WSU: Space, Time, and Einstein with Brian Greene

in the spring of 1905 a storm broke out in the mind of albert einstein at least that is how einstein himself described it later in life and the result of that storm of thinking is the special theory of relativity a theory that completely transforms our understanding of space and time and matter and energy i mean einstein found completely counter to experience that clocks in motion tick off time at a slower rate he found that objects in motion are contracted along their direction of motion he found that clocks that one set of individuals say are in sync relative to each other he found that if someone's moving relative to those clocks they would say 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 now you should say to yourself if there is such a unexpected nature to reality that we have missed through our everyday experience why have we missed it i mean why aren't we aware of special relativity right in our bones and the answer to that is when we look out at the universe we recognize that there are a huge range of scales that constitute reality and we humans only have access to a very small part of that totality so to give you a feel for that let's look at one axis a length axis and if you look at the scales that are out there in length atoms 10 to the minus 10 meter viruses 10 to the minus 8 meters red blood cells 10 to the minus 6. single celled organisms there we humans are 2 meters the earth 10 to the 5 meters solar system 10 to the 13 galaxy and on to 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 only the axis of length imagine we look at the axis of mass there too we will find a huge range of scales right so if we go back and look at atoms they weigh in 10 to the minus 26 kilograms go down to red blood cells 10 to the minus 15 humans well depends who you're talking to but about you know 100 kilograms or so solar system on to the entire universe the observable part 10 to the 52 kilograms a huge range of scales in mass we humans only have direct access to a small piece of it one more axis to look at is the axis of speed so we humans we walk around the world at certain ordinary everyday speeds sometimes you go into airplanes but there's a huge range of speeds out there the growth of human hair that's pretty small human speed typically space shuttle 10 to the 3 meters per second speed of light that's a number that's going to come up a lot in our discussion 10 to the 8 or so meters per second the point is in this spectrum of all possibilities in length in mass and also in speed we humans occupy a tiny tiny part so our experience the experience that has given us our intuition is built up from a very limited sense of what is actually out there so our intuition which really has come in some sense from evolution right so we evolved out there in the jungle and our intuition got built up in order that we can survive the survival value of understanding your environment is what matters we humans only have access to a small piece of the totality of what's out there and therefore it would be surprising if what we have experienced really does tell us about the physics at all possible scales in length in speed and in mass and it turns out that indeed it is the case that when you look at extremes of mass or length or speed the world operates the universe operates in ways that we are not accustomed to if you are looking at extreme say a very small size the new physics that comes into play is called quantum mechanics if you are looking at extremes of huge mass the new physics that comes into play is the general theory of relativity if you are looking at extremes of speed 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 be discussing so in a nutshell all of the discussion we are going to be having here is focused upon how the universe behaves 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 well we all know the answer to that but let's get all on the same page so if that car say has a speed of a hundred kilometers per hour we all know what that means if we look at how far it's gone divided by how long it takes it to get there so let's say this is one hour journey if it's going 100 kilometers an hour we know that it will have traveled 100 kilometers in that hour that's what speed is so in essence speed is nothing but distance that an object travels divided by duration now expressed in that language speed might seem like i don't know a kind of boring concept a pedestrian concept why concern yourself with speed the answer to that is clear if we recognize that distance is a measure that has to do with space duration is a measure that has to do with time so if we find as we are going to find going forward that speed has unusual features when the speeds involved get very big near the speed of light what we really will therefore be learning is that space and time have weird strange features that is what we are after okay good that's where we're headed let's start by first thinking about the non-strange features of speed the features that we all hold in our intuition so what are the basic features of speed well first off speed is a concept that even before einstein was known is a concept that is relative what do i mean by that well imagine we look at this car say it's going 100 kilometers per hour what we need to say is 100 kilometers per hour relative to the road why if that road is itself moving let's say it's on a boat the boat is moving then the car's speed 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 spinning around the earth itself is going in orbit around the sun with respect to the frame of reference our perspective right now that car is executing a pretty complicated motion not just 100 kilometers per hour so it's always vital when you're talking about speed to recognize that you can only ever frame the idea of speed for ordinary objects that we encounter as the object has this and that speed relative to this or that object you need to specify the reference in order that the speed that you are specifying has any meaning at all okay so that's a very basic feature of speed that it is relative another basic feature 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 with say a football george and gracie two characters that we are going to encounter often in our discussions and imagine that they're throwing that football back and forth at say five meters per second now that's all well and good completely understood but let's imagine that they go out to play a game of catch on another day and gracie is surprised to see that george has a hand grenade now she doesn't like hand grenades and so 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 it approaches her she can subtract her own speed from the speed of the oncoming grenade and that way have it approach her not say at the initial speed at which it was tossed which for instance might be just like the football five meters per second instead if she runs away at three meters per second she knows that now the grenade will approach her more slowly at two meters per second and that is a good thing when it comes to hand grenades similarly it's the case that if an observer like gracie is to not run away but say run toward an object that is being thrown at her the speed with which she approaches the object will be added to the rate at which it approaches her so if it was thrown at five and she's running toward it at three well we all know that that means it will approach her at eight meters per second speed is additive and subtractive you can change the speed with which an object is coming toward you by either running toward it or away from it now let me quickly mention just as a small footnote these basic calculations that we've done here we surprisingly are going to find that they are only approximate when you take into account some of the strange features of relativity but that's something that we will encounter later especially if you're doing the math version of this course but in terms of the basic idea that comes from our intuition you certainly would anticipate that if you run toward an object its speed will approach you more quickly if you run away from an object it will approach you more slowly third basic feature of speed that again we are all familiar with is this when you are executing a very special kind of motion what we call constant velocity motion motion that has a fixed speed fixed magnitude and a fixed direction then you can't feel that motion right you can't feel that motion for a very good reason if you're executing constant velocity motion you are completely justified in claiming to be at rest and the rest of the world moving by you in that sense constant velocity motion is very special because it is motion that is completely subjective there is no absolute notion of being in motion when the velocity is constant so let me give you a quick example of that so let's imagine that we have one and the same physical situation described from two different perspectives so what i'm going to imagine is having george and gracie floating in space okay floating in space now here's george's view of the events he looks out and he sees a character gracie coming toward him and she waves as she passes as does he his perspective is that he is stationary and she is rushing by him good now i'm going to show you exactly the same situation but from gracie's perspective so what does gracie say from her perspective she says that she is stationary out there in space she looks out into the distance and she sees george rushing by he waves she does too and he goes on his merry way is one perspective right and the other wrong absolutely not you are completely justified if you're not accelerating if you aren't changing your speed or changing the direction of motion to say that you are stationary now the reason why i'm emphasizing constant velocity motion is something that you have all experienced right if you are in a car and you take a sharp turn you feel your body being pushed this direction you know that you are moving if you are in an airplane and it's taking off as it accelerates as it speeds up you feel yourself pushed back into the seat you know that you are in motion but if you're not accelerating you don't feel the motion and in fact there is no way for you to detect the motion at all so for instance if you imagine that george and gracie were in two floating laboratories out there in space and they do experiments to work out the laws of physics they will work out exactly the same laws because there will be absolutely no remnant no experimental implication of their relative motion if that motion is at constant speed in a fixed direction no way to determine your state of motion because you are justified in saying that you are at rest now that idea that idea that you can claim to be at rest that there's no implication of constant velocity motion does not start with special relativity it does not start with albert einstein this is an idea actually that goes way way back it goes all the way back to galileo and galileo wrote a wonderfully poetic description of this idea let me show you a little visual representation of what he said and i'll just read his words too while this plays so he said shut yourself up on a large ship and there procured gnats flies and other small winged creatures he said let a bottle be hung up which drop by drop lets forth its water into another narrow neck bottle placed underneath then with the ship lying still observe how the winged animals fly with like velocity told all parts of the room how the distilling drops all fall into the bottle place underneath then he says having observed all these particulars make the ship move with whatever velocity you please so here it is the ship is going into motion and he says that so long as the motion is uniform by which he means constant velocity you shall not be able to discern the least alteration in all the four named effects nor can you gather by any of them whether the ship is moving or standing still that is the very same idea that constant velocity motion cannot be detected it has no impact on your observations old idea back with galileo so where then 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 just talking about the gnats and flies and the water dropping into the bottle place underneath that those things would not change if you go into motion so long as it's 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 the surprising new insight of einstein let's see what it means the speed of light is constant right that is one of the most famous sentences in all of science the speed of light is constant now what does it mean and why should you care well to get there let's think about how it is that einstein came to this idea that the speed of light is constant it's an interesting history where over the course of many centuries many people struggle to understand light and perhaps a good place to pick it up is in the 1600s to the 1800s when a whole group of physicists spent a lot of time put a lot of effort into trying to measure the speed of light and they did a pretty good job romer huygens bradley fouseau foucault these guys some of whom had very long hair made increasingly precise measurements of the speed of light with modern updatings we now know that the speed of light is 671 million miles per hour if you like those units it's 300 million meters per second or a little bit more precisely it's 299 million 792 458 meters per second but we will round that off to 300 million meters per second for the most part so that was good people understood the speed of light but even so physicists lacked an understanding of what light actually was and that's when two physicists michael faraday and james clark maxwell who through experiments mostly faraday through theorizing mostly maxwell they realized something quite amazing because they studied electromagnetic waves they studied ripples in an electromagnetic field and came to a stunning conclusion so maxwell did this mathematically based upon the experimental results of faraday and roughly speaking what maxwell ultimately concluded from the equations from the math was that an electromagnetic disturbance always travels at a particular speed regardless of the wavelength which is the distance between one crest and another and remarkably the speed that he found for electromagnetic waves again independent of whether they have a very long wavelength or if for instance they have a much shorter wavelength like this fella coming in here he 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 a stunning insight and again if you haven't studied electromagnetism it doesn't matter all that matters here is that maxwell had these equations and from the equations out came from a calculation a speed that was equal to the speed of light what was maxwell to conclude well naturally he said if the speed of the electromagnetic disturbances is equal to the speed of light then light itself must be an electromagnetic disturbance it must be an electromagnetic wave that was a great step forward now we had an understanding of what light actually is but even with the progress that that represented it still raised a profound mystery and that mystery is this as we described before whenever you talk about speed you need to say an object has this speed relative to that object you need to state things in that manner for speed to even have any meaning right but when it comes to the equations that maxwell was studying they didn't specify what speed 671 million miles per hour was relative to right so if you have sound waves the speed of sound is relative to the still air if you've got water waves the speed of the water wave is relative to the still water what was the thing relative to the speed of light was being calculated nothing seemed apparent so physicists dealt with this mystery by making up an answer they said maybe there is something called the ether filling space and when you talk about the speed of light you're talking about the speed of this electromagnetic wave relative to the ether experiments were done to try to find the ether and to make a long story short no evidence for the ether whatsoever so the puzzle remained this is where the genius of einstein comes into the story because einstein had this uncanny ability to look at something that everybody else had been staring at and see it in a new way and einstein said look if the equations are saying that the speed of light is 671 million miles per hour but the equations are not specifying what that speed is relative to maybe that's because you don't need to specify anything einstein said the speed of light is 671 million miles per hour relative to anything so long as it is traveling through empty space now this is a strange idea it is a maverick idea you might say it's a crazy idea because we are unused to any speed that isn't relative we are unused to any speed that for instance can't be changed by running toward it or away from it but that is what einstein was saying so let me give you just a little visual example of what this constant speed of light this fact that einstein was saying that you don't need to specify the reference for the speed of light it is just a number a law of physics here is what that would imply so imagine we have george and gracie out there again gracie has a meter that can measure the speed of light when she's standing still george fires his laser beam and she gets 300 million meters per second but now let's change things just a little bit and imagine that gracie runs away you would think the speed should be less because she's running away but no einstein would say it's still 300 million meters per second it is a constant it is a law of nature that the speed of light is that number relative to anything similarly if gracie were to run toward george you'd think the speed would go up because she's running toward the oncoming laser beam no 300 million meters per second again and the same thing would hold true if it's not gracie that's running but george so the source if the source is running you think the speed should be a little bit bigger 300 million meters per second not one iota bigger and similarly if george were to be running away you would think that the speed she should measure should be smaller than 300 million meters per second but it remains according to einstein the same fixed number a constant 300 million meters per second now if this is true right this is einstein's idea if it's true it's telling us as we mentioned before that speed has some very unusual properties when you're talking about speeds that are very fast near the speed of light 300 million meters per second or 671 million miles per hour is very fast fast enough to go around the earth seven times in a single second but einstein is saying at those speeds you begin to reveal a feature of nature that you would not anticipate based upon footballs or hand grenades or any of the ordinary objects of everyday experience so if it's true if speed has these weird features when you're talking about speeds near the speed of light then that would mean that space and time because speed distance over duration space over time it would mean that space and time have weird features that's why this is such a critical idea but of course the essential question is is it right is the speed of light actually constant we care about the speed of light being constant because speed again is a measure of space per time so if speed does something weird then that must mean as we've emphasized already that space and time must be doing something weird too and in this section we're going to describe one of the most startling implications of the constant nature of light speed which is that there is no universal agreement on what things happen at the same time that's where we're going now to get there let's start with the basic intuitive understanding of time right so over the course of many centuries we humans have gotten pretty good at learning how to measure time we have developed all sorts of clocks that through the ages have gotten better and better and better at measuring the time interval between one event and another with absolutely astounding accuracy now having said that we have still struggled for ages to really understand what time itself actually is we don't yet have an answer to that question but we do have certain basic understanding of the properties of time for instance we all agree that clocks that are properly functioning and properly set all of those clocks will tick off time at the same rate so they will all be in sync with one another they will all agree with one another we also generally agree that individuals that are measuring the duration of an event with properly functioning clocks will get the same answer we all agree on how long it takes for something to happen and we all agree generally speaking on what things happen at the same moment right those are the basic features of time as we experience time in everyday life here is the thing the constant nature of light speed says that all of that is wrong it tells us that properly functioning properly set clocks do not generally agree with one another it tells us that we generally do not all agree on what happens at the same moment in time and we generally will not all agree on how long it takes for something to happen now those are some pretty striking claims i'm not going to describe all of them right now but we're going to take on one of them want to describe how it is that the constant nature of light speed ensures that different perspectives of individuals that are moving relative to each other will not agree on what events happen at the same time and to do that i'm going to frame it in the context of a little story story goes like this imagine that there are two warring nations forward land and backward land and they've just come to an agreement they're ready to sign a treaty except each president stipulates that he does not want to sign the treaty before the other so the secretary general of the united nations needs to come up with a plan that convinces them that in the procedure that they are going to use each president will sign at the same moment here's the procedure that the secretary general comes up with he says look we're going to have you both sit at opposite ends of a table we're going to put a light bulb in the middle we'll turn on the light bulb when the light reaches your eye you sign the treaty you're equidistant from the bulb and therefore it should take the same amount of time for the light to reach you you should sign simultaneously so here is the setup the light goes off the flash goes toward each of the two presidents hits them and they sign the treaty and they are all very very happy good now they are all very happy that this agreement has been reached and a few months later they come to another agreement that they again they want to sign at the same moment except this time both presidents want to do it a little differently even though they have many many differences each of the presidents of forward land and backward land they both have a deep love of trains so they want to do the treaty signing ceremony on a train that is going right across the border between forward land and backward land so they set up the same scenario and here they are on the train train is going along there is a table again in one of the cars the presidents are equidistant on the train from the bulb the bulb will be turned on just as in the previous case and when each president sees the light he will sign the treaty okay so here we go the bulb goes off the light flashes goes toward each of the presidents and they sign the treaty and everybody on the train is again very very happy with the result but here's the thing just after they sign the treaty word comes that the people on the platform are fighting they're fighting because those folks from forward land claim that they have been duped they claim that their president from forward land signed the treaty first how could they come to that conclusion well here's how it goes so the people from forward land they are on the platform watching this happen and from their perspective look what happens when the flash goes off the president of backward land is moving away from the flash from their perspective so it takes longer to reach him than the president of forward land who's moving toward the flash so let me show you that again watch what happens when the flash goes off the president of forward land moves toward the flash president of backward land moves away from the flash the light has to travel further to reach the president to backward land than the president to forward land it has the same speed the speed of light is constant so if it has to travel further it's going to take longer to get there from the perspective of those people watching on the platform and therefore they claim that the two presidents did not sign at the same moment they claim that the president of forward land signed the treaty first let me show you one more variation on that so you can really see the detail of what's going on this time i'm going to draw a line where the flash takes place there is where the flash took place look how far the light has to travel to reach the president of forward land versus backward land it only has to travel this distance to reach forward land it has to travel this distance to reach the president of backward land light travels at the same speed if it has to travel further it's going to take longer to get there so we now have an interesting situation those people on the train are absolutely convinced that the two presidents signed at the same moment those people on the platform are just as convinced that they did not sign at the same moment so the big question is who is right and who is wrong and the answer is they are both right the reasoning of each group of individuals is absolutely impeccable for those on the train the presidents are equidistant from the bulb bulb lights up the light travels the same distance to each so they sign at the same moment perfect reasoning those on the platform they say the flash goes off and it doesn't have to travel as far to present a forward land as it does to the president to backward land speed of light is constant and therefore they do not get the flash at the same moment that reasoning is absolutely impeccable so what this is telling us is that the constant nature of the speed of light means that events which take 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 with respect to them now this relies of course on the constant speed of light because what newton would have said is he would say the projectile say the light will get an additional kick from the train moving in this direction and that additional speed will allow it to cover this distance in the same amount of time the light going this way would have its speed diminished and therefore it's traveling a shorter distance and when you take those two effects into account newton would say both of the presidents get hit at the same time regardless of your vantage point but because of the constancy of the speed of light we come to a very different conclusion this is what's known as the relativity of simultaneity and is one of the most startling implications of the constant nature of light speed the relativity of simultaneity strongly hints in fact almost necessarily requires that motion speed must be affecting time itself the rate at which time passes must be affected by motion that's the only way that we really can conclude as we have already that simultaneity depends upon your perspective i mean in the treaty signing ceremony if clocks on board the train agreed with clocks on the platform then everyone would agree on what happens at a given moment everyone would agree on whether the president signed at the same moment but they don't agree and so clocks that are moving relative to each other must tick off time differently now for us to make that idea precise to us to really understand how it is that motion affects the passage of time we need a way of measuring the passage of time we need of course a clock right now you can use for all that we are talking about here in relativity you should feel free to use any clock that you like right your favorite rolex your favorite grandfather clock any clock that you'd like to use i however am going to make use of a special kind of clock that's quite unfamiliar but as you will see it's a very powerful kind of clock for assessing the effect of motion on time so i should spend just a moment addressing the issue of what is a clock right so what is a clock a clock is any any physical system that undergoes cyclical repetitive motion and it does that cyclical motion it undergoes those cycles in a uniform way right so if you're talking about using the earth as a clock the earth spins around its axis in a uniform way and we use that to say every time it goes around once that's a day we can talk about the earth in revolution around the sun right it does that in a fairly uniform cyclical way and we call each revolution a year and on a more standard wrist watch if you have well i should say one of the old-fashioned ones that has a a second hand that's sweeping around it does that sweeping motion cyclically sweep after sweep after sweep and we call each of those sweeps a minute right so that is what a clock is conceptually the new kind of clock that i'm going to introduce has that same kind of feature cyclical motion a cyclical process happens over and over again but the process itself is a little unfamiliar because the kind of clock that i'm talking about is called a light clock what is a light clock a light clock is a contraption in which we have two mirrors that are facing one another and a ball of light will bounce in between them and every time the ball goes up and down you can think of that as tick tock right tick tock the ball is just going up and down so let me show you a quick visual of this kind of clock this light clock there it is so every time it's going up and down let's do it so it goes tick tock tick tock it's regular cyclical motion that you could use to measure how much time elapses between one event or another the reading is up there at the top of this light clock and what i want to stress at the outset is that this light clock however unfamiliar it is right it is unfamiliar you cannot go down to walmart and buy one of these light clocks but conceptually a light clock is no different from any other kind of clock which means any conclusion that we reach about the nature of time that makes use of this light clock as an intermediate part of the reasoning that conclusion applies to any clock it would just be harder to reach that conclusion with a clock that had a more complicated internal mechanism because as i'll show you in a moment the beauty of the light clock is that because the mechanism the tick tock mechanism is so simple we can very easily determine the effect of motion on the passage of time okay so to do that i'm going to want to introduce a second one of these light clocks because i'm going to want to compare the rate at which time elapses on one compared with the other not when they are stationary as they are here but i'm going to want to set one of them in to motion actually before i do that let me let me tell you what you're going to see just to prepare you because this is a great wonderful result that we're going to find and i want you to be fully prepared for when it comes imagine in your mind that i have one of these light clocks okay and i'm going to walk with it now it's in motion from your perspective think about the trajectory that the light will travel right 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 will have undergone a diagonal up and a diagonal down trajectory as i'm walking with the light clock so i'll show you this in a moment but let me just address one quick question first you might think well if you're walking with this light clock with the two mirrors won't the ball of light sort of miss the top mirror because you're moving as you're going along answer absolutely not what's the argument the argument is simply this from my perspective okay i'm undergoing constant velocity motion right same speed in a fixed direction which means from my view i can say that i am stationary and it's you and the rest of the world that are rushing by me and therefore from my perspective it has to be that the ball of light 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 you from your view too the ball of light cannot therefore fly out into space so what therefore would happen is this let's put these two clocks over to the side and let's look at that ball of light undergoing its motion in the moving clock diagonal trajectory up and down now notice something the amount of time that passes on the two clocks is different why is that well think about it look at the trajectory of the light in the moving clock because it's a double diagonal going up and down on the diagonal the trajectory that it follows for it to go tick and talk and let me show this in slow mo the trajectory that it follows to go tick and talk is longer right so let's take a look at that set these guys into motion this guy has already gone up and down he registers one this guy because he's going on a longer trajectory from your perspective and yet the speed of light is constant longer trajectory it's going to take it longer to get there which means this guy's gone tick-tock this guy has yet to reach the top of the mirror so if we let this guy continue on then notice that this guy is reached two this guy is just bouncing off the bottom he's far away from reaching two to let him keep on going and so on and so forth you see that the rate at which you've got time elapsing on the moving clock is slower than the rate at which time elapses on your stationary clock and it all comes down to the speed of light being constant the perspective that is from the laboratory observers those folks who are watching the moving clock rush by they see that the ball of light in the moving clock is still going bouncing up and down between the two mirrors but from the perspective of those of us in the laboratory the trajectory which the light needs to follow to go tick-tock in the moving clock the trajectory the path is longer because it's going along this double diagonal trajectory from our perspective and if the path is longer but the speed of light is the same that means that the tick tocks happen at a slower rate in the moving clock our clock's going tick tock tick tock the moving clock is going tick tock tick tock time itself is elapsing slower on 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 watching a moving clock they will conclude that time runs slowly on that moving clock and again i've used the light clock as a tool as an intermediate step because as we just saw i can easily see the effect of motion on the passage of time the same would be true if i used any other clock a rolex a grandfather clock because what we're talking about is how motion affects time itself and the conclusion is that from our stationary perspective a clock that's in motion will take off time at a slower rate we now know that time elapses more slowly on a clock that is moving relative to you and we're going to shortly calculate the rate at which that moving clock ticks off time compared to your clock but first i want to address two vital questions to this issue of the slowing of time and the first is if you are moving with that moving clock and someone on the platform watching you says that your clock is ticking off time more slowly do you feel that time is elapsing more slowly and the answer to that is absolutely no you don't because again it goes back to that very same point that i stressed at the outset right when we had say george and gracie out there in space so they're out there in space and they were passing each other and i emphasized that each could claim to be at rest and that the other is moving by them right so that very same idea here we're only talking about constant velocity motion fixed speed and a fixed direction that tells us that the person that you see moving with that moving clock that person can claim that they are at rest and it's you that's moving so from their perspective time is elapsing as it always does if you will the light clock relative to them is going up and down and up and down just as it always does you and the rest of the world are rushing by them so bottom line is nobody actually feels internally that time is ticking off more slowly because of the fact that everybody can claim to be the person the clock at rest okay now that being said if indeed you are watching a clock that relative to you is moving you should see that time on it is elapsing more slowly if you're not moving with that clock so the question then is why don't we ever notice that time ticks off slowly on a clock that's moving relative to us why did it take the genius of einstein to figure this out why don't we know this in our bones why don't we experience this in everyday life and the answer to that is the same answer that we've come to in analogous questions that we've encountered earlier it all has to do with the fact that everyday experience only taps into a small little part of how the world is configured and in this particular case everyday experience does not involve us watching objects that attain speeds near the speed of light which is where these effects kick in maximally so let me just give you a little demonstration where you can see that idea in action so this here is if you will your very own light clock that you can play with on your own and what you do with this light clock is you set the velocity of the clock at whatever value you want and it's all in fractions of the speed of light good okay now if you set the speed of the clock to be relatively small non-zero but relatively small let's look at the rate of ticking on that clock compared to what it would be doing if the clock were not moving at all and notice that the diagonal path here has hardly any diagonal to it at all because the speed of the clock is so slow compared to the speed of light that light goes up and down up and down with virtually no ability for the clock to move to the right during any of the tick tocks so the motion of the clock has very little effect on the passage of time when the speeds are slow but when the speeds pick up let's do another version of this let's put the speed i don't know 60 or so of the speed of light now the clock can move significantly between its tick tock because it's going quickly it's going at a speed on par with the speed of light half the speed of light a little bit more and now the diagonals truly are longer than the straight up and down and just to emphasize that point maximally in this little demonstration again play with this here we are at 99.9 percent of the speed of light so if i now turn this fella on here wow look at that look how far it didn't even get to do its first tick of tick-tock that's how fast this clock was moving relative to the speed of light so whereas when it's stationary it would go tick tock this one went to didn't even get the k of tick that's 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 which determines the rate at which time on it will tick off more slowly than on a clock that is stationary that's the key point but we want to go further because ultimately what we really want to do is to derive a formula for how much slower time ticks off on a moving clock compared to a stationary one and i'll derive that mathematically for those of you who are taking the math version of this course in the next section but let me give you the essential idea here and what is the essential idea well the essential idea can be gotten by a little bit of analysis on one of these moving light clocks so here is our little schematic version of a light clock that's in motion and if you think about the process of going tick-tock based on what we've described that little demonstration that we just did the key thing to think about in order to know how quickly this clock will go tick-tock is to look at the length of the trajectory so the light little photon if you will starts here and in order to go tick it has to travel that journey and to go tuck it has to travel that journey and since the speed of light is constant what is most relevant here is how long that journey is relative to how long that journey would be for the stationary clock and for the stationary clock it just goes up and down so if i just mark that for good measure let's give that guy a different color let's call this guy blue so if this length over here is say equal to l and this length over here is equal to d then this one will be d as well so this guy to go tick-tock the stationary one it goes up and down so it goes l plus l it goes to l to go tick-tock on the stationary clock and on the moving to go tick-tock it's d plus d equals 2-d now again since the speed of light is constant this is telling us that the duration for each tick tock on the moving clock let's compare that with the duration for this guy to go tick-tock on the stationary so that's the duration of tick-tock on the moving compared to the duration for tick tock on the stationary clock well that ratio is just the ratio of the distances because the speed of light is the same so that's 2d over 2l which is d over l so that is the essence of the issue tick-tock on the moving clock compared to tick-tock on the stationary clock is the ratio of the length of the trajectory in the moving clock from our perspective watching it compared to the length of the trajectory on the stationary clock so that is the key formula describing the rate at which the tick tocks happen on the two clocks but now let me just take this a little bit further and note the following point which easily can get confusing it's a very simple point so if ever you find yourself confused in this calm down take a deep breath think it through and you'll be able to work it out which is this if you're considering the amount of time that elapses between two events right if you're measuring the elapsed time between two events on any clock but in particular on a light clock you want to know the number of tick tocks that tells you how much time has gone by now if the duration for each tick tock is longer then less time will elapse on that clock longer tick tocks tick tock means less time will elapse so what that means is if we are looking at not the duration of tick tocks but the elapsed time so the elapsed time on and i'll fill in which clock in a minute divided by the elapsed time here and i'll fill in the clocks right now so if the duration of the tick tock on a moving clock compared to the duration of tick-tock on a stationary clock let's say this is a ratio of five to one that means that five times more time will elapse on the stationary clock where the tick tocks are faster compared to the amount of time that elapses on the moving clock so then this translates into the elapsed time on the stationary clock compared to elapsed time on the moving these are inversely related to the duration of the tick tocks 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 to go up and down in the stationary clock then that means the tick tocks are happening five times as slow in the moving clock which means five times more time will elapse on the stationary clock compared to the moving clock because the tick tocks are happening faster over here so that means that we can now take the little formula that we have indicated over here and translate that into the elapsed time in the stationary clock to the elapsed time and the moving clock is the ratio of the length of that diagonal that we have over here to the length of the straight up and down so now we've basically reduced the calculation of how much time slows on the moving clock compared to a stationary clock to really a bit of geometry geometry and trigonometry and i'll show you how that goes if you're taking the math version in the next section but let me give you the answer here is the answer that we will establish if this clock that we are looking at over here let's say this guy has a velocity that's heading over this way and this velocity is equal to v then that formula takes a speed v as input and tells us how much slower time elapses on the moving clock 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 of course being equals m c squared but this formula is just as important as e equals m c squared not quite as famous but it tells us that the time that elapses on a moving clock is slow relative to that on 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 we give it its own name we call it gamma again we'll derive that in the next section but i just want you to get a bit of a feel for this result before we do that we are going to look at two clocks and one clock you can imagine is here on earth which we will call the stationary one one clock is on a rocket ship so now we've gone from a train to a rocket ship because you want some of these speeds to be able to be really fast and what this demonstration will do it simply takes the formula the formula that i have told you that we will derive in just a moment so just remember what that formula is so this is equal to we claim 1 over the square root of 1 minus v over c squared the demonstration will take the v that you input into the demonstration calculate that and show us the amount of time that elapses on the rocket compared to the amount of time that elapses on earth okay let's do that all right so let's be conservative at first i've chosen the speed of the rocket ship to be 12 of the speed of light set this guy in motion and you can begin to see a bit of a time difference between them it's hard to see but there is a bit of a difference but now let's crank this up and let's go to 66 67 of the speed of light and now you really can begin to see that the elapsed time on the rocket ship is less from the perspective of those of us here on earth looking at our stationary clock and let's then be inspired and you should do this on your own again to feel the formula this formula for this object called gamma in your bones i'm now 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 begin to see the difference between these two it's dramatic here we are on earth and hour after hour is going by in the usual way but from our view that clock in motion time is ticking off very very slowly so this factor this guy called gamma this time dilation as we call it kicks in substantially for speed near the speed of light at everyday speeds time dilation is still there right so this guy over here where the velocity is in the denominator that kicks in at any velocity right but that number is so close to one for pedestrian everyday speeds that we don't notice it so clocks move around the world all the time we don't notice that they're ticking off time at a different rate only because that formula is such that gamma is very very close to one so the ratio of time on the moving clock to time the stationary is virtually indistinguishable from being equal to one another but the effect is there nevertheless having said that let me just emphasize one little loophole that we will come back to it's kind of a curious loophole which is that if you have individuals that are moving at relatively slow speeds but they're very very very far apart then that can amplify this effect so there can be big differences in time even at slow speeds if you're talking about observers that are very far apart in space we will come back to that and it's curious implications as we go forward but putting that to the side if you're talking about observers that are reasonable distances apart distances that might be planetary scales or even galactic scales it's only when they're moving relative to one another very quickly that this time dilation effect kicks in substantially but it is there all the time so in some sense we all carry our own time this shatters completely shatters the oneness of time that newton envisioned right newton envisioned that there's one clock out there in the cosmos ticking off second after second after second the same for all of us this shows directly that that's not true we each carry our own clock and our clock ticks off time at a rate compared to others that depends on the relative speed between us i have been talking as though time dilation is an established fact regarding how time itself behaves and there's good reason for that right we came to this idea of time dilation based upon an experimental fact that the speed of light is constant and then following in the footsteps of einstein we have parlayed that into an understanding that time on a moving clock ticks off more slowly than on a stationary clock good okay all that's fine but you know when you're talking about an idea that is as strange that is as counter to experience as time dilation well you just are happier you just are more convinced if you have some direct experimental support for that idea so the question is is there some direct experimental support for time dilation and there is there is a lot of experimental support i'm just going to give you two little examples that really help solidify the idea that this really is tapping into the true nature of time okay the first example is look the most flat footed straight forward way of verifying that time on a moving clock slows down it is an experiment that was undertaken in the 1970s and what happened in this experiment is very straightforward scientists took two atomic clocks they put one of those clocks on an aircraft and the other atomic clock they left back on the tarmac they flew this plane all around the world and they then landed the plane ultimately and compared the amount of time on the moving clock to the amount of time on the stationary clock and lo and behold when they compared the two clocks they found that different amount of time had elapsed on each in fact the time difference between them is exactly what einstein's ideas predict it's a touch more complicated to work it out relative to the formula that we have derived here the formula that we derive this gamma factor plays a part in the analysis but because this plane is flying around it's not at constant velocity gravity comes into the story it's a little more complicated but it absolutely establishes very directly that time on moving clocks ticks off at a different rate and when you undertake the detailed analysis taking account of all of the complexities that we're not going to talk about it confirms all of the ideas that we have described so that is look once you see that these two clocks show a different amount of elapsed time you i would think should be convinced that these ideas are correct but nevertheless let me give you one other piece of experimental evidence that's sort of fun and we're going to come back to it in a little while it has to do with a species of particles called muons you don't need to know what muons are but they're very much like electrons they're a little bit heavier but the essential feature 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 drop down toward earth but at some point in the journey the particle disintegrates it falls apart into other particles it breaks apart in essence and the question is how far can the particle travel before that disintegration kicks in and that's worth studying for a moment so the particle starts up here and drops all the way down to there and the issue is how far can it travel before it breaks apart now you all know what the answer to that is it must be the case that the distance it travels is equal to the velocity times the time and this time here is its lifetime right how long it lives before it disintegrates before it falls apart 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 minus 6 seconds and since scientists also know the velocity that these muons have in the upper atmosphere as they're coming down they know how far they should be able to travel and here is the remarkable thing when you do that calculation you find that the muon you would think should only have enough time before it explodes to go about 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 as measured in the laboratory the muon is in motion right that means that its clock is ticking off time more slowly which means that the muon from our view its clock is ticking off time slowly so it should be gamma times delta t as measured in the laboratory when it's at rest the way to think about this is it's as if the muon if you don't mind me putting it in slightly violent language the muon has a gun to its head right and when a clock that the muon is carrying ticks off 2.2 times 10 to the minus 6 seconds it pulls the trigger and it falls apart but if the muon is in motion from our view its clock is ticking off time more slowly so our watch will have long since gone by 2.2 times 10 to the minus 6 seconds and the mu1 still will not have pulled the trigger because from its view that amount of time has not yet elapsed so what this means is that the distance d that the muon should be able to travel taking this time dilation into account is now v times 2.2 times 10 to the minus 6 seconds its lifetime went at rest multiplied by gamma so the formula then is 2.2 times 10 to the minus 6 seconds times v divided by square root of 1 minus v over c squared and it's that formula this number is bigger than just the product of the two things in the numerator this fella over here makes it larger that explains that the muon can go all the way from here to here without disintegrating so let's get a feel for this result that muons travel further than you would have thought based upon newtonian reasoning because of this time dilation factor so this little demo what you do here is you can choose the speed of the muon there in the upper atmosphere and this will show how far the muon travels before it disintegrates so again at slow speeds not much of a difference from newton but then it really kicks in with the vengeance at high speeds and in fact this one lets you show the newtonian answer so that dotted line that you have on the bottom there if you can see it it'll be easier for you to see it on your own when you play with this so this dotted line is how far newton would say the muons will be able to travel before they disintegrate so that's just 2.2 times 10 to the minus 6 times their velocity but then if you take time dilation into account you see that the muons can travel much much further and that's how we can explain how they can reach from the upper atmosphere down to the surface of the earth where newton would have thought they would have disintegrated way before they hit earth's surface so that gives us two strong pieces of experimental evidence that time dilation is real straightforward direct experiments that can really only be explained by this idea that moving clocks tick off time slowly there's a wonderfully startling implication of time dilation that isn't often as fully emphasized as it might and i'd like to briefly describe it to you now it has to do with the following fact so we know from the formula for gamma that the effects of time dilation only really kick in in a significant way when the relative velocity that's being studied in a given situation approaches the speed of light that's all true but there is another way in which the effects of time dilation in which the effects of the relativity of simultaneity in fact can be amplified over very large distances over very large distances these effects can become significant even when the speeds involved are ordinary everyday velocities so how does this go well to set it up let's first think about time for a moment from the perspective of experience right so we all generally think of time as a kind of continuous unfolding a continuous flow but for the purpose at hand it's useful to also think about time in a different way is a kind of series of moments a series of snapshots one moment after another moment after another moment and any physical process can of course be described in this way a flower a wild animal running moment after moment after moment horse running and so forth it's just a series of snapshots that capture each subsequent moment in time in fact you can even go out into space if you will and think about the earth in its orbit around the sun again moment after moment after moment okay so what i'd like to do is start with that way of thinking about things and i'd like to compare my set of snapshots my sequence of events that are unfolding over time and want to compare my snapshots to somebody else's snapshots who's moving relative to me and to do that there's a related idea that i want to introduce which is the concept of a now slice and by a now slice what i mean is i consider the world and i think about all things that are happening right now like the stroke of 12 on a clock or at that moment my cat jumping or perhaps other events like a bird taking flight at this very moment say in venice or we can go cosmic on this too so we can imagine at that very moment a meteor just striking the surface of the moon or go even further away we can imagine a supernova explosion way in the far reaches of our galaxy now a now slice is a slice in this picture here where i put down all those events which i say happen at one moment in time and if i look at one now slice after another this is the unfolding of one moment after another after another so each of those events that lie on a given slice constitute those things that i say were real were happening at a given moment one now after another now after another now there are two points that i want to stress about this we give this picture a name that makes perfectly good sense we call this space time right because we have all of space in each one of these slices imagine the slice goes on forever includes everything that's out there in the cosmos at a given moment and along this direction of course we have the unfolding of time so we have space and time that's where the name comes from second point is common sense and everyday experience would tell us that every single observer in the universe regardless of their motion should agree on what is on a given now slice that's the newtonian view of how the world is put together but when einstein comes into the story that radically changes 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 they do not agree on what is happening at a given moment in time and that has a startling implication that i'd like to describe and to do that let me use a little metaphor here it's one that actually i used in my nova program fabric of the cosmos if you've seen that but if not it's a straightforward metaphor think about this whole expanse this whole space-time expanse as if it's kind of like a big cosmic loaf of bread and what these now slices are i'm basically cutting this loaf of space time into pieces which represent all of space at a single moment in time from my perspective if someone is moving relative to me they have a different perspective of what now is what is simultaneous and that means they carve up the loaf at a different angle from me so let me just show you that schematically so let me imagine i consider the bird's eye view of that picture just because it's easier for me to draw and let me write down my now slices in there so from the bird's eye view i will draw space at one moment of time space at say the next moment in time the next moment in time and so forth so these are all my now slices and just so that i have these labeled in a way that we all understand put it down here going to the right in this picture is what i consider the future and going this direction is what i consider the past now somebody is moving relative to me and let's say they are also interested in drawing space-time slices so let's draw theirs and because they're moving relative to me they will slice up this region of space-time at say a different angle relative to me they slice the loaf with a knife that's angled relative to my slice because their notion of simultaneity what's happening at a given moment say differs from mine now if we are dealing and this is the point if we are dealing with low velocities the person far away has a low velocity so we're not talking about velocities near the speed of light what that translates to in this picture is that the angle that we have here this angle is relatively small so in the vicinity of where that person is sitting low velocity motion has virtually no impact but the point and i'll show you an animation of this in a moment is that over larger and larger distances let's say i am over here and let's say somebody else who's doing the moving is far away over very large distances a tiny angle can get amplified into a very large difference in time a very large difference in our conception of what's happening at a given moment so let's take a look at that idea in in animated form let's imagine we are looking at a big expanse of space and time and we have a character an alien very very far away in space and we have a more familiar looking character a human being sitting still on a bench over here now if initially these two individuals are not moving relative to one another they share the same idea of simultaneity if there's no motion so they slice through the space-time loaf in the same way they both agree on what's happening at a given moment in time okay but now let's change things a little bit let's let our alien friend hop on an alien bicycle say and let's say the alien starts to ride away from me because of the relative motion between the alien and me or the guy on the bench the alien has a different conception of simultaneity a different notion of what's happening now and what that means is when the alien slices up the space-time loaf into all of space at a given moment the now slice the now slice will cut through at a different angle and again the point is small velocity means small angle but consider a small angle over larger and larger and larger distances between us and that small angle turns into a big change in time so in fact the aliens now slice actually sweeps in to the past and it can be a significant sweeping into the past when you put in some numbers as we'll do later on in this course you find that the sweep goes beyond when that guy was a baby goes further back in time than that and in terms of events on earth that the alien would claim to be happening right now from his perspective it might be hundreds of years ago say beethoven putting the final touches on the fifth symphony now the thing that's not completely obvious about this and does take some mathematics and if you're taking the math version of this course we will do the math if you're not taking the math version i hope this is sufficiently exciting that you might take the math version of the course but putting that to the side why did it swing to the past and not say to the future here's a quick way of thinking about it remember the treaty signing ceremony president of backward land right backward land was rushing away that president and he signed the treaty late right he was not the one who did it first he did it second if you recall so in essence if you are moving away you are sweeping to the past you are old news from that perspective but that also means thinking now from the treaty perspective the president of forward land if you are approaching if you're going forward your notion of simultaneity should sweep into the future and indeed that is the case so if the alien hops on the bike again but turns around say and doesn't ride away from earth but rides toward the earth then indeed the alien's notion of what's happening right now on earth does sweep from what we consider to be the present into what we consider to be the future and might include strange things from our perspective like this guy's great great great granddaughter maybe teleporting from one place in the universe to another so the point is the whole notion of what you consider to be real what you consider to be taking place right now is totally dependent on your emotion right so initially when the alien was not moving say relative to us let's put ourselves in the position of the guy on the bench from our view we agree with whatever the alien says is happening right now whatever's real we totally agree right now when the alien gets on a bike we don't suddenly discount the alien's perspective because he's on a bike so if the alien then says that other events are considered to be now to be real on his now slice at a given moment we should accord that statement the same status the same believability as when the alien wasn't moving relative to us so if the alien tells us that things in our distant past are real they are on his now slice at a given moment we need to take that into our perspective on what's real if the alien tells us that things in our future are on his now slice at a given moment we need to take that into account too so what this collectively tells us is that the traditional way that we think about reality the present is real the past is gone the future is yet to be that is without any real basis in physics what we're really learning from these ideas is that the past the present and the future are all equally real time dilation is one of the strangest counter intuitive concepts that you're ever really going to encounter and it's nice to have a kind of mental mnemonic a kind of shorthand way of thinking about this strange idea that perhaps makes it a little bit more intuitive i'd like to give you such an intuitive way of thinking about time dilation now and let me say you can justify the explanation i'm about to give you mathematically and if you're taking the mathematical version of this course we will justify it a little bit later on but if you're not don't worry about it but this gives you a nice way of thinking about why it is that time ticks off more slowly when a clock is in motion here's the idea forget about time for a moment let's just think about space and imagine that we have a car that's headed due north at 100 kilometers per hour now imagine that the car steers off and drives to the east without changing its speed now its motion in the northward direction will not be as quick as it was previously because some of the northward motion has been diverted into northeast motion so what that means is motion can be shared between dimensions and when motion is shared in that way motion that was fully devoted to one direction gets diverted to another direction so motion in the initial direction slows down let me just show you a little visual on that so here we have our car i'm going to show three versions of the car one's going due north the others are going northeast at various angles and there you see the point this car has traveled much further in the northward direction than these cars because these cars have diverted some of the northward motion into eastward motion so the idea is when you go in a different direction through space you divert some of your initial motion into that new motion in the new direction good okay now let's take that idea and apply it not to space but to space and time okay so right now here i am and you would say that i'm not moving relative to you say but of course i am moving right look at my watch right my watch is ticking second after second after second taking me forward in time forward in the time dimension if you will good now imagine that i get up and i start to walk einstein basically told us that as i walk i divert some of my previous motion through time into this motion through space which means i move through time less quickly much as over here this car goes less quickly in the northward direction because it's diverted some of the initial north motion into east motion when i start to move i divert my initial motion through time into this motion through space so my passage through time slows down that idea to me is the most straightforward intuitive way of understanding time dilation you can make it mathematically precise but putting that to the side if you want to think about why it is that a clock slows when it's in motion simply think to yourself when it's sitting still all of its motions through time when i see it moving through space it has diverted some of that motion through time into motion through space so it passes through time more slowly that's why time ticks off slower on a moving clock the constancy of the speed of light we've seen in the context of all of the ideas of special relativity has a dramatic impact on our understanding of time it also turns out remarkably that the constant speed of light has a dramatic effect on space and also on mass and we're going to talk about both of those but for now let's turn to the first one the implications of the constant speed of light for our understanding of space so first is why would we expect motion to affect space well it's pretty straightforward right 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've also learned that time is not constant so that means that space must in some way compensate for the non-constant aspects of time in order that their ratio stays the same allowing the speed of light to be unchanged so in schematic language in order to ensure that the speed of light is constant space must adjust itself in tandem with time so that the ratio for light stays fixed so the picture you should have in mind is something like this if we consider that time is not constant therefore space must change too in relation to motion so that the ratio space over time is such that the speed of light can remain unchanged and what we'd like to do is take that rough idea and make it explicit we'd like to now determine what the effect of motion is on space how are we going to do that 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 would you measure the length of a train well that's a pretty straightforward thing to do when the train is stationary right because if the train is stationary you take out a tape measure and you measure the length of the train right so let's get things going let's imagine that that is the situation and we're going to consider the length of a train from the perspective of somebody on the train so that would be our fearless train rider george from his view the train is at rest relative to him so he pulls out his tape measure and he simply stretches it from one end of the train to the other and that way he measures the length of the train and let's say he finds that the train is 210 meters long good okay that's all perfectly straightforward let's now imagine that our second character gracie she is on the platform so from her view the train is in motion so she has to use another approach to measure the length of the train right she can't really pull out her tape measure and measure the length of the train because the train's rushing by right so that's not a way that she's going to measure the train's length instead she does something more clever let's assume that she knows the train speed let's also assume that she has a stopwatch what she can do is the following she'll start the stopwatch just as the front of the train is passing her she will stop the watch as the rear of the train passes her so she knows how long it took for the train to pass by her she knows the speed of the train and she simply multiplies them together she multiplies velocity times time to get the length of the train right so let's see her do that there she is on the platform she has her stopwatch handy the front of the train goes by boom she starts the watch going when the rear of the train passes her boom she stops the watch she gets the elapsed time in this case 5.9 seconds she multiplies it by the known speed of the train in order to get the train's length that's her approach now here is the remarkable fact the two approaches george's approach where he simply used a tape measure gracie's approach where she uses this watch and the known speed of the train they yield different answers right so if you multiply this out just in this particular example 30 times 5.9 177 meters numbers i should say which i've made up just to illustrate the point which is that gracie has gotten a shorter length of the train compared to george now at first sight that is hugely surprising right but the question is does this actually puzzle george assuming that george has taken the discussion that we've already had to heart and he fully knows about time dilation with the notion of time dilation does the discrepancy in the length of the train from his perspective and from gracie's perspective does it puzzle him and the answer is no because from george's perspective here's what he says he says look i understand gracie's approach she's using length equals speed or velocity times elapsed time but i also know that gracie from my perspective right i'm now george gracie from my perspective is in motion clocks in motion tick off time at a slower rate if a clock is ticking off time at a slower rate it'll show less elapsed time and therefore it will in that multiplication yield a shorter length so from that point of view george understands why it is that gracie got a shorter length but the question remains who is right is the length of the train 210 meters as george says that it is or is the length of the train 177 meters as gracie says that it is now you can probably guess the answer to the question of who is right based on what we have discussed so far the answer is they are both right the answer is length itself is a concept that we need to rethink we normally think about 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 that idea comes from the following fact that we have emphasized repeatedly simultaneity is in the eye of the beholder right now to measure the length of an object you need to measure its front and its rear simultaneously at the same moment now if two observers have different notions of simultaneity they will therefore have a different notion of the length of an object no single result is solely right no result is wrong they're all equally good now having said that just a little bit of language we generally call the length of an object when you measure it when you are at rest relative to the object as george is in the case of the train we call that the rest length of the train we call it the proper length of the train but that just calls out a particular perspective the perspective of somebody 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 upon the speed of the object so the general conclusion then that we are reaching is that moving objects are shortened along the direction of their motion and let me stress that it's only along the direction of motion that the object will appear shorter the height of the object will not change at all and a little tiny argument can establish that for instance if you were to imagine that the train were going into a tunnel right and let's say it just barely fits now if it were the case that from one person's perspective the height of an object not in the direction of motion if that were to change let's say i were to say that the height of objects gets smaller well then from my perspective on the train the tunnel will be smaller i won't be able to fit i should smash into it right from the perspective of someone who's on the tunnel it'll be the train that will be shrunk in that direction and so it will fit now however weird relativity is it can't be the case that in one person's perspective there's a crash of the train into a tunnel and from another person's perspective there's not a crash that would really be a contradiction that would be a paradox that can't happen and therefore we learn that it can't be the case that dimensions that are perpendicular to the direction of motion they are not changed at all they stay fixed and so we describe the shortening of an object along the direction of motion we call that length contraction or we call that the rent contraction that is the language that we used and let's take a look at a simple example of that so here is a case where we're looking at a new york city taxi cab moving pretty quickly along and along the direction of motion the taxi cab is shrunken it is shorter along that direction than it would be when it is at rest so let's take a look at a demonstration which will give you a feel for the amount by which an object appears shrunken along its direction of motion when you are looking at it so this little demonstration here allows you to pick the speed of this taxi cab as it's rushing by you and again get a feel for this in your bones as the speed creeps up not much of an effect on its length but when the speed approaches the speed of light the object gets ever shortened along the direction of motion now you can ask yourself a natural question at this stage which is does the object in motion really shrink and if it does like what's the force that's squeezing on it now that's a natural question but it's pretty loosely phrased because the main point that we have come to is that the notion of length itself the notion of length itself requires a notion of simultaneity because again if you're measuring the length of an object in motion if you first say measure the back of the object and the object moves and then you measure its front let's say i'm measuring the length of a fish in a pond right if it's swimming along and i first measure its tail and i let the fish swim and then i measure its nose well i'll get a different length than i would if i measured the front and the back the nose and the tail at the same moment so it's critical in talking about the length of an object to commit to a notion of simultaneity and that requires choosing a frame of reference because observers in motion don't agree in what happens at the same moment in time and because of that different vantage points differ regarding what happens at the same moment we come to the conclusion that an object in motion has a length that depends upon its speed because its speed determines the degree of the lack of simultaneity between two perspectives so if we ask the question again do objects in motion shrink the best answer i can give you is yes and no it definitely is the case that an object in motion has a length from my perspective that is shorter than when that object is at rest but it's not as though someone has come in with a vice and squeeze crush the object down it simply is that the old idea that there's a universal notion of the length of an object needs to be updated by relativity the length of an object depends upon its speed when you measure it and that is a surprising result the length of an object depends on its speed now give you a couple other little examples of that to have in mind and both of these examples make use of the idea that we discussed earlier the relationship between observing something and measuring something so back then i described how we're mostly focused upon reality so we don't really care so much about human perception we care more about what happened in the world to be responsible for what we see but sometimes it's kind of fun to look at what we literally would see if some of the effects of relativity were visible to us and in this example here we are now going to look at a taxi cab but done more precisely so this is how a rushing taxi cab would literally look if you could see it rushing by at high speed notice that the taxi cab appears kind of twisted right in the last frames of that little video we could see the whole back bumper of the taxi cab even though ordinarily if a taxi cab was rushing by us we wouldn't be able to see the whole bumper we'd only see the part that was nearest to us the reason for that is a little bit complicated it makes use of the fact that when we look at something we are seeing light from the object light takes different amounts of time to reach us from different points on a three-dimensional object because those three-dimensional points are a different distance from our eye if you take that into account then that little video gives you a really good sense of what it would be like to literally see an object rushing by near the speed of light second example puts us inside the taxi cab itself shows us what it would be like to look out the window of a taxicab that's rushing through a city near the speed of light and as you can see the world around you not only does it have the length contraction that we've described if you take into account the finite light travel time the differences from one point to another you see that space has a kind of warp distorted curved look the world around you seems to be curving in around you when your speed approaches the speed of light so again if you could find a taxi that could travel near the speed of light and if you had really good eyes so you could actually see the world around you as it was rushing by you at very high speed that is 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 that we've already done with time so once you know that time has weird features you know that space must have weird features too in order that in tandem they can keep the speed of light constant we understand now the resolution of the poll in the barn paradox at least qualitatively let's now take a look at the numbers the mathematical details which will allow us to reconcile these two apparently contradictory perspectives so what we want to do is to first take a look say at the perspective of team barn we know that they say that the poll fits and what we want to work out is how does team pole come to a different conclusion from the perspective of team barn team barn is like i can't understand how they came to this different conclusion they said the poll doesn't fit let's understand mathematically how it is that they resolve that headache that conundrum and they do finally understand this claim of team poll that doesn't fit so to set this up let's start by considering the perspective of team barn let's try to figure out how they make sense of the strange conclusion that they hear from team pole and to do that let's record a little data to begin with so according to team barn the pole is rushing by at 12 13 the speed of light pretty fast and remember that the pole its rest length we are told is equal to 15 feet okay good now let's draw a little picture of this so we know what we are talking about we have the pole it's rushing along and it's rushing along in this direction to my left as i face the board and the speed of this guy is at v equals 12 13 c okay now what does this mean from the perspective of clocks that are being carried by the pole from the barn's perspective right so let's imagine that the pole has clocks at the front and the back and let me draw those for good measure so let's say we have a clock here and we have a clock over here and we know that the clocks from the barn's perspective will not be in sync with one another right so if the pole is rushing this way we know that the leading clocks will lag behind in time they will be late right and let's calculate how far behind this clock is relative to the clock at the rear and that we know how to calculate the time difference so the difference between those two clocks this little formula we take v times the distance between those two clocks divided by c squared so plugging in the data that we have at hand velocity is equal to 12 over 13c we've got the distance between those clocks and again this formula is the distance as viewed in the frame whose clocks we're discussing so this will be 15 feet and since we're using feet of course we'll take c to be one 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 when team barn looks at these clocks if for instance this clock over here on the right is reading 13.8 nanoseconds then this clock over here on the left will be 13.8 nanoseconds behind so it will be reading zero when this one's reading 13.8 nanoseconds now qualitatively what does that mean qualitatively that means that according to team barn team poll is assessing the location of the rear of the pole before it assesses the location of the front of the pole that means that the pole moves between when the rear position is assessed and when the front position is assessed and that's why the front has time to get out of the barn according to team barn that is why according to team barn team pull comes to this weird conclusion that the poll doesn't fit now let's make that quantitative we know that there's this 13.8 nanosecond difference in the reading of the clocks that is not the time difference according to team barn regarding when the front and the rear are assessed because of course this clock needs to catch up to that one 13.8 nanoseconds but it's ticking off time slowly because it's a clock in motion according to team barn so if we calculate gamma in this case which will allow us to convert that time difference into the time difference according to team barn what is gamma in this case well again 1 over the square root of 1 minus 12 over 13 squared and if you work that out that gives us a nice answer of 13 over 5. so what team barn says is you take this 13.8 nanosecond difference in the clock readings and you multiply it by 13 over 5 to figure out how long after the assessment of the rear of the pole will the front of the pole's position be assessed by those in team pole so let's do that so if we take 13 over 5 times the difference in the clock readings and let me just write that as 12 over 13 c times 15 feet divided by 1 foot per nanosecond just to get our full answer so the 13s go away the 5 goes into 15 3 so this gives us 36 nanoseconds so according to team barn 36 nanoseconds after the rear of the polls position is assessed the front of the polls position will be assessed by those in team poll how far does the poll move between those two assessments well that's just velocity times time right so we have 36 nanoseconds how fast is this pole traveling well it's going 12 13 c so that's 12 13 foot per nanosecond using our usual formulation and therefore if you just plug in the numbers and calculate this out you will get 33.23 feet approximately as how far the pole is going to move between the assessment of its rear location and the assessment of its front location so what does this mean let's draw a couple of pictures to see what this implies for the measurement of 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 get inside of the barn and now let's look at the motion of the pole from the barn's perspective so according to the barn folks gracie and her friends they say that the poll fits inside period end of story but they also recognize that the clocks that are attached to the pole are not in sync relative to each other and they say that the pole people first measure the location of the rear of the pole and only later measure the location of the front and we've calculated that there's a 36 nanosecond time difference between when the rear and the front are measured which means the pole moves during that interval in fact we've calculated how far it moves the distance between these two locations we calculated that as 33.23 feet so that's how far the pole moves between when the rear of it is measured and when the front of it is measured so of course the pole doesn't fit inside the barn during that interval it travels this distance so the front of the pole gets outside of the barn so in this way team barn is able to clear their headache right they didn't need any excedrin they didn't need any advil they just did a little calculation and in that calculation they recognize that team pole first assesses the rear location and only later assesses the front and by the time they assess the location of the front it has slipped outside of the barn that's how team barn explains this weird sounding conclusion according to team poll that the poll does not fit inside so that gives us our nice explanation according to team barn of the weird conclusion of team poll good now what we want to do is the same kind of analysis but from the perspective of team poll we want to understand mathematically how it is that team poll can explain the observations the conclusions the claims of team barn that the poll does fit and we can do really just the same calculation but it's worth doing a second time same essential ideas but now let's look at this from the perspective of team paul and try to understand how team paul explains this headache inducing claim that team barn is saying where the team barn says the poll fits team paul says how could they say that and we want to understand how they clear that conundrum by analyzing 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 of reference are not in sync even though the barn folks say that those clocks are in sync right so just so we can get a picture of what's going on here if we draw a little schematic here of the barn so the barn has clocks all along its length let's just draw the clock say at one end and the other and according to team pole it is the barn that is rushing along 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 is rushing in that direction according to team poll this is then a leading clock it's in the direction of the motion leading clocks lag behind they are late so this clock will lag behind this clock and we can calculate how much it will lag behind in order to understand the reasoning of team barn from team pole's perspective so what is the time difference between those clocks well we know what to do we take the velocity which is 12 13 c we multiply it by the distance between the clocks from the perspective of team barn that's the way this formula works that is 10 feet and we divide through by c squared which is just one foot per nanosecond in the approximation that we are using so this gives us 120 divided by 13 nanoseconds and if you just work that out that's approximately 9.2 nanosecond difference between these two clocks what that means is according to team poll if say this clock is reading 12 noon then this guy over here will be lagging behind and will be 12 noon minus 9.2 nanoseconds so qualitatively what this means is according to team poll team barn is first going to assess the location of the front of the pole say they will claim it's inside at 12 noon and then they will allow some time to elapse before this clock catches up to 12 noon and in that interval the barn will move to the right allowing the rear of the pole to slip inside and that's why according to team pole team barn says that the front and the rear of the polar inside at the same moment whereas from team pole's perspective those are not the same moment because the clocks in the barn are not in sync now we can make that quantitative of course by figuring out according to team paul how long will it take for this clock to catch up to 12 noon it's not 9.2 nanoseconds because clocks in the barn frame of reference are ticking off time slowly according to team poll so we have to multiply that 9.2 nanoseconds by gamma to get the amount of time that will lapse according to team poll between when those two clocks when i should say this clock catches up to 12 noon so let's do that what is gamma gamma is 1 over the square root of 1 minus 12 over 13 squared 1 minus v over c squared work that out that is 13 over 5. so now if you allow me i'm just going to take that 13 over 5 and multiply it by the time difference that we calculated above so if you don't mind i'm just going to do it right over here 13 over 5 times 120 over 13. 13s cancel 5 into 120 24 so that gives us 24 nanoseconds so again according to team paul the assessment of the front and the rear of the pole happened 24 nanoseconds apart that is this clock and this clock differ from one another and it takes 24 nanoseconds for this clock to catch up to 12 noon now in that interval the barn is moving over to the right how far does it move well that's just velocity times time so we have 12 over 13 c so let's do that 12 13 feet per nanosecond multiplied by that 24 nanosecond time difference and if you just calculate that out it comes to about approximately 22 feet so according to team pole the barn is moving 22.2 feet between the measurements of whether the front and rear of the pole are inside namely more precisely it is moving 22.2 feet between when this clock and this clock have the same reading so we can see what that implies by drawing a diagram similar to the one that we had over here but now this is from the perspective of team paul let's draw a version of the barn and let me as before give some doors for this pole to come inside and from the perspective of team poll what we therefore have learned is the following from their view the poll does not fit inside and again that's the end of the story according to team paul but team pole also recognizes what we have now calculated which is that the clocks in team barn are not in sync with one another so according to team poll what happens is the barn folks first assess the location of this end of the pole they say it's inside but then they wait 24 nanoseconds before they assess the location of that side of the pole so let me just draw that so imagine that we now look at this situation 24 nanoseconds later so let me draw another schematic of the barn now according to team pole the barn has moved in those 24 nanoseconds which means that even though according to team pole this end of the poll did not fit inside the barn at the same moment as that end of the poll they recognize that according to team barn they do fit because according to team barn what happens is they wait 24 nanoseconds later according to team pole the barn moves over we've calculated how far that is how far was that ah that's 22 feet so let me just record that for good measure so if i draw a little dotted line here a little dotted line over here this now is 22.2 feet so clearly this diagram is not to scale but the idea is correct so according to team paul what happens is the barn moves over 22.2 feet between when team barn assesses this location and that location and then of course according to team barn the pole will fit inside it's not measuring according to team paul the front and the rear at the same moment whereas 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 coming back with the vengeance so that is the explanation according to team paul of how it is that team barn comes to this weird conclusion that the pole fits inside whereas they know that it doesn't fit inside both of these pictures are absolutely correct they're just two different perspectives that allow us to understand how it is that not only do these two frames of reference come to a different conclusion as to whether or not the poll fits inside but we now also understand how each team understands the other team's claim even though they don't agree with it again coming from the relativity of simultaneity so that's the mathematics behind this way of resolving the apparent paradox of the pole in the barn what we want to do now is to get some feeling for these ideas by working with some demonstrations which are good to play around with these things on your own and that's what these demonstrations are for so let's take a look quickly at one but you should study this in more detail this is very similar to the demonstration that we had earlier except now you will note that it's adorned with a little bit of extra detail so we now have doors on the barn one moment in time from the perspective of the barn but not one moment in time from the perspective of the poll and you know that because there's going to be a flash i should have said that when it was going watch for the flash and again do this on your own the flash is at one moment in time from the polls perspective the poll says that and that those are same moment in time from the polls perspective very different moments in time from the barnes perspective and similarly you should work on the poll perspective two where you'll see again the relativity of simultaneity but now in mathematical form fully explains this poll in the barn paradox the twin paradox is the most famous of all paradoxes in the special theory of relativity but before we get into it let me just stress the most important point there are no paradoxes as i said earlier in special relativity if there were the theory would collapse there are however situations where it seems like there's a paradox it seems like we have two perspectives that we can't somehow meld together into a coherent story but that generally means we have to think the situation through with detail think it through with our understanding of the essential physics and when we do that all paradoxes fall to the side the same will happen here okay let's set it up the paradox or the seeming paradox will involve a couple of characters they're twins george and gracie and the scenario is one that may have occurred to you in our earlier discussion of time dilation and space travel because we are going to imagine in this case that gracie goes into a spaceship she travels out into space we're going to have her go pretty fast she's going to turn around and come back here is the issue from george's perspective stay at home george on earth looking at gracie from his perspective he knows how time works in special relativity he says that her clock must be running slow so that when she comes back he will say that he will be older not as much time will have elapsed on her slow moving clock compared to him gracie however she looks back at george and she says to herself look i understand special relativity too and it's george from my perspective that's moving i'm stationary and therefore it's his clock that's ticking off time slowly so when i return it should be the case that i gracie am older because less time will have elapsed on george's clock that is the issue let's see that in visual form so here is the question there is george first character and let's bring in his twin these are fraternal twins gracie she gets into her spaceship and we're going to send her off into space so let's do that and let's say she's going fast so there really will be some kind of significant time dilation going on here she goes off into space she's going to reach some point that we will call p the turnaround point and she will then come back okay so then the question that we are faced is when these two guys compare the amount of time that has elapsed according to each will it be the case that george is older than gracie will his clock have ticked off more time will it be the case on the other hand that it is gracie that is older will it be that more time has ticked off on her clock or you could even imagine a resolution that puts these two together maybe it's the case that each of them will have aged each will be the same age as the other when they return that is the question we want to figure out which of those three scenarios is correct to resolve this seeming paradox that each says that the other's clock must be ticking slowly each therefore says that they should be older okay so i'm going to offer you some resolution to this paradox the deep question is who is right which of these scenarios is correct now you might guess that they are both right because that has been often times what we have encountered so far in special relativity each perspective is right and you just have to reconcile them with your understanding of how special relativity works so that's a natural guess but that guess will not fly in this case that will not work because at the end of the journey right george and gracie are together right so they can face one another they can be standing right next to each other in fact they can even go into the same frame of reference and there can no longer be any mismatch in their observations at that point one cannot look at the other and say that you are older and the other looks at the other person says you are older that's a contradiction that's a paradox that can't happen so we cannot rely upon the kind of resolution that we have found earlier somebody here is right somebody here is wrong how do we figure out which well i am going to give you the answer it is gracie who turns out to be younger 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 cuts to the heart of the matter where does the contradiction come from where does the apparent paradox come from it comes from george saying that he can be viewed as at rest gracie moving therefore her clock ticks off time slowly and of course gracie can say i am stationary and it's george who's moving and therefore i can claim that it's his clock that is taking off time more slowly is that valid in this situation well remember the only time that you can claim to be at rest and the rest of the world is moving by you is if you are going at constant velocity constant speed in a fixed direction that is manifestly not true in this case for gracie gracie going out into space and then coming back she has to turn around and when she turns around she has to accelerate she slows down and then she speeds back up to get back to george on earth she feels that acceleration she knows that she is moving she is no longer moving at constant velocity she is no longer justified in saying that she is at rest and the rest of the world is moving by her so her perspective her reasoning is negated by the fact that she is accelerating she is not in an inertial frame of reference throughout the whole version of this journey and therefore we cannot trust her conclusion george on the other hand is in an inertial frame he is at rest on earth's surface he does not move subject to the issues of whether earth surface is an inertial frame but those are the complexities that we are not worrying about for the discussion we are having george's perspective therefore is valid he is constant velocity throughout this entire journey his conclusions are unassailable they are absolutely correct and he says that gracie's clock is ticking off time more slowly and therefore when she returns he will be older period end of story that is the resolution or i should say a resolution of the twin paradox we will come to other explanations as we proceed further with this deep scenario that lets us really sink our fingers into a lot of the details that we have been developing but as a first pass to this scenario that is the explanation gracie is accelerating for part of the journey her perspective therefore cannot be taken into account she cannot say she's at rest george is the only one who can say that and his conclusion that he will be older is correct perhaps the most famous equation in all of physics is equals mc squared it comes right out of the ideas that we have been developing and i'd like to first give you a sense of where it comes from just motivating e equals m c squared for those of you who are taking the mathematical version of this course i will do a full derivation in short order but let's just begin with the ideas that lead us to the possibility that energy and mass e and m might be deeply related they might have some deep connection that is ultimately embodied in einstein's famous equation equals 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 jousters so it's going to be a kind of joust but different from the one that you might have in mind we're going to have two individuals perfectly matched so they're going to be on identical horses they'll have identical masses and they're going to hold a lance but not one that has a sharp end we're going to imagine that there is a large metal ball at the end of the lance and the way this joust will work when the two combatants cross by each other just as they are passing each is going to lunge outward toward the other smashing their spherical balls together trying to knock the opponent over so let's take a quick look at what that joust would look like this initially was meant to be another george and gracie animation but you know i contacted gracie she said talk to my agent getting kind of diva like so we have a new character whose name is evil george so it's george versus evil george there they have their lances with the metallic spheres they slam those into each other and because they are perfectly matched we are assured that it will be a draw a tie okay now george takes a course in special relativity and he starts to think he says to himself from my perspective i'm stationary right and therefore evil george is coming at me and let's assume that this joust is happening at very high speeds the horses are moving near the speed of light let's just say to be dramatic and george says to himself that means evil george is in motion from my view that means evil george's watch his clock is ticking off time more slowly that means from my perspective evil george as he goes by he may be moving quickly on his horse but his movements will be slowed down slow motion so george says this is going to be a piece of cake because as evil george goes by he's going to lunge at me so slowly that i will easily be able to knock him over and win so that is his view in his mind so just to peer into georgia's perspective as he thinks about this relativistically he says evil george's lance is approaching me slowly so i should win and yet he doesn't win it's still a draw so the question is what was george leaving out evil george thrust the lands at him slowly he should be able to knock him over because he is going to thrust it quickly what has he left out well if you think about it the amount of impact that you receive from a joust of this sort depends on two things not one it depends on the speed of the lunge that's absolutely the case but it also depends on the mass of the sphere at the end so what this tells us because it has to be a draw again right because it can't be that you change your frame of reference and you turn a draw into a win all observers in space time agree on the events they may not 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 george with the same force even though the lance is going slowly it must be that the mass the mass of the sphere at the end of the lance must increase to compensate for the slow push that evil george uses so that suggests to us that energy of motion must be able to increase the mass of an object and in fact we can go a little bit further we know the degree to which evil george's lance has slowed down it's just the time dilation factor gamma that we have encountered over and over again so it must be the case that the mass at the end of evil george's lance increases by the same gamma factor to precisely compensate for the slowdown with which evil george is thrusting his land so that suggests to us that the mass of an object must depend upon its speed the mass depends upon its speed and the way it depends on the speed is the mass that it has when it's at rest multiplied by the gamma factor now that is a remarkable formula because if we look at a little demo over here look at what this formula is telling us as the velocity of an object gets larger and larger this is telling us that its mass becomes bigger and bigger so you can play with this on your own of course but there you see it as the velocity of an object gets larger its mass grows larger and larger until as it approaches the speed of light the mass grows without bound now this has a number of vital consequences number one we have spoken a lot in our discussions about the speed of light and implicitly we've always been using the fact that nothing can go faster than the speed of light but you may have recognized i have never really established that for you now i have because if you think about it when you try to speed up an object you've got to push on it right now if its mass gets larger because its speed increases from your initial push to get it go faster still you have to push it harder and harder and in fact as the speed of the object approaches the speed of light its mass gets bigger and bigger and bigger which means you need to push harder and harder and harder to get it to go faster still until at this point you need an infinite push to get it to go beyond the speed of light 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 take the energy of motion the energy of evil george's motion it turns into if you will the mass of an object that he is carrying so this interchangeability between energy and mass is itself pretty vital but let's take it one step further still when an object is at rest it still has mass of course m at v equal to zero what we call the object's rest mass so using the interchangeability of mass and energy we're led 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 energies are in traditional units you may recall it's kilograms times meters squared per second squared if you're not familiar with that formula don't worry about it 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'd have to multiply mass by something with the units of meter squared per second squared meter squared per second squared well meters per second that's a speed what speed would we multiply by in order 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 amazing realization that energy and mass are interchangeable sort of like dollars and euros with the conversion factor between energy and mass being nothing but the speed of light squared e equals m c squared thanks for coming along on this ride into the wondrous world of special relativity these wild ideas that come out of einstein's thinking about space and time matter and energy let me leave you with just one thought as you reflect back on all of the material that we've covered all of the results that we have found all of them fundamentally come from one idea the constancy of the speed of light right that is where the relativity of simultaneity came from remember forward land and backward land it's the constant speed of light which makes it so that those on the train and those on the platform do not agree on what happens at the same moment the constant speed of light is what makes clocks tick off slowly as they are moving by us right we use the light clock again constant speed of light as it travels along the double diagonal that is why time ticks off slow when a clock is in motion we then parlay that into length contraction where lengths of objects in motion appear shortened along the direction of motion and finally we've gone further and we've come to this stunning realization 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 really able to think it through to its logical conclusion sometimes that results in a revolution in the way that we think about things so go back to the rest of the course review it try to get a feel for these wondrous ideas because special relativity is truly one of the crowning achievements of our species

Channel: World Science U

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Keywords: Brian Greene, Space, Time, and Einstein, Free online courses, Physics, Albert Einstein, special relativity, Quantum Mechanics, General Relativity, black hole, speed of light, Relativity of Simultaneity, How Fast Does Time Slow?, Time Dilation, The Twin Paradox, Implications for Mass, laws of physic, masterclass, World Science U, WSU, World, Science, Festival, New York City, NYC

Id: PWo5SnrrOU4

Length: 151min 26sec (9086 seconds)

Published: Wed Jul 22 2020