Okay, so that's introduction to Alice and Bob and frames of reference. Let's do another few examples here, few more examples to perhaps clarify things a little bit or just see if we understand them correctly. Okay, let's add another frame of reference here. We'll add Earth, so let's just say we've got Okay, here's Earth, and we're going to imagine that Alice and Bob are back in their spaceships again, and they start off perhaps in a race or something. Maybe not a race, but they're just flying around and so here is Alice in her spaceship and she's going with velocity v. And here's Bob in his spaceship and we'll say he's going with velocity, if I can write it well here, 2v. Okay, whatever the velocity is, Bob is going twice as fast as Alice. So, this is now from the perspective of Earth. There's a, and we're on Earth, we're the observer. We're watching Alice go by us at velocity v. Bob goes by us at velocity 2v. We also on Earth would have our lattice of clocks, again an imaginary lattice or grid of clocks, but they'd all be synchronized. So that's what it means to observe something. We say later on, where are Alice and Bob out here some place. To figure that out from our perspective on Earth, from our frame of reference it means our clocks out here take pictures at that point where Bob is, or where Alice is, and so on and so forth. So we can pinpoint their location and we can pinpoint their time and that gives us a space time event. Well let's do a couple of space time diagrams here. First of all, let's look at what this looks like from Earth's perspective. So we'll do that over here, maybe. Okay, so we'll call this t sub E for the Earth frame of reference and x sub E for the Earth frame of reference in terms of the position, so time and position, with respect to Earth. And we don't need the tick marks here, we'll do sort of a, maybe not quite as precise here but we'll get the essential details. So Alice's is going along at velocity v, and as she goes along each of the clocks here, Earth's clocks, take photos, the Earth's frame of reference clocks take photos. And so we'd see perhaps something like this where, again, we'll do red here for Alice going by, so you've got Alice perhaps going like this, and so that would represent her path along the x axis in time, and then Bob, of course, is going faster. And so this is Alice. Bob is going faster than Alice. Twice as fast, actually. So, his path, or his path will be along the x axis here. But the representation of that in our space time diagram, remember that lower slopes mean faster speed. He covers more ground in a given amount of time than Alice does. In fact, he covers twice as much. So at t = 1 second right here he's gone twice as far. So he's about right here. There, that was one second. When Alice has gone this far at this time, Bob has gone twice as far. So he's about right there. And at this time, when Alice has gone this far again, he's gone twice as far. He's going twice the velocity, so it'd be maybe right about there. And so on and so forth. So this would represent Bob. His plot in the space time diagram. And lower slope means faster velocity. So that'd be the Earth perspective. What would Alice see from her perspective? As far as she's concerned, she's sitting here, she may say as far as I'm concerned, we'll pretend she doesn't know if her engine is on or anything like that. She sees Earth going backwards away from her. And she sees Bob going forwards away from her to the right. So Earth is receding back to the left, Bob is going out to the right because of course Bob is going twice as fast as she is. So from the perspective of her cockpit, she see's Earth receding at velocity v, she sees Bob going away from her at velocity v as well. Because the difference between their velocity, Alice and Bob's velocity, is just v. 2v- v leaves us just v. So if this Bob was going 20, we'll say 20 meters per second, then Alice is going 10 meters per second, that means Bob is going away from Alice at a net velocity of 10 meters per second, or if Bob is going 30 meters per second, Alice would be going 15 meters per second, so the net velocity difference between them is 15 meters per second. So for Alice do her plot here. So t sub A, for Alice, and x sub A, for Alice's perspective. She sees Earth going backwards to the left in the negative direction at velocity v. And that would look something like this. There's that line there so that would be what Earth would look like from her perspective and she sees Bob moving away from her at the same velocity but in the other direction. And so, it looks something like this. Roughly speaking. So, there's Bob from Alice's perspective. Bob moving away at constant velocity, Earth moving the other direction behind her at constant velocity. What about Bob's perspective? Let's do one for Bob here. Let's try to make it straight, straighter okay. So there's that. This is X Bob and T Bob. So this is from Bob's perspective. His frame of reference. So, what does he see? Well, he's moving away from earth at twice, at 2v, whatever v happens to be. So from his perspective just sitting in his cockpit, he sees Earth going behind him. Receding in the distance backwards at 2v. And what about Alice? Well, he sees her receding behind him. Remember, he's going faster than she is. So he sees her receding by him at a velocity v. So again maybe if Earth is going, we'll use for example here a very ridiculously small velocity, say 30 meters per second. So if Bob is going 30 meters per second, Alice is going 15 meters per second so he'll see Earth going backwards at 30 meters per second and Alice going backwards at 15 meters per second and you'll get something like this. So roughly like this. So this would be Earth from his perspective receding backwards at 30 meters per second or twice what the velocity is and then Alice. If I can sort of get this in here like that. Just eyeball it there. Alice would be something like that from his frame of reference. So, if you look at this you say they are like mirror images of each other. Bob's perspective and earth's perspective. Which makes sense because Earth's Alice is moving that way at v and Bob at 2v. If we flip the perspective to Bob he sees earth moving behind him at 2v, this one and Alice moving behind him in the other direction at v. So, it helps just to play around with these diagrams a little bit, make up little examples for yourself. See how these things work. I'm going to do one more thing on these diagrams, then we'll do one other example here to have you sort of think about things. But if I were to plot, let's look at Alice's plot here. Where would Alice be on this plot? Okay, this is really her frame of reference. She's sitting in her cockpit, she sees Bob going away from her in that direction, she sees Earth going the other direction. How would you plot her on here? Well, think about it. From her perspective, she's just sitting in her cockpit at position 0. So as time goes on she's always at position 0 from her perspective. So on a space time diagram, the plot for whatever observer or really whatever frame of reference we're talking about is just going to be at position x = 0. And so at time t = 1, or whatever here, this is her position right here and then the next time here and here and here and here. So, in other words, that's her plot on her space time diagram. She's sitting at position 0. Similarly for Bob, same thing, he's sitting in his cock pit. Did I, I can't remember, did I lift up Alice before? Anyways, so here's Bob. He's sitting in his cock pit at position zero. So, if he's recording his position, there's a little clock there too that if time goes on it just takes a photograph of himself I guess like some of those arcade machines and it says, okay, here I am I'm at that position, position zero the whole time. So, that's his plot on there on his diagram. Same thing on Earth, if we were the Earth observer and we had our lattice of clocks here and we said lets record our position as time goes on, that's pretty trivial, we're all at position 0 and so our position here, or the Earth observer's position here would be there as well. But note, of course, from another perspective, our position, from Bob's perspective it's like that, from Alice's perspective, it's like that, and so on and so forth, different frames of reference. So the physical events are exactly the same. We've got an earth Alice moving away at v, Bob moving away at 2v, or from Alice's perspective, Earth moving backwards at v, and Bob moving forwards at v. Or from Bob's perspective, Earth moving backwards at 2v, Alice moving backwards at velocity. It's the same exact physical situation, but we have different perspectives, and so we end up with different space time diagrams. We also have a fancy word, somewhat fancy word for our plots here, these are called world lines. World lines. So it's in this case, this is Alice's world line. From the perspective of and observer on Earth, from Earth frame of reference. This is Bobs world line it is tracking his position and time through the frame of reference of earth. Through the Earths space time diagram. And the Earth observer is just, the world line Earth observer is just a vertical line here, it never moves. Same things for here for Alice, got the world line above from Alice's perspective, the world line of earth from Alice's perspective and the same way for Bob as well. When we talk about world lines, so we talk about Bob's world line from Alice's perspective talking about Alice's world line from Bob's perspective and so on and so forth. Couple other things just to know about world lines and space time diagrams if you haven't figured out yet. But anytime you have a vertical line, let's say I plotted something, let's use this one down here, Bob's diagram here. So if Bob did a plot of some event and it looks like this. Like that. That is the world line of an object in Bob's frame of reference that does not move. Because look at this, at every given time it's at the same position. So a vertical line represents something that is not moving, so that might be maybe. Maybe the tip of Bob's, the front tip of Bob's spaceship. As far as Bob is concerned, he's in the cockpit. That's not moving from me. It's a fixed distance from him here. Maybe he's sitting in the cockpit right here, you know one meter ahead here or a number of meters not one meter probably a little more than that. So 10 meters ahead of him is the front of his spaceship, and it's not going to be moving as far as he's concerned. It's always 10 meters ahead unless something terrible has happened to his spaceship. So, vertical line represents something that just stays in the same place in that frame of reference, just like Bob in the cockpit. It's staying in the same place at the 0 mark, maybe this is the 10 meter mark at there. Same thing for all these other things. We can draw a vertical line, that represents an object sitting at that location that is not moving in that frame of reference. But of course in another frame of reference it is going to be moving, the front of Bob's space ship. To Alice is going to be a line like that in fact. And actually not quite exactly because if you think about it this is 10 meters here. Then from Alice's perspective, if this is Bob exactly in his cockpit, then the front of his ship is also going to be moving up just sort of 10 meters ahead of where Bob and his cockpit is. So that's a vertical line, what about a horizontal line? Horizontal line as we've emphasized represents everything that goes on in that reference at that given time. So if I draw horizontal line, we'll come over here to Earth's perspective here. If I draw this horizontal line right there more or less. All right, that represents the time we call that t E = 3 seconds sort of 3 minutes or whatever it is so, this is the t = 3 line and if I look at everything along that line, that represents everything that happen from Earth's perspective at t = 3 seconds. So, Alice was at this position right here at t = 3 seconds, whatever that mark is on the x axis. Bob was at this position right here at t = 3 seconds. Our clocks took a photograph at that point in that time and noted their position. So horizontal line represents everything that happens in that frame of reference at that particular instant in time. Vertical line represents something that is not moving in that frame of reference. It's staying in the same position. Diagonal lines represent things that are moving. If it's tilted to the right, it's moving off to the right. Tilted left, it's representing something moving backwards. If it tilts down more, it means it's moving faster. If it tilts up more, it means it's moving slower. So, that's just application of where the frame of reference is, the definition of what a frame of reference is, and then looking at it from our spacetime diagram perspective