Okay, in this video clip what we want to do is explore the light clock in a more quantitative fashion. In the previous video clip we did it in a qualitative fashion. Also bring out a few other points here that maybe got lost over in the previous clip. So let's see what we've got here, I want to leave a lot of space here to do some work. So we have it way over here on the left hand side but I think you can see that there and we'll point out the details of course. So Alice's clock, this is her clock as she sees, the special light clock we talked about, the bottom here the top here. One tick after one o'clock we will call delta t sub a, delta as you may or may not know. The Greek letter delta is often the symbol for a change in something or perhaps elapsed time, that's why we're using it here there's no special meaning to it or anything like that. It's just Delta t sub A is one tick for clock which is the round trip travel time for the light beam light pulse to get to the top and back down again. In fact, lets write an equation then for delta t that we can do here. So weve got delta t sub A is going to be the travel time up and back down again. We know if we have a distance to cover. And we have a certain velocity if we're going 120 kilometers an hour. Or if you want to 120, not 120, if we want to go 120 kilometers and we're going at 60 kilometers an hour or if we want to go 120 miles we're going at 60 miles per hour, we take the distance, divide by velocity we're going, the speed we're going, and that gives us the elapsed time. So 120 kilometers, yeah 60 kilometers a hour, 120 divided by 60 is 2 hours, same thing for 120 miles and 60 miles per hour, 2 hours in either case. So delta t a then is going to be the round trip distance. And we're going to call this distance L, from the bottom mirror to the top mirror. So it's just going to be 2 times L, L up and L back down again. L plus L is 2 L. Divided by the speed of light c. That's how fast it's travelling up and then back down again. Okay, so that's an equation we're actually going to use it a little later on here, but we have delta ta is 2L over c. Now let's look at Bob's perspective again. So here is the diagram. And just to bring out one thing that perhaps wasn't quite as clear in the qualitative analysis we did before. Bob of course is seeing Alice's clock moving here. So this is Alice's clock as Bob sees it. This is Alice's clock as Al sees it, just sitting next to her. But Alice is moving now with respect to Bob at a velocity. And so these are really three snapshots of Alice's clock as Bob sees it. So the first snapshot is when the light beam leaves the bottom mirror and the upper mirror is up here. We didn't draw it in there. So that's one snapshot of Alice's clock. Here's the second snapshot right here where the light beam hits the top mirror. And then the bottom here is down here in that position. And the third snapshot is back down again when the light beam hits the bottom mirror again and the top mirror would still be right up here. So, snapshot one to the clock, snapshot two to the clock, snapshot three of the clock we just didn't draw it in the top mirrors there or actually the bottom mirror in the middle we left that off as well. So it's just Bottom mirror up to the top mirror and back down to the bottom mirror. This distance here is L, this distance. It's going to be the same for both of them. You might ask, there's so much relatively weirdness going on here. How do we know L is the same? Maybe Bob see L as something different. Alice's clock shrinks or expands or something like that. Well there actually is an argument that we will consider later on to show that L with not change. That this vertical distance as Alice sees it will be the same for Bob as well. But we'll leave that at a later time. We'll just have to take that, as an article of faith at the moment. But what can we see here? Well, here's the path of the light beam, bottom mirror up to the top mirror, back down again. That vertical distance L. We'll call this path length D. D up to the top, D back down by symmetry this length and this length are the same because there's a constant velocity motion as it goes along. And then finally the bottom of this whole triangle we got here, we'll give call it x. What is x? Well, think about this a minute. The round trip time is up and back down again. By definition we're calling that delta TB. One tick of Alice's clock according to Bob is going to be delta t sub B, one tick of Alice's clock according to Alice is delta t sub A. And as we saw qualitatively they are going to be different we'd like to get a quantitative relationship between those two quantities. That's our goal here to be able to say delta tB equals something something Involving delta tA. So, this distance here, we'll call X. What is that? Well, that's just the total distance Alice's clock travels from here, to here, to here. And, that's just going to be the velocity of the clock, the velocity of Alice and her clock going by. Times the time elapsed. In other words, the time it takes for the light beam to go up and back down again, which by definition is delta P sub E. The tick time of Alice's clock or going back up and back down again as observed by Bob. So, X here, this total distance X, is just v times delta t sub B. So let's write now an equation. Let's block that off a minute. Let's write an equation for delta t sub B here, see what we can write down. We actually have a couple things we can do. Well, delta T sub B by definition, the tip time for us at clock, as Barbara views it, is this travel time B. Or actually not the travel time, the distance D plus another distance D. So, the total distance covered is two D divided by the velocity, which of course is C. Again, the principle of light constancy, as we talked about, Einsteins's principle plus the principle of relativity led him to the conclusion that the velocity of light, the speed of light, was constant for all observers. No matter how they were looking at it, whether its coming towards them or away from them or whatever. It all should be measured at some angle here like this always measured at c, and this is the sort of root source of all the weirdness, or most of the weirdness, that you get in this special theory of relativity. So velocity c up, velocity c down that way. The distance is two d so the tick time that Bob sees is 2 d over the velocity c. And right off that you could say, well hey, are we done? Because we have an equation for delta t a, we have an equation for delta t b. We can see right off the bat because d is bigger than l clearly, you know here's l hypotenuse here of the little triangle is D, D is greater than L. Therefore delta t B is greater than delta t A. The tick time that Bob observes is longer than the tick time that Alice observes. So Bob observes Alice's clock running slowly. In other words, that's the qualitative analysis we did before. And here we see it somewhat quantitatively. But we'd like something better. Again, we've got D here, we've got L here. We're not quite sure what those are. We'd like to have an expression involving delta t b and delta t a. And also maybe involving C in and maybe B as well, but not L and D. Because we could have different clocks, different lengths here, and so on and so forth. Different velocities. So what about another equation we can get out of here that maybe could help us? Well if you remember your, not even really trigonometry. But probably algebra, early algebra. The Pythagorean theorem. Right triangles. And what we have here is a right triangle. We have l as one side, we have this distance here as the other side, and d is the hypotenuse there. So we know if we knew what this side here was, we'd have d squared = l squared + this side squared. Pythagorean theorem so the question is though, what is this side right here? Well we said this distance here was x, which was the velocity of Alice and her clock times the tick time as Bob observes it, the time to take to go from there to there. So that's good, and this is x over 2. So this divided by 2 essentially is this side. So let's just start writing that down here. Let's say, okay, Pythagorean theorem says that D squared = L squared + x over 2 squared. We'll just write that first. Okay, and now at this point especially for those of you doing maybe a more quantitative approach. Taking the more quantitative approach to the course. Or even if you're just auditing or taking a more qualitative approach as well. Little challenge here for you would be to take what we've got so far. We've got this equation, this equation, and this equation. And we also know, we'll fill this in a minute, but we know X down here, what that is involving V and delta tB. And can you put them together, manipulate them a little bit and come out with a relationship just involving delta tB, delta tA. The velocity v, and the speed of light c, because that's our goal here. So if you want to pause the video clip at this point, spend a few minutes working with that see how far you can get with that. And then come back and which we'll do right now we'll plow on here and persevere. So D squared equals L squared plus X over 2 squared. We can fill in some things here. We'd like to, remember, get rid of the D squared and the L squared. The D and the L here. And, in a minute, we'll plug in this for X. But, let's focus on D first here. From this equation here. If we rewrite this. Pull the c over here, and divide each side by 2. You can see that D = c delta tB over 2. That's good news because now I can square that and plug it in here. We'll do that in just a minute. This is one here I can rewrite and it tells me that L multiplied by C here. L = c, Delta TA over 2. That's good news because I can square that and plug it in for X square, and then the X there can be this, let's plug all that in and see what I've got. So D here if I square that is going to be c square times delta tB square, all over 2 square is four of course, equals elsewhere, so it'd be c squared delta tA squared over 4. And I've got x squared over 4. Right, x squared 2 squared is 4. And x squared is v delta tB. So that's going to become + v squared. Delta tB squared over 4. And now were getting someplace. Because we see, I know have an equation, no more of Ds, no more Ls. So those are out of the picture because we replaced them with equivalent expressions involving delta tB and delta tA. I've just got cs and vs in there, c squareds and v squareds, so it's looking good. I have a delta t B squared over here, and other delta t B squared over there, so I'm going to try to combine those terms. Everything is divided by 4, so if I just multiply the whole thing by 4, I can get rid of the 4s, just cancel the 4s out, so these 4s will go away. And I'm just left with C squared delta t D squared equals C squared delta t A squared + V squared delta t B squared. Let's bring all the delta t B squared on this side. So I'm going to take this term, subtract it from each side. Bring it over here. So we'll write this as C squared. Delta tB squared minus that term, v squared(delta tB) squared = c squared delta tA squared. Again, for some of you if you've been doing a lot of things like this recently, you can almost do this in your head and jump ahead here, but we're going to take it step by step, try to make case slowly here, not to go too fast, lose anybody. And again, pause it, ponder things if you have to. I've got c squared delta tB squared, v squared delta tv squared. Let's factor out the delta tB squared there. So I've got delta tB squared times c squared minus v squared equals c squared,delta tA squared. Just doing some algebraic manipulations here. And this is starting to look pretty good here. We don't need this anymore right? So just to get a little room here because I want to bring this up here. And rewrite this here. So remember the goal was write something like delta tB equals some other stuff involving delta tA. So let's divide both sides by this. We're going to bring this over here to the denominator. So we're going to write this as delta tB squared equals c squared times delta tA squared. All over c squared- v squared. Now we're really getting some place. We have a little bit more to do here, we've got some squares in there we would like to get rid of. And so I'm going to rewrite this just slightly, this right hand side as c squared over c squared minus v squared times delta t a squared, I didn't really do much of anything there I just moved the ta squared out here so that we could focus on the c squared over c squared minus v squared. And now let's bring it back to the center of the board. So we've done all these manipulations here. And we don't need this any more either. We'll leave up our original delta tB there. And so let's bring it back to the center board so we can work with this. So we've got delta t B squared = c squared over c squared- v squared delta t A squared. Let's simplify this. We're going to use a little trick you may, or probably may not remember. Something we talked about in the math review video clip, in the introductory set of video clips, we're going to rewrite this as, keep that c squared up there, we're going to write this as c squared one minus V squared over C squared is still delta tA squared, and of course delta tB squared over here. Okay? As you convince yourself that this is the same as that, C squared minus V squared is the same as C squared, as 1 minus V squared over C squared. Multiply that by 1, we got the C squared. Multiply this times this, the C squareds cancel, I'm just left with V squared. So just a different way of writing this. And it looks actually like it's worse, and in general it would be. But, look what we've got here, now we can cancel this C squares. I got a C squared in the numerator, a C squared in the denominator. Get rid of that, get rid of that. And so we can write as delta tB squared equals 1, so we have a 1 here now on top over a 1 minus V squared over C squared times delta tA squared. But now let's take the square root, get rid of these squares. Since all our quantities are positive, you may remember when we do square roots, you gotta do a plus, minus thing to be precise about it. In this case, everything's going to be positive so we can write delta tB. What we're doing is we're just taking the square root of each side. The square foot of this is delta tB, equals the square root of this. 1 minus, V squared over C squared, times this whole thing, times delta tA. And one more thing here just slightly more. Again we'll want to bring this up in the center so we can look at it. So hopefully we've got all that done. Just bring it up here. And rewrite it just slightly more because you may remember if I've got the square root refraction. Numerator, denominator, that's the same as the square root of the numerator divided by the square root of the denominator, but square root of 1 is just 1. Again, everything being positive here. So, we can rewrite this as delta tB equals 1 over square root of 1 minus V squared over C squared times delta tA. And that is our final result for our quantitative analysis of the light clock. It's going to be a very important result as well. Let's just analyze a little bit and make sure that things check out as much as possible. So here's our final result. First of all, we said that delta td was greater than delta tA. We argued it qualitatively. We saw it somewhat quantitatively here. We knew that d was bigger than l and therefore delta tB must be a bigger number than delta tA. Again the idea is that the moving clock ticks more slowly from Bob's perspective. Does this result here also indicate that? Is delta tB greater than delta tA? If it isn't, after all that manipulation, then probably we made a mistake some place along the way. Or our earlier analysis is incorrect. So let's just think about this a minute. And let's also think about a couple extreme cases here. What happens if V is 0? Okay if V is 0, that means Alice's clock isn't moving at all. It's just like Alice and Bob are side by side, and they're sitting there and they've got their clocks next to them. And they should be running the same with identical clocks. If V is 0 here, this becomes 0, so I've got 1 minus 0 square root of 1 is 1. I get 1 divided by 1 is just 1. I get delta tB equals delta tA, as we would expect. In other words, if they're just sitting there next to each other, there's no difference in their tick rates assuming that they are identical clocks. And the math is checking out for us there. So that's okay. What happens if V, let's think about another extreme case. What happens if V equals C, if Alice is actually going at the speed of light? Well, if V equals C here, I've got C squared over C squared then. That's 1. 1 minus 1 is 0. I've got a 1 divided by 0 situation. This is infinite. So that's a problem. But it's an indication from the mathematics, that Alice's velocity can never be equal to the speed of light. In fact, what we do, more arguments later on this essentially that, nothing can travel at the speed of light, because you get these contradictory situations that delta tB equals infinite times delta tA. So rule that situation out. What about everything else with velocity between 0 up to but not quite the speed of light, any of those situations. Well V then is going to be less than C for sure. So this quantity right here, V squared over C squared, is going to be the square of a fraction which gives you another fractional number. Something like 0.8 or 0.792 or something like that. So I'm going to have 1, essentially to say it another way, this number's going to be between 0 and 1. If you're not sure about that, get out your calculator. Just do some examples here with the v and c, and you'll see you always get a number that's like, again, 0.8, 0.32, or whatever. So we've got 1 minus a number between 0 and 1. That means this whole thing inside here is going to between 0 and 1, a positive number. The square root of a number between 0 and 1, is always between 0 and 1. So, I am going to have 1 divided by a number that's between 0 and 1. Now there's a number smaller than 1. Give you an example. I might have something like this. 1 divided by, perhaps say 0.8. If the square root of this thing comes out to 0.8, between 0 and 1. Well, 1 divided by a fraction like that, this is 0.8, or eight-tenths, that's 10-eighths, a number bigger than 1. So, our little analysis here essentially says this quantity right here is always going be greater than 1. Could be equal to 1 actually, it's going to be greater than or equal to 1. It's equal to 1 if V equals 0. But if V has any magnitude at all, it's going to be greater than 1, and therefore I'm going to have a number greater than 1 times delta tA, that means delta tB is going to be greater than delta tA. And this special quantity here, this 1 over the square root of 1 minus V squared over C squared, occurs a lot in the special theory of relativity. And therefore, it's given a special name. We say it equals the Greek letter gamma, a special symbol for it. And one of our next tasks will be to explore gamma a little bit. And especially in terms of time dilation, and then a little bit later on in something called length contraction as well. But we want to get a feel for gamma and this V squared over C squared business here to see when this really matters. When time dilation actually occurs so that we could see it perhaps in the real world. But remember the basic idea here is that this is, I just mentioned it, we call this time dilation. Time dilation. Another way to say that is moving clocks run slow. They tick more slowly then if they're not moving. So again, if Alice and Bob are right next to each other with identical clocks, they both looked at them and said just fine. When Alice goes in motion however, as some velocity V passing by Bob with her light clock. As far as she's concerned, her clock right there is just ticking away nicely, nothing weird about it. Bob, however sees her light clock acting like this, where it goes up and comes down again at a diagonal. And therefore, he sees her clock ticking more slowly. He tells Alice, hey Alice, something's wrong with your clock because it's not working like it was before. I've got my clock right next to me, it's ticking away just fine. Your clock is now ticking more slowly. Each tick takes longer time than it did before when I compared it to my clock. So that's the whole idea of time dilation. And really you can reverse the process. You could say Alice looking at Bob's clock is going to get the same result and also she's going to see his clock ticking more slowly. And so you might be thinking, well, what's the deal with that? How do we understand that? So we'll have to figure that out, although really it all goes back down to the relativity of simultaneity. The idea being, again a rough analogy might be if you're looking at somebody off in the distance and they look small to you, and yet you know their regular height, if their right next to you. And they all they looking at you, you'll be looking small to them and we think nothing of that because its just part of our everyday experience. This type of time dilation is not part of our everyday experience because for it to really become a factor, velocity has to be very high, up near the speed of light. And, typically we don't deal with things like that, so we don't see it. Velocity here is very low compared to the speed of light. This is a very, very small number. This square root then is pretty much the square root of 1, and it's just 1 divided by the square root of 1, maybe just slightly, slightly different. But the difference is so slight, for all practical purposes as we observe things, delta tB equals delta tA and we don't observe that affect. But when you get the speeds up closer to the speed of light then this time dilation business comes into play, and so we'll want to explore that a little bit more as we proceed.