In previous video lectures, we've been talking about how the speed of light is an ultimate speed limit. And in this video, we want to actually look at a case where it seems like You can go faster than light. And so we're going to see how this works out. This is sort of a fanciful situation as you can see. We're going to have a trip from San Francisco to St. Louis, in the middle of the United States, Midwest, and then all the way to New York there. This is also a nice example because things like this actually happen in sort of in astronomical observations. Where it seems like things are moving faster than the speed of light especially when you have these very energetic jets that are shot out of quasars. If a little bit about astronomy which, a very energetic, galactic nuclei where there is a black hole in the center and a lot of interesting things going on. It seems like when you observe them over time, that they're actually moving faster than the speed of light. But it seems like if you really analyze it, not only seems like, but it is such that it's not really going the speed of light. Situations like that aren't too difficult to analyze, but they require a little trigonometry so we're not going to tackle that in here. Instead, we're going to use this example. Again, San Francisco to St. Louis to New York. So we'll explain what's going on here. It also allows me to do at this point to introduce a book to you that, for those of you who want to take it to the next level after this course. Especially using a quantitative approach, it's an excellent text book to do that with. I will give you a list of other books as well. In fact I have a copy right here. It's called Space Time Physics. I'll write this Up here by Edwin Taylor and John. So spacetime physics by Edwin Taylor and John Wheeler, really a classic in the field. It's in the second edition. We didn't use him for a textbook in this course. We could have, but the material we're covering is certainly less than half of what's in this book. It's not overly expensive, but you can get it online probably for about $60 US new or used maybe a little less than that. But obviously it costs some money to do that. But it is the type of textbook if you want to go beyond what we're doing in this class. It does things a little differently than we've done. For one thing they like to just get rid of c completely. We've talked about how you can treat c as one, but we keep it around in our formulas. They like to just take C out of the formulas. And sometimes I find especially for introductory courses that causes more confusion. But once you've had this course and worked your way through it perhaps a little bit, then spacetime physics by Taylor and Wheeler is a classic text in the genre as you will. So and the reason I bring up this is is a example that they do in their book. I've modified it slightly in a couple things, but it's essentially still the same thing. So, what's going on here? Well imagine we have in our spaceship or somebody is in the spaceship, maybe Bob or Alice and they fly by the Golden Gate bridge in San Francisco. Europe. And when they do that, and sort of velocity B, they have a flash of light at that point. And we take a snapshot as it were if you want to do. And then they keep flying across United States, and St Louis is about in the middle there. St Louis, you may or may not know, has a famous Gate Way Arch There and so they fly by or maybe even fly under the gateway arch there. And when they do that, they let, have another flash of light so that's flash number two at time t two whenever they they reach there, and then they're flying on to New York city at that point. So, what's going on here? What's going to happen is, the observer at New York City is going to observe these flashes of light coming by. So the first flash of light contains an image of the spaceship and the Golden Gate Bridge, moving by there. And so that's represented by this green flash, which, traveling along here, across the United States towards New York. Meanwhile of course, the spaceship is also traveling that direction, but not as fast. V is going to be less than C, of course. So it's moving along too. So then when it gets to Saint Louis and the Gateway Arch. The Gateway Arch because it was like the Gateway, Saint Louis is many respects was like the Gateway To the western United States, as Europeans moved across, and others as well. So the gateway arch there. And in flash number two here, they take a picture of the spaceship and the gateway arch. And that also then goes in this red flash of light, which is color coded here. So this red flash of light contains the image of the gateway arch and the ship at the gateway arch. And so then at that point you have these two flashes of light containing these two. Images, one from, originally from San Francisco, moving across, and one from St. Louis. And of course, the one from San Francisco is going to be ahead of the one from St. Louis, because obviously, the initial flash is moving faster than the space ship. And so when the space ship gets to the gateway arch and makes a second flash of light, the first one has already moved beyond that certain distance. And we'll see exactly. How much in a second here. Okay, and then the idea here is these two flashes of light are going to approach the observer in New York who is going to observe them, see them. And is going to measure a time difference between the arrival of flash #1 and flash #2, with flash #2 just slightly behind flash #1 there. So let's analyze this a little bit. The distance from San Francisco to St. Louis is going to be the velocity of the ship, times the time difference it takes to get there is actually velocity times distance or velocity times time gives us a distance. So if this was t1 and this was a t2, so it's just the difference in time how long it takes to get there and what velocity the ship is traveling at and we'll just call that as usual delta t. So that's the time and v velocity of the ship times delta t is the distance from San Francisco to St. Louis. And then we're also interested though in what is the distance between these two flashes that contain. Green one contains the image of the golden gate and the spaceship. And the red one contains the image of the gateway arch and the spaceship. Well, we know that when the second flash occurs, the green flash has gone this far And that distance from here to here is just going to be the speed of light times delta t. because the ship went from here to here in delta t. And so, the speed of light is going to just go a little bit farther because it's going faster. So it's going from here to here in delta t. So this is snapshot two, when that flash of light is a little ahead of that one. And so we can say that this distance here between the two flashes of light is the total distance from San Francisco to our first flash. Which is going to be c delta t, because that's again how far it's traveled, minus this distance, which is just v delta t. So, the first flash of light has gone c delta t, minus the spaceship distances covered, leaves us the remaining distance. And that's going to be the distance between the two flashes of light there, as they keep traveling on toward New York. So, c delta t minus v delta t, we rewrite that as c minus v, times delta t. So that's the distance between them, and so these flashes keep going on here. Eventually, actually very quickly, they get to the observer in New York, and the observer in New York is going to measure the time difference between the two flashes of light. Well, what is that going to be? Well okay, here comes first flashes light in It reaches that. The second flash of light is falling along behind it at speed c and a certain distance behind it. So, once the first flash of light gets here, it takes the second flash of light just that amount of distance and time to catch up with it. And that's just going to be the distance here divided by its velocity. So it's going to be the time difference between the arrival of flash number one and number two, is simply going to be the distance between them divided by the speed of light. It's how long does it take this rear Flash to catch up with the first one. So that means it's going to be c minus v delta t all over the speed of light. Okay, so again, do your little Imagination here of thought experience. So this one comes in, the green one comes in at this point and, with just the observer here. And at that instant in time the red one is still about right here traveling that way and is traveling at C still, and so it's just going to take, going at speed c it's going to take this amount of time. The remaining distance to be covered divided by speed. And you get the time difference between the arrival of flash number one and flash number two. Now here's where it gets a little interesting. Okay, because you're the observer say in New York, and you see the two flashes arrive. The first flash arrives and you see the image of the spaceship at the Golden Gate bridge. And then, the second flash arise and you see the image of the spaceship at the gateway arch in St Louis. You say, wow, okay it went from San Francisco to St Louis pretty quickly there. In fact it only took this amount of time to go from San Francisco to St. Louis because that's the difference in arrival time of those two flashes. And they're just like images coming [INAUDIBLE] images coming to your eye here in New York. And so you see first one comes in, you say hey I see the spaceship at the Golden Gate bridge. And the next one comes in a little bit later and I say I see that same spaceship now at the Gateway arch. And so, gee, I wonder how fast that spaceship was travelling to get from San Francisco to St Louis. Well, it's simply the amount of time that I perceived between those two images coming in, I see, here it is in San Francisco. And now, here it is in St Louis, divide it by the distance it had to cover between San Francisco and St Louis. Well, what is that distance? That distance is simply v delta t, so let's put this together now. The observer in New York is going to see a time difference between the two flashes arriving and those images are just going to contain, first the San Francisco image and then the St.Louis image. I say okay it went from San Francisco to St. Louis, my perception says That's how fast it occurred and therefore, its velocity, we have a little room here so let's see. Let's make a little room in here. We can squeeze it in the middle. The velocity here, I don't want to erase this cause we're going to use it here. So let's call this the, apparent velocity okay? This is the velocity of the spaceship according to the observer, based on these two images coming in. The first image when it was at [INAUDIBLE] bridge San Francisco, followed by the second image when it was at the Gateway arch in Saint Louis. And the apparent velocity is going to be simply the distance between here, present. Say, hey that's how far it seemed to travel. That's v delta t divided by the time difference. Distance divided by time is velocity. And so now, we're going to erase this a little bit to make a little room. Here is the time difference, c minus v, delta t divided by c. We can cancel the delta t's here and can bring the c actually up to the top, lodge buttons becomes c Times v over c minus v were our answer. That is in [INAUDIBLE] unit, right? We've got velocity is on the bottom, we got [INAUDIBLE] of velocity squared. Well sort of a velocity squared on the top, therefore the units will be a velocity in terms of the cancellation. So it checks out, this is a velocity. So, this is the apparent velocity of our spaceship to an observer in New York City there. Well, what is that. Okay, if we use units of C in terms as we've been doing one light second per second or one light year per year. See this becomes one. So, this time becomes V Over 1-v, okay. Now let's put in one of our typical speeds here. What if the velocity is say, 0.8c. Not a typical speed in real life of course but for our third experiments with special theoretary. Let's put 0.8c in there,okay? So that means we're get 0.8c, Divided by 1-0.8C, but remember C is just 1 here in our units. So it just becomes 0.8 divided by 1-0.8. So this equals .8 over 1-0.8. Again assuming C is 1 light year per year, 1 light second per second. Those are the same units. Well 1- 08 is .2 and what I get here is 4, and remember what the units are this is the velocity. So, if C is in units of 1, that means the units here are light years per year or light seconds per second. So let's just say light years. Per year. We'll chose those. Think about that a minute. That's four times the speed of light. That's 4 c. One light year per year is c, the apparent velocity according to the observer in New York is four times the speed of light. And in fact, if you choose another value for v here. If you make it say, .9c and put .9c in here. You'll get 9 times the speed of light, okay. And essentially, depending on how high you make c you can get very high values that are much greater than the speed of light for the apparent velocity that's going on here. It is really is sort of an optical illusion is what's going on. It's because the spaceship while the flash of light move that way the spaceship is also moving at a fairly high velocity in that direction. And at quote unquote normal velocities, if it's much, our normal everyday velocities, this would be far smaller than the speed of light. But it does show you can get situations where it seems like you can have things that are traveling faster than the speed of light. And what we're going to do in our next video clip actually, is look at a situation that shows you definitely can't have things traveling faster than the speed of light, or all of causality breaks down. So that's what we're doing next in terms of things that are faster than light. To see that, yes, you really can't have situations like this. You can have seeming situations, perhaps but in actuality you can't have things going faster than the speed of light.