplies the speed by the time. to get the correct length of the train, so let's see how she does it there, she is on the platform, she has her stopwatch at hand, the front of the train passes by the boom, she starts the clock when the back of the train passes by her pen, stops the mire, she gets the elapsed time, in this case 5.9 seconds, she multiplies it by the known speed of the train to get the length of the train, that is her approximation.

Here's the notable fact that both approaches are George's approach, where she simply used a tape measure. approach where she uses this clock and the known speed of the train give different answers, so if you multiply this just in this particular example 30 times 5.9 177 meters numbers, I must say that I have made up just to illustrate the point which is that Gracie has gotten a shorter train length compared to george now at first glance, which is hugely surprising, but the question is: does this really faze george, assuming that george has taken seriously the discussion we've already had and that he fully knows about time dilation with the notion of time dilation he makes the discrepancy in the length of the train from his perspective and from Gracie's perspective it baffles him and the answer is no because from George's perspective this is what he says look , I understand Gracie's approach she is using length equals velocity or velocity multiplied by elapsed time, but I also know that gracie from my perspective I am now george gracie from my perspective is in motion moving clocks keep time at a faster rate If a clock keeps time at a slower rate, it will show less elapsed time and therefore in that multiplication it will produce a shorter length, so from that point of view, George understands why Gracie got a shorter length. short, but the question remains who is right is the length of the train 210 meters as george says it is or is the length of the train 177 meters as gracie says it is now you can probably guess the answer to the question of who is right based on what that we have discussed so far the answer is they are both right, the answer is length itself is a concept we need to rethink.

We normally think of length as the length of an object, but in fact, the length of an object depends on its speed when you measure it. Now where does that idea really come from? from that idea arises from the following fact that we have repeatedly emphasized simultaneity is in the eye of the beholder right now to measure the length of an object it is necessary to measure its front and its back simultaneously at the same moment now if two observers have different notions of simultaneity, therefore, will have a different notion of the length of an object.

No result is the only correct one. no result is wrong. They are all equally good. Having said that, just a little bit of language that we generally call the length of an object. When you measure it when you are at rest relative to the object like George is in the case of the train, we call it the rest length of the train, we call it the proper length of the train, but that only indicates a particular perspective, the perspective. from someone who is not moving relative to the object, but fundamentally you can measure the length of an object when it has any speed relative to you and you will get a different answer depending on the speed of the object, so the general conclusion we are coming to coming is that moving objects get shorter depending on the direction of their movement and let me emphasize that it is only along the direction of movement that the object will appear shorter, the height of the object will not change at all and a small argument can establish that, for example, if you were to imagine that the train enters a tunnel, and let's say it barely fits now, if it were the case that, from a person's perspective, the height of an object was not in the direction of motion, if That will change, let's say.

If I said that the height of objects becomessmaller, then from my perspective on the train the tunnel will be smaller. I won't be able to fit in. I should crash into it from the perspective of someone in the tunnel. the train that will shrink in that direction and therefore fit now, as strange as relativity is, it cannot be the case that from one person's perspective there is a train crash into a tunnel and from another person's perspective person there is no collision that would really be a contradiction, it would be a paradox that cannot happen and therefore we learn that it cannot be the case that the dimensions that are perpendicular to the direction of motion do not change at all, they remain fixed and That's how we describe it.

The shortening of an object along the direction of motion we call it length contraction or we call it rental contraction, that's the language we use and let's take a look at a simple example of that, so here's a case where that we are looking at. In a New York City taxi that is moving fairly quickly along the direction of motion, the taxi shrinks and is shorter in that direction than when it is at rest, so let's take a look at a demonstration that will show you will give you a feel for the amount by which an object appears to shrink along its direction of motion when you look at it, so this little demo here allows you to choose the speed of this taxi as it runs past you and again feel this in Your bones as the speed increases does not have much effect on their length, but as the speed approaches the speed of light, the object becomes shorter and shorter in the direction of movement.

Now you can ask yourself a natural question at this stage: what does the object do in The motion actually slow down and does it like what is the force that is squeezing it now, that is a natural question, but it is formulated quite vaguely because the point The main thing we've come to is that the notion of length itself requires a notion of simultaneity because again, if you're measuring the length of a moving object, if you first say measure the back of the object and the object moves and then you measure your forehead, let's say I'm measuring the length of a fish in a pond, right?

If it is swimming and I measure its tail first and let the fish swim and then measure its nose well, I will get a different length than I would if I measured the front and back, nose and tail at the same time. Therefore, when talking about the length of an object it is essential to commit to a notion of simultaneity and that requires choosing a frame of reference because moving observers do not agree on what is happening at the same moment in time and because to those different points of view. differ with respect to what happens at the same moment, we conclude that a moving object has a length that depends on its speed because its speed determines the degree of lack of simultaneity between two perspectives, so if we ask the question again, do the objects in motion reduction the best answer I can give you is yes and no, it is definitely true that an object in motion has a length from my perspective that is shorter than when that object is at rest, but it is not as if someone has entered with a vice and squeeze crush the object is simply that the old idea that there is a universal notion of the length of an object must be updated by relativity, the length of an object depends on its speed when you measure it and that is a result amazing.

The length of an object depends on its speed. Now I give you a couple more small examples to consider and both examples make use of the idea we discussed earlier, the relationship between observing something and measuring something, so back then I described how we are mainly focused on reality, so that we don't really care that much about human perception, we care more about what happened in the world to be responsible for what we see, but sometimes it's fun to look at what we would literally see. if some of the effects of relativity were visible to us and in this example we are now going to look at a taxi, but done more precisely, this is literally what a running taxi would look like if you could see it speeding by Speed ​​note that the taxi appears a little crooked in the last frames of that little video, we could see the entire rear bumper of the taxi, although normally, if a taxi ran past us, we wouldn't be able to see the entire bumper we would only see the part closest to us .

The reason this is a bit complicated makes use of the fact that when we look at something we are looking at light from the object, the light takes different amounts of time to arrive. us from different points on a three-dimensional object because those three-dimensional points are at a different distance from our eye. If you take that into account, that little video gives you a pretty good idea of ​​what it would be like to literally see an object. running close to the speed of light the second example places us inside the taxi itself shows us what it would be like to look out the window of a taxi that runs through a city close to the speed of light and how you can see the world around you not It only has the length contraction that we have described, if you take into account the finite travel time of light, the differences from one point to another, you see that space has a kind of distorted curved appearance, the world around you seems to be curving around you when your speed approaches the speed of light, so again, if you could find a taxi that could travel close to the speed of light and if you had very good eyes to be able to see the world around you as it passes running past you at very high speed, that's what you would see, so these are some very strange effects that the constant speed of light has on the nature of space, but they all follow directly from the analysis we have already done with the time, so once you know that time has strange characteristics you know that space must also have strange characteristics so that together they can keep the speed of light constant now we understand the resolution of the survey in the barn paradox at least qualitatively now Let's take a look at the mathematical details that will allow us to reconcile these two seemingly contradictory perspectives, so what we want to do is first take a look at the barn team's perspective, we know that they say the survey fits and what we want to work out is how works.

The pole of the team reaches a different conclusion than the perspective of the team barn team barn it's like I can't understand how they came to this different conclusion they said the survey doesn't fit, let's understand mathematically how they solve that headache that enigma and finally understand this statement from the survey team that doesn't add up, so to set this up, let's start by considering the barn team's perspective, try to figure out how they make sense of the strange conclusion they hear from the team pole, and to do so, let's record some data to start . So according to the barn team, the pole goes quickly at 12 13, the speed of light, pretty fast and remember that the pole, its length at rest, they tell us, is equal to 15 feet, okay, now let's do a little I draw this so we know what.

We're talking about we have the pole, it's running and it's running in this direction to my left as I look at the board and this guy's velocity is v equals 12 13 c. Well, what does this mean from a watch perspective? that are being carried up the pole from the perspective of the barn, so let's imagine that the pole has clocks on the front and back and let me draw them just in case, so let's say we have a clock here and we have a clock here and We know that the clocks from the barn's perspective will not be synchronized with each other, so if the pole runs this way, we know that the main clocks will be delayed in time, they will be late, so let's calculate how far behind this clock is. is relative to the clock on the back and we know how to calculate the time difference, so the difference between those two clocks in this little formula we take v multiplied by the distance between those two clocks divided by c squared, so we enter the data we have. in the hand the speed is equal to 12 over 13c, we have the distance between those clocks and again this formula is the distance as seen in the chart whose clocks were discussing, so it will be 15 feet and since we are using feet, of course We'll take c to be a foot per nanosecond squared, so here we have our answer: 12 times 15 is 180 divided by 13 nanoseconds and this is about 13.8 nanoseconds, so what that means is that when the barn team looks these clocks, if, for example, this clock ends. here on the right is reading 13.8 nanoseconds, so this clock here on the left will be 13.8 nanoseconds behind, so it will be reading zero when this one is reading 13.8 nanoseconds now qualitatively, what does that mean qualitatively?

That means that according to the barn team survey, it is evaluating the location of the back of the pole before evaluating the location of the front of the pole, that means that the pole moves between when the rear position is evaluated and when the forward position is evaluated and that is why the front has time to leave the barn according to the barn team that is why according to the barn team survey it comes to this strange conclusion that the survey does not fit now let's do that quantitatively, we know that there is a 13.8 nanosecond difference in the clocks reading which is not the time difference according to the barn equipment regarding when the front and back are evaluated because of course this clock needs to hit that 13.8 nanoseconds, but it's ticking time slowly because it's a moving clock according to the barn equipment, so if we calculate gamma in this case, which will allow us to convert that time difference to the time difference according to the barn equipment, what is gamma in this case?

Well, again 1 over the square root of 1 minus 12 over 13 squared and if you calculate that, that gives us a nice answer of 13 over 5. So what the Barn team is saying is that you take this difference of 13.8 nanoseconds in the clock readings and multiplied by 13 over 5 to calculate how long after evaluation the back of the pole the front will remain. The pole position will be judged by those on the team pole, so let's do it, if we take 13 over 5 times the difference in clock readings and let me write that as 12 over 13 c multiplied by 15 feet divided by 1 foot times nanosecond just to get our full answer so that the 13 goes away and the 5 goes into 15 3, so this gives us 36 nanoseconds, so according to the team barn, 36 nanoseconds after the back of the position of the polls, the team members will evaluate the front of the position of the polls. polls how far the poll moves between those two evaluations, well that's just speed multiplied by time, so we have 36 nanoseconds, how fast does this pole travel?

Well, it goes 12 13 c, so that's 12 13 feet per nanosecond using our usual formulation and so if you just plug in the numbers and calculate this, you're going to get 33.23 feet approximately how far the pole will move between the evaluation of its rear location and the evaluation of its front location. So what does this mean? Let's make a couple of pictures to see what this means for measuring the position of the pole, so let's draw a little schematic representation of the barn and, for good measure, let's make some doors so that the pole can enter the barn and now let's look at the movement of the pole from the perspective of the barn, so, according to the people in the barn, Gracie and her friends say that the survey fits within the period at the end of the story, but they also recognize that the clocks that are connected to the pole are not synchronized each other and say that the pole people first measure the location of the back of the pole and only then measure the location of the front and we have calculated that there is a time difference of 36 nanoseconds between when the back and the front are measured , which means that the pole moves for In that interval, in fact, we have calculated how far the distance between these two locations moves, we calculate that it is 33.23 feet, so that is the distance that the pole moves post between when you measure the back and when you measure the front, so of course. the pole doesn't fit inside the barn during that interval, it travels this distance, so the front of the pole comes out of the barn, this way the barn crew can relieve their headache as they didn't need any excess.

They don't need any advice, they just did a little calculation and in that calculation they recognize that the team pole first evaluates the rear location and only then evaluates the front and when they evaluate the front location, it has slipped out of the barn, that's how The team barn explains this strange-sounding conclusion according to the team's survey that the pole doesn't fit inside, so that gives us our good explanation according to the team's survey for the strange conclusion of the team's survey.Well, now what we want to do is the same type of analysis, but from the perspective of the team survey, we want to understand mathematically how the team survey can explain the observations, the conclusions, the team's barnstorming claims that the survey does fit and we can actually do the same calculation, but it's worth doing it a second time. essential ideas, but now let's look at this from Team Paul's perspective and try to understand how Team Paul explains this headache-inducing claim that Team Barn says where Team Barn says the survey fits. team paul says how could they say that and we want to understand how they solve that puzzle by analyzing the clocks in the barn frame from their own perspective and of course the key idea will be that, according to team paul, the clocks in the barn frame barn reference are not synchronized even though the barn people say that.

Those clocks are synchronized correctly, so so that we can get an idea of ​​what's going on here, if we draw a little sketch here of the barn, so that the barn has clocks along its length, let's draw the clock, let's say, on a extreme and on the other. and according to the team's pole, it is the barn that runs at a speed v in this direction and that speed v is equal to 12 13 of the speed of light and since the barn runs in that direction according to the team's survey, this is so a leading clock is in the direction of motion leading clocks are lagging, they are lagging, so this clock will lag behind this clock and we can calculate how much it will lag to understand the reasoning of the barn equipment from the perspective of the pole team, so what? is the time difference between those clocks well we know what to do we take the speed which is 12 13 c we multiply it by the distance between the clocks from the perspective of the team barn that's how this formula works which is 10 feet and we divide by of c squared, which is just one foot per nanosecond in the approximation that we're using, so this gives us 120 divided by 13 nanoseconds and if you calculate that, there's about 9.2 nanoseconds difference between these two clocks, which That means depending on the team. survey, if you say this clock reads 12 noon, then this guy here will fall behind and it will be 12 noon minus 9.2 nanoseconds, so qualitatively what this means is that, according to the team's survey, The barn team will first assess the location of the front of the The post says they will claim it is in at 12 noon and then they will allow some time to pass before this clock reaches 12 noon and in that interval the barn will move towards the right allowing the back of the pole to slide in and that's why According to Team Pole Team Barn says the front and back of the pole inside at the same time, while from Team Pole's perspective those are not the same moment because the clocks in the barn are not synchronized.

Now we can do that quantitatively, of course. When calculating, according to the team's survey, how long it will take for this clock to reach 12 noon, it is not 9.2 nanoseconds because the clocks in the barn's reference frame are ticking slowly, according to the team's survey, so we have to multiply those 9.2 nanoseconds. by gamma to get the amount of time that the team surveys will elapse between those two clocks, when it should say this clock reaches 12 noon, so let's make what gamma gamma is 1 over the square root of 1 minus 12 over 13 squared 1 minus v over c squared, calculate that to be 13 over 5.

So now, if you'll allow me, I'm going to take that 13 over 5 and multiply it by the time difference that we calculated earlier, so if no No matter, I'm going to do it right here 13 over 5 times 120 over 13. 13 cancels 5 over 120 24, which gives us 24 nanoseconds, so again, according to Paul's team, the evaluation of the front and back of the pole. It happened 24 nanoseconds apart, that is, this clock and this clock differ from each other and it takes 24 nanoseconds for this clock to reach 12 noon. Now in that interval the barn moves to the right. How far does it move? Well, that's just speed. multiplied by time, so we have 12 over 13 c, so let's do that 12 13 feet per nanosecond multiplied by that time difference of 24 nanoseconds and if you just calculate that, it comes out to about 22 feet, so according to the post on the equipment, the barn moves 22.2 feet. between the measurements of whether the front and back of the pole are in, that is, more precisely, it is moving 22.2 feet between the time this clock and this clock have the same reading, so we can see what that involves drawing a diagram similar to the one I had here but now this is from the Paul team's perspective, let's draw a version of the barn and let me, as before, give you some doors for this pole to enter and from the survey team's perspective, which what we have learned is the following from your opinion, the survey does not add up and again that is the end of the story according to team paul, but the pole team also recognizes what we have now calculated, which is that the clocks in the team barn do not They are in sync with each other, so according to the team's survey.

What happens is that the people in the barn first evaluate the location of this end of the pole, they say it's inside, but then they wait 24 nanoseconds before they evaluate the location of that side of the pole, so let me draw it, so imagine that now we look this situation 24 nanoseconds later, so let me draw another schematic of the barn now, according to the team post, the barn has moved in those 24 nanoseconds, which means that although according to the team post, this end of the survey did not fit inside the barn at the same time. By the time the survey is over, they recognize that according to the team barn, they fit because, according to the team barn, what happens is they wait 24 nanoseconds later, according to the team pole, the barn moves.

We have calculated how far away it is, how far away it was. Oh, that's 22 feet, so let me record that just in case, so if I draw a little dotted line here, a little dotted line here, this is now 22.2 feet, so clearly this diagram is not to scale , but the idea is correct, so according to the team. paul, what happens is that the barn moves more than 22.2 feet between the time the barn team evaluates this location and that location and then of course, depending on the barn team, the pole will fit inside; is not measuring, according to team paul, the front and back at the same time, while of course, according to team barn, this moment and this moment are the same moment, according to team pull, they are not the same moment, the relativity of simultaneity returns with a vengeance, so that is the explanation according to team paul of what that barn team is like. comes to this strange conclusion that the pole fits inside, while they know it doesn't fit inside.

Both images are absolutely correct, they are just two different perspectives that allow us to understand how not only do these two frames of The Reference come to a different conclusion about whether the survey fits or not, but now we also understand how each team understands the statement of the another team even though they disagree with it again, coming from the relativity of simultaneity, so that's the math behind it. One way to solve the apparent pole in the barn paradox what we want to do now is to get a feel for these ideas by working with some demos that are good for playing with these things on your own and that's what these demos are for. so let's take a quick look at one, but you should study it in more detail.

This is very similar to the demo we had before, except now you'll notice it's embellished with a little extra detail, so now we have doors on the barn at a moment in time from the barn's perspective, but not at a moment in time. time from the perspective of the survey. and you know because there's going to be a flash, I should have said that when I was going to watch the flash and again do this on your own, the flash happens at a certain time from the polls perspective, the poll says that and that those are the same moment in time from the perspective of the surveys very different moments in time from the perspective of Barnes and similarly, you should work on the perspective of survey two, where you will again see the relativity of simultaneity, but now in mathematical form it fully explains this survey in the barn.

Paradox The twin paradox is the most famous of all the paradoxes of the special theory of relativity, but before I get into it let me emphasize the most important point: there are no paradoxes, as I said before, in special relativity, if the If there were, the theory would collapse there. However, there are situations where it seems like there is a paradox, it seems like we have two perspectives that we somehow can't merge into a coherent story, but that usually means that we have to think about the situation in detail, think about it with our understanding of the situation. situation. essential physics and when we do that, all the paradoxes will fall to the side, the same thing will happen here, okay, let's set up the paradox or the apparent paradox will involve a couple of characters, they are the twins george and gracie, and the scenario is one that may have It occurred to you in our previous discussion about time dilation and space travel because we're going to imagine that in this case Gracie gets into a spaceship, she travels into space, we'll make her go pretty fast, she'll turn around and come back.

Back here is the problem from George's perspective, stay home, George on Earth, looking at Gracie from her perspective, he knows how time works in special relativity, he says his clock must be running slow so that when she comes back he says he will be there. Older, there won't have been as much time on his clock, which moves slowly compared to him, Gracie, however, looks at George and says to herself, Look, I understand special relativity too, and from my perspective, George is the one. that is moving, I am stationary and therefore it is his The clock is ticking slowly so when I return it should be that Gracie is older because less time will have passed on George's clock, that is the problem, let's see that in visual form, so here's the question: George is the first character and Let's bring her twin, these are fraternal twins, Gracie, she gets on her spaceship and we're going to send her into space, so let's do that and say that it goes fast, so there will actually be some kind of significant time dilation here. she goes into space, she's going to reach some point that we'll call p the tipping point and then she'll come back okay, so the question we face is when these two guys compare the amount of time that has passed according to each.

Could it be the case that George is older than Gracie? Will her watch have read longer? Could it be the case, on the other hand, that grace is greater? Could it be that more time has passed on their clock or could you even imagine a resolution that brings these two together, perhaps it is the case that each of them has aged, each will be the same age as the other when they return, that is the question we want to find out which of those three scenarios is correct to solve. This apparent paradox is that each says that the other's clock should be running slowly, therefore each says that they should be older, so I am going to offer you a solution to this paradox.

The deep question is who is right, which of these scenarios is correct now. you can guess that they are both right because that has often been what we have found so far in special relativity, each perspective is correct and you just have to reconcile them with your understanding of how special relativity works, so it is a natural assumption , but that assumption won't fly in this case that won't work because at the end of the trip, george and gracie are together, so they can be in front of each other, they can be next to each other, in fact, they can even enter the same frame of reference and there can no longer be any mismatch in their observations at that moment one cannot look at the other and say that you are older and the other looks at the other person and says that you are older that is a contradiction that is a paradox that cannot be happen so we can't rely on the kind of resolution we've found before someone here is right someone here is wrong how do we find out which one?

Well, I'm going to give you the answer. It is Gracie who turns out to be the youngest. George turns out to be older and I'm going to give you three explanations for that. Let me start with explanation number one, which is a simple explanation that really gets to the heart of the matter. Where does the contradiction come from? Where does the apparent paradox come from? of george saying that he can be seen as at rest, gracie moving therefore his watch keeps time slowly and of course gracie can tell that i am stationary and it is george who is moving and therefore i can claim that it is your watch that is keeping time more slowly.

Is that valid in this situation? Remember that the onlyThe moment you can claim to be at rest and the rest of the world is moving alongside you is if you are going at a constant speed in a fixed direction, which is manifestly not true in this situation. In case gracie gracie goes out into space and then comes back, she has to turn around and when she turns around she has to accelerate, she slows down and then accelerates again to get back to george on earth, she feels that acceleration, she knows she's going moving it is no longer moving at a constant speed it is no longer justified to say that it is at rest and the rest of the world is moving alongside it so your perspective your reasoning is negated by the fact that it is accelerating it is not in a frame of inertial reference throughout the entire version of this trip and, therefore, we cannot trust its conclusion.

George, on the other hand, is in an inertial frame, he is at rest on the surface of the earth, he is not moving, subject to the question of whether the surface of the earth is an inertial frame, but those are the complexities that we are not concerned with. concern for the discussion we are having. George's perspective, therefore, is valid. He has a constant speed throughout this trip. His conclusions are unquestionable. They are absolutely correct and he says that Gracie's clock is ticking. more slowly and therefore when she returns he will be older. End of story, that's the resolution or should I say a resolution to the twin paradox.

We'll get to other explanations as we go through this deep scenario that really lets us sink our fingers in. in many of the details that we have been developing, but as a first step to this scenario, that is the explanation. Gracie is speeding for part of the trip. Therefore, her perspective cannot be taken into account. She cannot tell that she is at rest. George is the only one. Who can say that and is his conclusion that she will be older correct? Perhaps the most famous equation in all of physics is equal to mc squared. It comes directly from the ideas we've been developing and first I'd like to give you an idea. where does simply motivate e equals m c squared come from?

For those of you who are taking the mathematical version of this course, I will do a full derivation in a little while, but let's start with the ideas that lead us to the possibility that energy and mass e and m can be deeply related, they can have some deep connection that is ultimately captured in Einstein's famous equation equal to m c squared. Now, to do that, I'm going to start by telling you a little story, a story that I like to call the parable of the two joustors, so it will be a kind of jousting, but different from the one you may have in mind, we will have two perfectly matched individuals, so What they will be on identical horses, they will have identical masses and they will be holding a spear but not one that has a sharp end.

Let's imagine that there is a large metal ball at the end of the spear and the way it will be held when the two combatants cross paths. Just as they pass, each will lunge towards the other breaking their spherical balls trying to knock the opponent over, so let's take a quick look at what that joust would look like. Initially this was meant to be another George and Gracie animation, but you know, I contacted Gracie and she said to talk to my agent and she went all diva, so we have a new character whose name is Evil George, so It's George versus Evil George.

There they have their spears with metal spheres and they hit them against each other and because they are perfectly combined, we are sure that there will be a tie. Well, now George takes a special relativity course and starts thinking, he says to himself from my perspective, I'm stationary, right? and so evil george comes towards me and let's say this joust is taking place at very high speeds, the horses are moving close to the speed of light, let's say let's be dramatic and george says to himself, That means evil George is on the move from my point of view, that means evil George's clock, his clock is ticking. out of time more slowly, that means that from my perspective, evil George as he passes by may be moving quickly on his horse, but his movements will slow down in slow motion, so George says this will be a piece of cake because a as evil George passes by.

He will lunge at me so slowly that I will easily be able to take him down and win, so that is his vision in his mind, so just to look at Georgia's perspective as she thinks about this relativistically, she says that evil George's spear is coming. approaching me slowly. so he should win and yet he doesn't win, it's still a draw, so the question is what did George leave out. Evil George slowly pushed the lands to him. You should be able to take him down because he's going to push you quickly. left it out, if you think about it, the amount of impact you receive in a joust of this type depends on two things, not one, it depends on the speed of the thrust, that is absolutely the case, but it also depends on the mass of the sphere.

In the end, then, what does this tell us? Because it has to be a draw again, because you can't change your frame of reference and turn a draw into a victory. All observers in spacetime agree on events that they may not. We agree on when and where they happen, but it can't be that from one perspective it's a draw, from another perspective it's a win, so we know it still has to be a draw, so how can it be that evil George hits to George with the same force? Even though the spear goes slowly, it must be that the mass of the sphere at the end of the spear must increase to compensate for the slow thrust that evil george uses, which suggests to us that the energy of the movement must be able to increase. the mass of an object, and in fact we can go a little further, we know the degree to which Evil George's spear has slowed down, it's just the gamma time dilation factor that we've found over and over again, so that it must be the case that the mass at the end of Evil George's spear increases by the same gamma factor to compensate precisely for the deceleration with which Evil George pushes his earth, suggesting to us that the mass of an object must depend on its speed. and the way it depends on speed is the mass it has when it is at rest multiplied by the gamma factor.

That's a remarkable formula because if we look at a little demonstration here, let's look at what this formula tells us as velocity. of an object gets bigger and bigger, this tells us that its mass gets bigger and bigger, so you can play with this on your own of course, but there you see it as the speed of an object increases, its mass grows more and more until it approaches the speed of light the mass grows without limits now this has a series of vital consequences number one we have talked a lot in our discussions about the speed of light and implicitly we have always been using the fact that nothing can go faster than the speed of light, but you may have recognized it, I've never really established it for you now I have because if you think about it, when you try to accelerate an object you have to push it right now If its mass increases because its speed increases from the initial push to make it go faster, you still have to push it harder and harder, and in fact, as the speed of the object approaches the speed of light, its mass It gets bigger and bigger, which means you need to push harder. and it becomes increasingly difficult to get it to go even faster until at this point it takes an infinite push to get it to go beyond the speed of light.

There is no such thing as an infinite push and that establishes that the speed of light is a speed limit for any object that has mass, the second consequence here is that this result strongly suggests that energy and mass are interchangeable, that you can taking the energy of motion, the energy of evil George's motion that becomes, so to speak, the mass of an object that it carries. so this interchangeability between energy and mass is in itself quite vital, but let's go one step further when an object is at rest, of course it still has mass m at v equal to zero, which we call the rest mass of the object, for what we use mass interchangeability. and energy, lead us to anticipate that the object at rest still also has energy, so let me motivate the formula for how much energy the object at rest has.

We know what the units of energy are in traditional units, you may remember, they are kilograms per meter. squared per second squared if you're not familiar with that formula don't worry, but this is the unit within which energy can be specified. Now mass, of course, has units of kilograms, so for energy and mass to have the same units, we would have to multiply mass by something with the units of square meter per second square meter per square second well meters per Second, that is a speed. By what speed would we multiply to have some universal way of translating between energy and mass?

Well, of course, the speed involved would be the speed of light, so this is one way of thinking about Einstein's surprising realization that energy and mass are interchangeable, sort of like dollars and euros, and that the factor of conversion between energy and mass is nothing more than the speed of light squared and is equal to m c squared. Thank you for joining us on this journey into the wonderful world of special relativity, these crazy ideas that arise from Einstein's thinking about space and time, matter and energy, let me leave you with just one thought as you reflect on all the material we have covered, all the results we have obtained.

I discovered that they all come fundamentally from one idea, the constancy of the speed of light, that's where the relativity of simultaneity comes from. Remember the earth forward and the earth backward, it is the constant speed of light that makes those on the train and those on the platform not agree on what is happening at the same moment the constant speed of light is what makes clocks move slowly as they pass by us right, we use the light clock again the constant speed of light as it travels along the double diagonal which is Why time moves slowly when a clock is on motion, then we convert it to a length contraction where the lengths of moving objects appear shortened along the direction of motion and finally we have gone further and come to this amazing understanding that energy and mass are interchangeable , which is just to show that if you focus on an idea, if you focus on some new feature of the world and are actually able to think about it to its logical conclusion, sometimes that results in a revolution in the way we think . things, so go back to the rest of the course, review it, and try to get a sense of these wonderful ideas, because special relativity is truly one of the greatest achievements of our species.