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 animation form and visuals, and to give you some intuition for the … 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. The story I first wrote down 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 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 president from the two countries wants to sign the treaty before the other president, so 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 … Oh, there you go. They’re going to sit at opposite ends. It’s was harder than you think. Opposite ends. Oh, it’s like I got jazz hands here. Opposite ends of a long trade, and in between them is going to be a light bulb. And the idea is that the bulb will initially be in the off position and then the secretary will turn on the bulb sending light, heading left and right. And since the speed of light is constant, isn’t effected 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 entourages, if that’s the right plural of entourage.
There’s the two presidents, president of Forwardland facing forward, Backwardland is facing backward, that’s where the names come from. The bulb is 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 Forwardland signed first and the president of Backwardland 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 Forwardland is running toward the light, so it hits his eyes first. The president of Backwardland is running away, so it hits his eyes second.
Let me show you this a little bit more slowly. Again, speed of light is constant. It’s just that the president of Forwardland is heading toward the light, so look, the light doesn’t have to travel as far to reach his eye. Boom hits it first. President of Backwardland, 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 with the speed of light being constant. It is just that the president of Forwardland 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 of Forwardland has made up some of the distance by heading toward the light. President of Backwardland 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 of Forwardland 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 present of Backwardland, according to those on the platform because the president of Backwardland 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, as 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, while according to those on the platform, those events are not simultaneous. And in a sense, 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 just referring back to The Elegant Universe, I remember I said, writing as a young man, whatever it was more than 20 years ago, that if you took one thing away from the book, 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 and 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 Forwardland signing the treaty and the president of Backwardland 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 happened 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 Forwardland facing forward, Backwardland facing back, and in between we have our little light bulb. And I don’t know, the little light bulb gives off some light that’ll stream both left and right towards each of the two presidents. And we want to work out the time difference between it hitting those two presidents.
So to do that, let’s first work out the amount of time that it takes that beam of light to reach the president from Backwardland. How do we do that? Well, bear in mind that once the light is emitted, it streams toward the president of Backwardland, So certainly it has to cover the distance between the bulb to the presidents. 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 two and L over 2 being the two distances the light needs to travel.
But because the train is moving while light is in transit, the president of Backwardland is actually going to move a little bit toward the right while the light is in transit. How far will the president of Backwardland move? Well, if the speed of the train is V and it takes a time TB 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 the distance that the president of Backwardland 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 Backwardland in the eye, must be L over two plus V times TB. And that allows us to solve for TB. And we can just write TB times C minus V equals L over two and therefore, TB is equal to L over two 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 Forwardland. The only difference here is that the distance that the light needs to travel is now L over two minus the distance that the president of Forwardland travels while the light is in transit. Because Forwardland, the president of Forwardland, is going toward the light, makes it easier, it decreases the distance that the light needs to travel. And 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 C plus V. Only difference is that sign L over two. And therefore, TF is equal to L divided by two times C plus V.
All right, so we have the two 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. So let’s do TB minus TF. What do we get? So we get L over two times, one over two times C minus V minus one … Whoops, I already have two there. We don’t want to double it up. One over C mine is V times one over C plus V, and that gives us then L over two. 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 plus V. And you C plus V times C minus C squared minus V squared. So it’s C plus V from the first term minus, and I have to multiply above and below, denominator and numerator, by C minus V on the right. And now we can just get L over two times upstairs to V divided by C squared minus V squared.
So our formula then is LV 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 those 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.
Now, there’s another way I can write it. Let me also write this as L times V over C divided by C times one minus V over C squared. All I did was pull some Cs out on the top and the bottom to write it in that forum. 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 than 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 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 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 yield significant discrepancies in what observers claim to happen at the same moment. So it’s 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 change my mind every time I sit down to do this, but I think I’m going to focus upon relativistic mass, or at least sometime 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.