Okay, now let's spend a few minutes talking about the so-called Galilean Transformation. The name goes back to, of course, Galileo himself in the early 1600s. He didn't formulate quite like we'll formulate here but the basic idea he had was the same. This is a question, given the location and time of an event in Bob's frame of reference, we're talking about Alice and Bob again, what is the corresponding location and time in Alice's frame of reference? And we're assuming here as usual that at t = 0, our time origin, Alice and Bob are side by side, XA = 0, XB = 0, so that when we start the clocks, they're side by side. Going back to our, we'll put Bob on top this time, Alice here. And so, as they're moving perhaps with respect to each other, maybe this time Alice is moving forward. And so at t = 0, they're, in terms of their synchronized clocks, they're right together here. And then at a later time, of course, then you can figure out what the times and positions are of however they have to be moving. So, that's the question we want to address. We'd like to ideally have a little equation that we can plug numbers into so they don't have to go through the analysis every time. So, here's the situation at t = 0. Here's Alice and her lattice of clocks there, all synchronized. Here's Bob in his spaceship with his lattice of clocks all synchronized. So, the XA axis, the XB axis, so again they measure position in any place along there. Then let's assume that two seconds later, Bob is moving at velocity, v = 3. Whether it's meters per second or whatever, doesn't really matter here, we'll just call it velocity, v = 3. So two seconds, times the velocity, means that he's moved six meters, right? Maybe 3 meters per second times 2 seconds, 6 meters, and that's what we've represented here, where each position of the clock represents 1 meter. So from Alice's perspective, he has moved six meters. So one, two, three, four, five, six. You can see, he's in position six right there. His clocks have moved forward as well. The clocks would extend outward there. Now, again, going back to the question. Some location and time of an event in Bob's frame of reference. So, let's get an event on here. Let's say at position two, he sees a flash of light at time, t = 2. So, a flash of light right here occurs right next to his clock there. Maybe right in front of me. Takes the photo and says, I saw a flash of light. In position two, at time t = 2. What would Alice say the position is, okay? What's the location in Alice's frame of reference? Well, we can see here if we just count off by the diagram, one, two, three, four, five, six, seven, eight. Right, but we don't want to have to draw a diagram every time and count like that. So let's think about this, well, Bob sees it at position two, one, two there, he has moved how far? He's moved six positions, and we know that from the velocity times the times. Two seconds, three meters a second, he's moved six meters, if meters is what we're using for our units of length here. And so we can actually say, this little equation here, we can say that XA, okay, of the flash, okay, in other words, we know the position of the flash in Bob's frame of reference is at position two. What's the position of the flash in Alice's frame of reference? Well, it's simply going to be the distance Bob has moved plus where he sees the flash occurring. So it's actually going to be, we'll write like this, we'll say, will be his position, XB of the flash. And I'll erase that, so we can get a little more room here, but for the moment we'll squeeze this in. It'll be XB of the flash plus the distance he's moved. The distance he's moved is simply velocity times time, so it's plus velocity times time, okay? So that's the equation for any given event that happens in Bob's frame of reference, any place along his x axis here where he says, I saw a flash of light at a certain time, in a certain location. If I want to find out, okay, where is that flash of light in Alice's frame of reference, where does she see it occurring? Because note that when this flash occurs, she could also take a picture right there, right? She has a photo clock, a camera there, that takes a photo of the clock and the flash and says the flash to me occurred, at, position one, two, three, four, five, six, seven, eight. And we'll deal with time here in a second. Okay, so this is the basic so-called Galilean Transformation. It's a transformation because it allows us [COUGH] excuse me, to transform between Alice's frame and Bob's frame in terms of event occurring. So if Bob measures an event on his measuring stick as it were and clocks, then we know that the position in Alice's frame of reference is given by this simple formula here. Let's erase our question again. So we've answered part of the question. We've answered given the location of an event in Bob's frame of reference, this is how we would find it in Alice's frame of reference, assuming that our origins are set properly there. Again, we'll deal with the time in a minute. But let's do an example here where. Okay, just to make this a little clearer perhaps here, what about the other way around, okay? Let's say because you might say, well, what about something that happens in Alice's frame of reference and we want to figure out what it is in Bob's frame of reference. This is the equation, if I want to go from Bob's frame of reference to Alice's, okay, a flash that Bob says it occurs at a certain position. Just add the velocity times time, just how far Bob has traveled, okay, along Alice's perspective there. But what if we want to go the other direction? So let's say in this case, we have a flash at, so our first here is we had a flash, Bob saw it a XB equals 2. What if I have a flash at, let's say for Alice's perspective, let's do it right here at three, okay? So she sees a flash or maybe the flash, we could say the flash occurs right in between them there. But it occurs to her at time t = 2 and XA equals 3. What is the position of the flash from Bob's perspective? Well, let's think about this and then we can write the equation like this. So that X Bob of the flash, is going to be. Well, we can just read off the diagram here. To Alice it occurs at position three, to Bob it occurs at position negative three. One, two, three, going backwards here, so how is that working? Well, let's see if something like this works, sort of we know that this was equation for going from Bob's perspective to Alice's perspective, where we added velocity. Well we know that from Bob's perspective, Alice is actually going the other direction, negative velocity. So, maybe what if we subtracted it? Let's just see if that's going to work here. So what if we did this if we said, it's going to be XA flash, in other words, the position of the flash as Alice sees it, we're looking at this flash now because this is what Alice measured here. And we want to find out, we might sort of confirm that yes we can get negative three because we can see that's where Bob has it. What if we do -vt? Well, again this is at two seconds we're imagining. Velocity of Bob is three here. But since it's in the other direction, it's really negative three. That's why we have the minus sign there. And what does this work out to be? Well, according to what we've said, it's in position three for Alice. So this would be position 3 -, 3 times 2 is 6. 3- 6, of course we get -3. So it works in that specific case that yes, flash that Alice saw at t = 2 seconds at position three is position negative three for Bob according to our equation there. And you can do some more examples, in fact, let's just do two more examples here. But I recommend that you play around with this a little bit. Draw some diagrams like this, plug in the numbers, convince yourself that this actually works. So that was one example we did here. So our two equations are, if I know what Alice observes, and I want to find out where it would occur for Bob, then I use this equation. Position of Alice- vt, where v is taken as a positive number here, and that'll give me the position in terms of Bob's perspective. Or, if I know what Bob saw, his position, I add vt and I get Alice's position here. So let's just write two more examples, maybe. What if we said XA = 6? So, that means that Alice has observed, we'll just do t = 2 seconds for all our examples here. And Bob moving relatively three meters per second that way. Or from his perspective, Alice three meters per second the other direction. So, Alice observes it at position six. One, two, three, four, five, six. She observes it right here. That equals two seconds. While it equals two seconds, Bob is also right there. So, we'd better get an answer that corresponds to Bob being right at that position of the flash. So, let's see what we get. So, in this equation, we're going to get XB of the flash equals XA of the flash, which Alice says it's 6- vt. And vt being 3 times 2 is 6, so I get 6- 6 = 0. That's telling me as far as Bob is concerned, the flash occurs at position zero. And that's exactly where he is, he's at position zero there in his spaceship, and he sees the flash occurring. Now, let's do another one where Bob measures something. So what if Bob says, I see a flash again at t = 2 seconds. And the flash is going to be at negative one. So, he sees it right there. And his clock records it. A little photograph there. And therefore, what is Alice's position going to be? When is she going to observe that flash? Well again, we can count from the diagram. One, two, three, four, five. So, that's the answer we're looking for here. Let's see if it actually works. So here's the equation this time, we're given Bob's coordinate and we're going to add vt and that should give us Alice's coordinate. So we say that XA of the flash is this equation, Bob's coordinate that he observed which is negative one we're told. -1 + vt. Again, vt being 6, 3 times 2. 3 meters per second times 2 seconds. Again, if we're doing this at different times, we'd have a different time here. If we're doing it 5.5 seconds, then it'd be 5.5 here. Velocity would still stay the same, for example, always you can change the velocity as well for different examples. But we're doing t = 2 seconds, velocity 3, and so it's a + 6 and 6- 1 here is, oops hold it here, wrong way. Remember it's at negative one. So, actually, got ahead of myself there. That was correct. We get an answer of 5. 6- 1 = 5. Is that correct? Well, let's see. So that's Alice's position. One, two, three, four, five. Yes, Bob occurs at negative one, Alice occurs, therefore, the same flash from Alice's perspective occurs at five. So that is the idea here of the Galilean Transformation, that any given event, and these flashes we've been playing around with represent events that occur along the x axis, perhaps from Bob's perspective. And then we can figure out what Alice would see, where the flash would occur from her perspective. Or if we're given the flash, the event, some place along Alice's frame of reference here, Alice's string lattice of clocks, then we can figure out what it is in Bob's perspective. Now, one other factor here that we've been glossing over a little bit because our initial question said, given the time and location of one event, say from Bob's perspective, what about the time and location from Alice's perspective or the other way around? We've been working with the locations. We have two equations now. So here are the two key equations. There's this one right here. This assumes, it's helpful to fix this in your mind so that you get the minus signs. The standard situation here is Alice, stationary, Bob, moving to the right at a certain positive velocity. And then, given that situation, we can say okay, if I have something in Alice's perspective and I need Bob's location, this is the equation, I subtract vt from Alice's location. Or, here's the other one right here, if I'm given something in Bob from Bob's prospective, his frame of reference, I add vt to get Alice's location of that, okay? It can get confusing here, because they're essentially the same, except you got a plus here and a minus here. So you have to know who's moving in the positive direction, who isn't. So we've assumed, for example, that Bob is the one moving in the positive direction. Alice, from Bob's perspective, is moving in the negative direction. Given that situation, then these are the equations here. If you switch Alice and Bob around, then this becomes B, that becomes A, that becomes A, that becomes B. So it's good to fix in your mind the standard situation, and then know the equations for that. So what about time though? Well, here's the basic assumption of the Galilean Transformation, really the assumption of physics from Galileo's time in a few years after Galileo. Newton thought deeply about this as well, the basic assumption from that time essentially until about the time of Einstein was that Alice's time was Bob's time. That assuming Alice's clocks were all synchronized and Bob's clocks were all synchronized, then they'd always be synchronized compared to each other as well, so that at this point, there's a flash. Alice's clocks and Bob's clocks would read the same thing. You wouldn't have to worry about that. You just have to figure out the difference in locations according to where these events were occurring. What Einstein did is he questioned this assumption. He suddenly realized, in one of his famous phrases, that time was suspect. And that's what we'll be getting into a little bit more next week, as we go on here. But before we do that, we have one more video clip for this week. And then we also have to talk about some things next week to learn more about waves. Because they're, especially light waves, are central to the idea of the Special Theory of Relativity. But Einstein suddenly realized in 1905 actually, about six weeks before he sent his Special Theory of Relativity paper off for publication. Didn't take him long once he got the essential insight that there's something weird going on with time. That this basic assumption that everyone had taken for granted pretty much, that Alice's time would be the same as Bob's time was not correct. So, that's something we need to delve into further as we go along and understand further in our overall goal, of course, of getting a deeper understanding of what Einstein is talking about and some implications of this. But, we need the background, we need that solid foundation. And the Galilean Transformation is part of that solid foundation. And based on this then, we'll see how things change over the next couple of weeks, as we delve into Einstein's work on the matter.