Relativity of Simultaneity: Deriving the Time Discrepancy

[00:00:00]hey everyone welcome to this next episode of your daily equation today I'm going to focus upon an equation that captures the notion of the relativity of simultaneity a mouthful right what does that mean it means that two individuals that are moving relative to one another will not agree on what things happen at the same moment of time they will not agree on what events what things happen simultaneously and again it'll come out again from this notion of the constant nature of the speed of light an idea that is so incredibly fruitful at least in the mind of an Albert Einstein okay and then after I explain the idea in in animation form and visuals and to give you some intuition for the ia well intuition is probably too strong a word actually if you think about it many people have asked me do I have an intuition for these ideas in relativity and the answer the most truthful answer is not really I can follow the chain of reason I can do the mathematics but do I have sort of a deep inner intuition in my bones for these ideas I don't think that I really do so I don't know if I can get you to that point but at least I want to show you the chain of reasoning so at least you can explain it to yourself in your own mind for why it is that these weird implications are true okay so to give you that level of understanding I'm going to begin with a little story a story I first wrote down in in my book the elegant universe so some of you may be familiar with the story but it goes like this there are two nations that have long been at war they are called forward land and backward land and and that's not a value judgment by any means as you see it comes right out of the scenario that I'll be describing now the two nations finally have come to a peace treaty but neither from the two countries wants to sign the treaty before the other presidents they need some scheme that will ensure that they each sign the treaty simultaneously so they consult with the United Nations secretary-general and they come up with the following plan the two presidents are going to sit at opposite how do I orient my hands so they look like ah there you go you're gonna sit at opposite ends it's harder than you think opposite ends it's like I got you know jazz hands here opposite ends of a long trade right and in between them there's going to be a light bulb and the idea is that the bulb will initially be in the off position and then the Secretary General will turn on the bulb sending light heading left and right and since the speed of light is constant isn't affected by the direction in which it moves and since each president is equidistant from the bulb the travel time to each president's eyeball will be the same when they see the light each president picks up the pen signs the treaty and that should be the means of ensuring that they sign simultaneously now one little additional detail that the secretary-general puts into the scenario a crucial one as we will see the secretary-general says hey when you sign the treaty why don't you let the inhabitants of the two nations that live on opposite sides of a railroad track why don't you let them watch the signing ceremony by doing it on a train that goes right along that track and both presidents say sure yeah why not let's do it on a train so that everybody can witness this historic event okay so now let's take a look at a little animation that will show you what happens and we'll begin with the perspective of those on the train the two presidents and their respective entourage 'as if that's the right plural of entourage there's the two presidents president of forward land facing forward backward land is facing backward that's where the names come from the bulb was in the off position and then we turn on the bulb and when we do so the light streaks left and right heading toward each of the presidents again same distance same speed of light so it hits them at the same moment they pick up their pens they sign the treaty and everybody is so thrilled with the outcome but then surprisingly shockingly unexpectedly all of these folks on the train get word that riots have broken out you can see it here on the platform because the inhabitants of the two nations do not agree that the presidents signed at the same moment in fact they claim that the president of forward land signed first and the president of backward land signed a second let me show you why they come to this conclusion let's rerun the scenario but now not from the perspective of those on the train but from those on the platform here's what they see the light goes off and the president of forward land is running toward the light so it hits his eye first president of backward land is running away so it hits his eye second let me show you this a little bit more slowly again speed of light is constant it's just that the president of forward land is heading toward the light so look the light doesn't have to travel as far to reach his eye boom hits at first president backward land boom hits him second because he's running away from the light light has to travel farther to reach his eye again this has nothing to do with the speed of light changing it has everything to do the speed of light being constant it is just that the president of forward land is making it easier for the light to reach his eye by heading toward the light light doesn't have to travel as far the president forward line has made up some of the distance by heading toward the light president of backlund is making it more difficult for the light to reach his eye by racing away on the train from that beam of light light has to travel farther so in other words we have two perspectives that I can summarize right here on the Left the perspective of those on the train light goes off heads in both directions equidistant hits both eyes at the same moment simultaneously perspective of those on the platform president before Lent is heading toward the light therefore it hits his eye first light doesn't have to travel as far and you see that the ball of light is still in transit to reach the president a backward land according to those in the platform because the president tobacco man is running away from the light light has to travel farther therefore to reach his eye what do we make of this the question of course is who is right those on the train on the Left perspective those on the platform the right perspective and the answer is I suspect many of you no doubt would guess or know or thought through they are both right it is not that one perspective is right and the other is wrong both perspectives are right even though they come to the shocking conclusion that according to those on the train the events of the two presidents seeing the light and signing the treaty are simultaneous well according to those on the platform those events are not simultaneous and in a sense on I'm actually tempted to just stop our discussion right here because that is such a deep insight into the nature of time again before Einstein the view was that there is a universal notion of what happens at a given moment a universal notion of what things happen simultaneously but Einstein showed that the constant nature of the speed of light which is vital to this story because the light is having to travel different distances according to those on the platform at the same speed and therefore it doesn't complete its journey simultaneously according to those on the platform those two events are not simultaneous and you know just referring back to the elegant universe I remember I said writing as a young man you know whatever it was more than 20 years ago that if you took one thing away from the this scenario of the treaty signing ceremony and how it allows you to come to the conclusion that simultaneity is in the eye of the beholder in the eye of the observer if you took that away I'd be satisfied if that in fact was the only point you took away I feel the same here if in this whole series the one thing that sticks is the relativity of simultaneity and this little story that allows you to think through why it's true I will feel like this whole series has served its purpose but I'm not going to stop here this is called your daily equation for a reason which is I want to now just spend a couple of minutes won't take long to derive mathematically the time discrepancy according to those on the platform between these two events between the president of forward land signing the treaty and the president a backward land signing the treaty and that is not hard to do we're going to get a formula that expresses that time difference in terms of the speed of the train that's vital the train wasn't moving everybody would agree on what things happen at the same moment you need observers in relative motion and also as we'll see what comes into the formula is the distance between the two presidents and that has an interesting implication which I will finish up with now to get there let me use my iPad here and let me try to bring it up on the screen good all right so now you and I we can now work this out together how to do that all right let me just draw a little picture so imagine that I have my train I'm not going to really draw a train but schematically and I've got the president of forward land facing forward backward land facing back and in between right we have our little light bulb and I don't know the little light bulb gives off you know some light that will stream both left and right toward each of the two presidents and we want to work out the time difference between it hitting those two presidents so to do that first work out the amount of time that it takes that beam of light to reach the president from backward land how do we do that well bear in mind that once the light is emitted it streams toward the president of backward land so certainly it has to cover the distance between the bulb to the president so let's give that a name let's say that the entire distance between the two presidents let's call that equal to L and therefore you've got L over 2 and L over 2 being the two distances the light needs to travel but because the train is moving while the light is in transit president to backward land is actually going to move a little bit toward the right while the light is in transit how far will the president to backward land move well if the speed of the train is V and it takes a time t be for the light to reach the president then the distance that it will have to travel is half the length of the Train plus V times TB so it's got to cover half the length of the train it's also got to cover at the distance that the president of backward land raced away while the light was flying toward him and therefore it must be the case that C times TB that is the total distance that the light travels before it hits the president of bakwin land in the eye must be L over 2 plus V times TB and that allows us to solve for T V then we can just write T B times C minus V equals L over 2 and therefore TB is equal to L over 2 times C minus V okay now let's do the same calculation for the amount of time from the perspective of those on the platform for the light to reach the president of forward land the only difference here is that the distance that the light needs to travel is now L over 2 minus the distance that the president of forward land travels while the light is in transit because forward land president of Portland is going toward the light makes it easier decreases the distance that the light needs to travel and well how far does the train travel while the light is in transit V times TF and again just as above we can solve that TF now it's C + V only differences that sign L over 2 and therefore TF is equal to L divided by 2 times C + V all right so we have the the 2 times TB over here and TF over here and to get the discrepancy between those two events that is the lack of simultaneity of those two events we just can subtract those let's do t be minus TF what do we get so we get L over 2 times 1 over 2 times C minus V minus 1 whoops already have a 2 there I mean don't want to double it up 1 over C minus V times 1 over C + V and that gives us then L over 2 and I'm going to combine those by putting them over the same denominator C squared minus V squared how do I get that well I'm going to multiply the left term by C + V and you see plus B times C minus V get C squared minus V squared so it's C + V from the first term - and I have to multiply above and below denominator numerator by C minus V on the right and now we can just get L over 2 times upstairs - V divided by C squared minus V squared so our formula then is L V divided by C squared minus V squared and that's it that's the formula that we are looking for this L is the L measured by on the platform so you don't have to worry about length contraction per se it's built into the very definition of L and the formula is quite nice though there's another way I can write it let me also write this as L times V over C divided by C times 1 minus V over C squared all I did was pull some seeds on the top and the bottom to write it in that form and writing it that way is particularly nice because V over C in everyday life is a small number and therefore this time difference this is the time difference between the two events is minuscule and that's why in everyday life we are not aware of the relativity of simultaneity but if the over C is large then that relativity of simultaneity that time difference will grow larger and larger but I want to finish up by just noting one other curious fact that's kind of fun to bear in mind because there's this factor of L that comes into the time difference formula you can also make this expression large not only by making V approach C that's sort of the usual thing in relativity you say hey how can you make the effects bigger you got to go to higher and higher speed you can also make it large by letting L whoops let me get rid of that guy over here that's making this guy L goes big big separation between the two events of interest and so if you imagine that even if you have slow speeds V over C is a small number but two observers are very far apart maybe on you know opposite sides of the universe there's some subtleties that even come into talking about the notion of time over such large distance scales but put that to the side on very very large distances even small velocities can you of significant discrepancies in what observers claim to happen at the same moment so sort of a second lever arm for making the effects of relativity larger and larger okay that's the equation that I wanted to get to today the equation that captures the relativity of simultaneity and I think next time I'll pick it up probably I keep guessing and I make the wrong guess as to what I'll do next I kind of changed my mind every time I sit down to do this but I think I'm going to focus upon relativistic mass for at least some time soon I'll do relativistic mass maybe it will be the next episode in any event that's it for today that is your daily equation looking forward to seeing you next time take care