On to part four of our space time diagrams revisited. And so what we've done here is we've expanded our diagram from last time where we had Alice's plot space time diagram, her frame of reference. And then we figured out what Bob's space time diagram, the axis, would look like if we put it also on the same plot as Alice's diagram, and this is what we got from last time. Note that just to put some numbers here, we've chosen v equals 0.5c for the relative velocity between the two frames of reference. Bob in other words is moving at velocity 0.5 c in the positive direction, which means gamma is 1.2. That will be important in a minute or two here. And we also mentioned last time that we had to be careful about the tick marks on the axes for Bob here. We can't just draw them in, assume they're in the same location, because we have to go according to the Lorenz transformation equations, so we know that Xa or Xb and Tb do not give us the same numbers in Alice's frame of reference, Bob to Alice or Alice to Bob here. So here's what we need to do. We need to say, okay, here's Bob's frame of reference in sort of standard form, and here are the points on his axes here. So, for example, the point xB equals 0, tB equals 1 is this point right here, okay? That's just on his tB axis. And this would be the point. X would be equal to zero, t sub B equals 2, and so on and so forth, and the other points. This would be the point x sub B equals 1, t sub B equals 0. And so on and so forth. And so what we need to do is take each of the standard tick marks on Bob's frame of reference here, in standard form, and translate them, transform them using the Ren's transformation equations, onto the skewed axes here. We're not going to do all of them, we'll just do a couple of examples, so we can get the idea here, and then we're going to see what implications that has for how we can read things off the diagram. And in particular, we'll be able to see time dilation and length contraction just from this diagram alone here, which of course makes sense, because the whole thing really is based on the [INAUDIBLE] transformation equation. So let's take a look at this. Bob's space time point XB = zero, TB = zero. We're going to use Lorentz transformation equations to figure out, we're going to be able to plot this point from Bob's frame of reference there onto Alice's plot here. And, we expect it's going to appear on the T sub B axis, because all these points are on the T's of the axis. In Bob's frame of reference, we show that this is how the T's of the axis looks like when we put on Alice's plot. And therefore, we know this point is going to be on here some place we just don't know exactly where, and that's what we need to figure out. So, if we look out our equations up here, so it's actually the top appear that work well this time. So, we'll erase that and say ok. Because knowing I've got x sub b equals zero, so I can put zero in for the x sub b's here and t sub b equals one. So what I get for that if I do that. We're going to put these in here. So I get x sub a equals gamma. Now x sub b is simply zero. Plus v times t sub B, which is 1. So, I just get v times 1, and obviously, zero plus v is just v, so this becomes gamma v. And in fact, let's put some numbers in here for it. If gamma is 1.2, and V is .5 C, here. 1.2, or really, .5 C, and remember, C is the speed of light, of course. And, then we're using it in units of light years per year. So, it just has a value of one, very nicely for us. So, this becomes .5 x 1, and times 1.2. So, one half of 1.2 is .6. And this would be in units of light years, or whatever light type units we happen to be using there. So that gives us the x sub A coordinate of this point in Bob's reference frame, but transformed to Alice's. And then t sub A Is simply going to be gamma times. Now t's of b is 1 plus v over c squared times zero in this case. X would be a zero. And very simply that just gives us gamma, and gamma is simply 1.2. Okay, so let's plot that now. So we say okay we found out that this point right here, that point there. From point of reference, when I put it on Alice's plot is going to appear at x of a equals 0.6. So, this is one, two, three, four. So, it's about 0.6 is about right there. And t sub a is going to be headed value 1.2, so If I'm going to move it up here, it's about right here is where it is. Now, when I'm putting tick marks on here, remember, tick marks need to be parallel to the appropriate axis. So, all of the tick marks here are for tA are parallel to the x axis. Because, really, these represent many lines of simultaneity. There are all this indicates. All along there is a line of simultaneity for t of a equals one, t of a equals two, t of a equals three, and so on and so forth. And they need to be parallel to the x of a axis in this case. So tick marks on the time axis need to be parallel to the x of b axis, so here is the xb axis. Tick mark the location according the results there are that, so haven't got that quite right, but something like that. Draw it big there. So, that point right there appears in that quad. Let's do another one though. We won't do too many of these. Let's do one more here for the T sub B axis. Let's change this, not T sub b equals one, but T sub b equals two. In other words, now we're looking at that point. Where does that point occur on this axis? Okay. We're on Alice's plot really. And so, we come up here. This can very easily change our equation. XB is still 0, so the 0s still are there. But the 1 just becomes a 2 here. And this 1 over here becomes a 2 for our second case. And we see we get two gamma v. 2 gamma v and of course it's not going to be 0.6 anymore, so we'll get rid of that for a minute. And down here we get 2 gamma, 2 times gamma plus 0. So we've got 2 gamma times v. 2 times gamma, 2.4 times one-half, essentially, gives us 1.2, and 2 gamma is simply 2.4, so now we have another point, in other words, the point Xb equals zero, T sub b equals two. Bob's plot over there for his frame of reference transforms onto Alice's frame of reference into The point okay, x of a equals 1.2 so here is 1.2 approximately and 2 sub a equals 2.4, so we sort of go up here roughly speaking, it looks like it's some place right around there. Okay, so that's that tick mark. Again this tick mark is parallel To the x and b axis, because it marks out lines of simultaneity there as we were talking about in the last video clip. And we could continue doing this. We could do [INAUDIBLE] B and so on and so forth. And you can see the pattern here. We've got one there. The next one will probably be, well will be something like this roughly speaking. And so on and so forth. So that gives us the tick marks on the tb axis. Now let's do the x of b axis here. So in this case, let's do this point right here. We're assuming by the way the origins coincide. So the 0 point here matches up with the 0 point there of course. But this is xb equals 1. So now we're looking at the point x sub b equals 1, t sub equals 0. Okay, because now we're on the x sub b axis here. So x B equals 1, t sub B equals 0. Where do the x sub A and t sub A coordinates, analysis plot for that. And so again, we have a situation here where, just plug in the numbers. So, these are the things that are going to be changing here. We'll see what we get. Okay, so x sub a gamma times x sub b, so x sub b is one this time. So, there's a one here, and t sub b is zero, so that's very nice, we just put a zero on there. And for this one, t sub B is 0. And we get a 1 over here. So the top one is just gamma times 1 + 0, so this just becomes gamma. And the bottom one becomes 0 + B over c squared times 1, so that's gamma. Times v over c squared. And putting some numbers in here from before, gamma is 1.2. Gamma's 1.2. And, c squared is going to be 1, for our units of light years per year. So I've got gamma times v, and that's going to be 0.6, 0.6. So now we have this point right here, that's what we're working on now, and that one's going to come over here, and Alice's coordinates to Xa equals 1.2, so about right there, and Ta equals 0.6, so up here like this, about right there. I'm a little off on my plot here, but so that's the location and the tic mark now is going to be parallel to the tB axis. So that's, again roughly speaking, you want to draw something like that. I'm a little off there. I don't want to make it too big, but that's what that tick mark looks like, okay? Let's do one more here. So, xB = 2, because then we'll see the pattern. xB = 2, so now we're looking at this point right here, on the x sub B axis t sub B is 0, and so we come up here, the zeros stay the same, this becomes a two, this becomes a two, and we get two gamma and two gamma v over c squared in those cases, so we get 2.4 and 1.2, 2.4, 1.2. And so we go X sub a equals 2.4 So over to about right here at 1.2. So about right there, and we try to draw it roughly parallel to that. And then we get other results as we go along. So the next one would be something like about right here, probably. And so on and so forth, okay. Now what that means here is, there are lines of simultaneity and lines of same location, and in terms of Bob's axes here. And so again they're parallel to the t sub B axis, so they would be roughly like that, like that, like that, okay. And, extending down too, of course. All the way down there, okay, and then the lot. So, those are lines of same location. Hope I said that right. Lines of same location, here. For a given value of x of b, so this is 3. This is 3 light years on the x of b axis right there, and anywhere on this line here, as far as Bob is concerned, those would be events that happen at 3 light years for any given time. And then the lines of simultaneity for Bob on this plot are parallel to the x of b axis so it sort of looks like this. And, try to get it in here. Of course, it goes the other direction, too. Okay. That way, you get the idea, there. This is really something you need graph paper for, to get it looking right. Okay. So, now let's see what we can get out of this result here. Let's look at a case where we have a certain point, a certain space in time event. And let's focus on Alice's Coordinates here for a minute, and let's do her a few lines of and same location. So, lines of simultaneity here. Well, sort of there. Like that. Don't want to make it too messed up here. Something like that. So I can see it's getting a little messy there. But lets see if we can see what's going on here. So, what this means is any where along this more or less black straight line here, that's the line of simultaneity for Alice that's t sub a equals 1. Okay so let's see what that intersects with Bob's t sub b axis, in particular right here. Right there. If that space time event occurred, a flash of light say, Alice would measure it at t's of a equals one, right there, and her x of a would be maybe .4 or something like that down here. So that would be, for Alice it would be x of a equals 0.4, t's of a equals 1. That would be the flash of light in her coordinates for the flash of light. Look at what Bob's coordinates are for the flash of light. Okay, so let's figure that out. So for this flash of light for Bob, clearly it's happening on his time axis, so that's x of b equals zero for him. It's happening right where he is. So it would be xb equals zero. Okay, because the t sub B axis is by definition x sub B = 0, his location. And what's the time for him? The time for him is, well note where the t sub B = 1 point is, it's right here. It's right there. And that's less, the flashlight is occurring less than that time. It looks like, you know, from here to here is one, for one light, not one light year, one year for Bob from there to there. Tick mark of one, and therefore, you know maybe it looks like about 0.7, roughly speaking. 0.75, somewhere in that region. So we say tA, just very roughly here. We're trying to get a rough idea. 0.7, 0.8, someplace in that vicinity, just trying to read it off our somewhat imprecise graph there, but it's precise enough to see. That clearly when Alice measures that event at one year on her clock, her synchronized clocks, Bob, his clock reads 0.7. Whoops sorry this is not A here, B. Bob's clock is 0.7. What do we have there? That's time dilation. Alice sees Bob's clock running slowly then more slowly than her clock. And we can do a precise, obviously it's going to be a 1.2 factor here, we know from earlier results, so I don't know if, 1 divided by 1.2 is probably not quite 0.7 but it's in the vicinity there. So that's time dilation that we can see here as well. And it's a little harder to see on this plot, but you can also see length contraction. Okay? Because a given length is going to be shorter to Alice than to Bob's. Let's see if we can just get a feel for that. Let's see, given length, right, it's going to be shorter for Alice compared to Bob. The proper length in Bob's frame of reference, say, for his spaceship. So let's look at this and see that, well, what is length here? Let's say just this distance here, right there, is a standard length of one to Alice, right? A standard length of one to Bob is this vertical line here on this plot. Clearly, this line is longer than that line. So in other words, if Alice is comparing two points say, the beginning and end of Bob's spaceship. She'll get a shorter value, a shorter measure, than Bob will. It's a little messy there, not quite so precise. But hopefully, you get the idea that this actually shows not only time dilation. It shows length contraction, as well. And in fact, one other thing you can see here, and we'll do this the next video clip, is you can see the relativity of simultaneity and leaving clocks lags, that's what we'll work on next.
