Spacetime diagrams, part three. And now, we get to bring in the special theory of relativity. So we've got one of our standard situations here. Let's revisit it. We'll have Bob traveling in a spaceship in the positive x direction to the right at velocity V. Alice will be observing. And if we wanted to plot Bob's world line in Alice's frame of reference, it looks something like this, of course. Depending on what the actual velocity was, but the slope here is simply going to be 1 over the velocity, as we've talked about and remind ourselves in a previous video clip here. These dots are not important, I was just sort of trying to get a nice plot there. So we can see, actually, if we wanted to calculate the velocity here, and we're assuming from now on that our units are going to be, say, in light years, for x, and years or time or perhaps light seconds for x and seconds for time. But that gives us c, of course. The speed of light being one light year per year or one light second per second. So, again, if we were to plot a light beam going out from the origin, it'd be at a 45 degree angle. Slope of 1 here. And we can see Bob, at least on our plot here, is traveling such that, looks like in two years, say, he goes one light year, right? And in about four years, he goes two light years. So it looks like he's going at about 0.5, one half the speed of light there. So this slope would be, actually the slope is 2 here, and therefore, velocity is one-half the speed of light. And so, that would be Bob's world line in Alice's frame of reference. Straight line, because he's going at constant velocity there. If we were looking at it from Bob's perspective, from his frame of reference, of course, he's sitting in his cockpit. As far as he's concerned, he's at rest in his lattice of clocks and so on and so forth, and he sees Alice receding behind him. So, it looks something like this to him, if he was to plot Alice's world line in Bob's frame of reference. You would see Alice receding backwards, so the slope here if we were to put that in, slope equals minus 1 over V. It's sloping downward. And so, essentially, Alice is going at minus 0.5 the speed of light to the left as far as Bob is concerned. And so, we can see there are symmetrical situations there. That's how Bob sees Alice in his frame of reference. That's how Alice sees Bob in her frame of reference. Now, for any given point though, remember, we have the Lorentz transformations. So, given, say, in Bob's situation, a given spacetime point, like this one right here at negative 2, 4, if I wanted to find the corresponding point in Alice's frame of reference over here, it would not be negative 2, 4, right? So, this point here, negative 2, 4, I plug it in to the Lorentz transformation, that would go from Bob's frame of reference to Alice's frame of reference. So, this is sort of our standard 1 here. We could plug in the coordinates for Bob. That point he measures in his space-time point, in his frame of reference. And we could find the corresponding coordinates in Alice's frame of reference. So we need, obviously, we need V so we can calculate gamma and do the calculations and get the actual results there. Or if we had a point in Alice's frame of reference, she sees a flash of light, an event, or something happens at a certain x and t value in her frame of reference, we could figure out the coordinates that Bob would see it in his frame of reference using our second set of transformations. And remember the minus sign here. So our standard situation, our standard that perhaps we memorize, is this one where we have the frame of reference, in this case Bob moving to the right, Alice observing. And then, if we have the opposite situation where something that's moving to the left, we just change the minus sign. And be careful that we have everything set up correctly there. So, that's nice. What we'd like to do is somehow could we put Bob's plot on the same plot as Alice's plot? And because it's just not a 1 to 1 correspondence, you can't just take this and put it over here. The Lorentz transformation tells us it's going to be a little bit more complicated than that in terms of going from a certain point in Bob's plot to a certain point in Alice's plot. In the previous video clip, we had those skewed plots, as it were, those non-square plots, to show you how it might work. And this is going to be very similar to that, and we're going to do it semi-quantitatively. We're not going to go into all of the details to do it precisely, but we want to get the basic idea here. Because, as we'll find, it's going to be useful later on, which is why we're doing it, of course. So, let's start off and just say, okay, here's the t sub B axis in Bob's frame of reference, right? And, actually, remember that would be the line of same location, that is, xB equals 0. So that's actually Bob's location in his frame of reference. If we were to draw his world line on here, Bob's world line, it would just look like this. It would go right along the x axis there. The x sub B axis, because that's where he's located in his frame of reference. He never moves from that position. Same thing for Alice in her frame of reference. Her position would be, her world line, would be right along the t sub A axis, at x a equals 0. So, our question here is, then, if we want to draw the t sub B axis, in the context of Alice's plot here, what do we have to do? Well, essentially, we just have to plot Bob's location. The tB axis in his frame of reference is his location. Well, we've actually all ready done that. This is Bob's location at any given time in Alice's frame of reference. So actually, we can even just say sort of like this. Erase the Bob there. Put a little arrow on here. This is the tB axis. Bob's t sub B axis in Alice's frame of reference. That's his line of same location. In Alice's frame of reference, his line of same of location is that, that's the world line he's on. Here, it's just a straight vertical line. Okay? So, you see it's skewed a little bit. Maybe you can see now the similarity to what we are doing in the previous video clip. Could we, actually, sort of qualitatively, could we do it quantitatively as well? And, yes, if we look at the Lorentz transformation here, and we say we want tB equals zero, okay. So, we're going to say we want tB equals, I'm sorry, not tB equals zero. We want the tB axis is Xb equals zero. That's Bob's location, okay? So, we're concentrating on the tB axis here, Xb equals zero. Let's look at our equations here. And I picked a nice one, it's this one right here. Okay? Let's just write that down over here. In fact, XB equals gamma XA minus vtA. I've got XB equals zero. The Lorentz transformation says that's equal to gamma times XA minus vtA. That's a true equation. And if xB = 0 then we can say this thing equals 0 and that also means that, just to be clear about let's just put an equal sign over here. This means gamma XA- gamma vtA = 0. This will play like through gamma there. And then bring this to the other side. So what that means is gamma xa equals gamma vta. And those of you who have done a fair amount of algebra recently you can jump ahead and see this in your head already. But what's our result then from this? Well, the gammas cancel. We have a gamma on each side, so we can get rid of those. And we're just left with xA = VtA. Or another way we can write that is tA = 1 over v X sub A. Just divide each side by V and switch the order there. Okay? The reason I wrote it in this form here. Is because this is our standard form for writing, you may recognize that as an equation of a line. Way back when we did a little math review on that. We're not going to push this too hard here. But quantitatively for those of you who are familiar with this essentially the t value here is my vertical axis, the x values are the horizontal axis, and the 1 over v is the slope of any line that I've drawn here. So this is telling me that when xB = 0 for Bob, which is his tB axis, the equivalent line on Alice's frame of reference is of this form, t sub A =1 over vX sub A. And that is exactly this line here with slope 1 over v It goes through the origin, for those of you who remember about the y-intercept of the equation of a line. The y-intercept is 0 here. Again, we're not going to push that too hard for those of you that's very fuzzy on it or can't remember what we're talking about there. The point being here is that. By setting xB = 0, that is the tB axis in Bob's frame of reference. And using the reference transformation we can find the coordinates of the tB axis, an equation really for those coordinates for the world line. In Alice's frame of reference it turns out be a line like this with slope 1 over v. For whatever the velocity of transformation is there between the two frames of reference. Okay, os we've drawn part of Bob's frame of reference now, and Alice's frame of reference. You also might say, before we go on to the x axis here, this xb = 0 is a line of same location, what about other lines of same location. Okay, so over here lines of same location are, do it in green here, that's a same location, that's xB = 1, 2, 3, and so on and so forth or 1.5 or whatever you want to do. Could we transform these lines of same location and also put them on this plot here? In other words we're starting to take Bob's grid, the vertical lines here at least. And reproduce them on Alice's plot. Well to do that just to indicate that this line is, if that's 1 right there, x = 1, we'd say, xv = 1, and by the Lorentz transformation, that equals this equation. Gamma xA- v t sub A, and just a little bit of similar algebra here. We could get an equation out of this. Somewhat similar to this, although this time we have a y-intercept value in there. And what we'd find is, we'd get a line something like this. Let's get rid of the slope part here, now. We've seen that. So, we'd get. Well, not quite like that. Let's get a little straighter than that. Try to get it in here. Something like that. In other words, it would be parallel to the tB axis. Just like the lines of same location here are parallel to the tB axis, when we transform it to Alice's plot, the lines of same location will be parallel to the tB axis. So it would be diagonal lines like that, parallel to this tB line we have here. This side can do it quantitatively, mathematically. There. Okay? So the vertical lines over here in frame of reference, his grid, space time grid, is transformed into lines at an angle. Slope 1 over V actually. And the lines are the same location will transform into parallel lines over here. Okay so that's part of the grid. That sort of takes the lines of same location in Bob's frame of reference. And transforms them into lines of same location in Alice's frame of reference. Now note that Bob's lines of some location are at an angle here. Alice's lines of same location are of course still vertical lines here. So they're different. A line of same location for Bob is not a line of same location for Alice. Okay, now let's look at, okay, so we did this. Let's look at now the xB axis. We like to take the xB axis here, and what does it look like if we put it on Alice's plot? So x-b axis. So, we want the x-b axis here. And the xB axis, of course, is when tB = 0. Time, t-sub-b = 0. Here, ok? So t-sub-b = 0. And if we look at our equations over here, we can use different ones, but this one in particular is a nice one to use. Because we've got the tB all by itself here. It's going to = 0, so we can say this = gamma Times tA- b over c squared xa, okay. And again, we'll just say that equals 0 because for this special case for the tB = 0, the x sub b axis. Well we can solve this equation for xA in terms of tA, in fact, we'll multiply it out this time because we should be able to see that if we've got something times something equals 0, that means either this is 0, or this something is 0. Clearly gamma is not 0. It's greater than or equal to 1. That means this must be equal 0, so this implies that tA- v over c squared, x sub A = 0. And from that, we get t sub A = v over c squared X sub a. And that is our result for it. And again if you look at that you say in Alice's frame of reference with t being the vertical axis. X sub a being the horizontal. This is the equation of a line with slope v over c squared. Okay. And no y intercept axis. So what that will look like if you actually put in some numbers here and plotted it. We'll just give you the result here, is it looks something like this. Let me see if I can get this more or less out. Something like that. That would be the X sub V axis because remember sort of X sub V axis would be in tB = 0. So, we plug in T sub B = 0 using what ever lengths transformation equations And got an equation that is true when t sub b equals zero, okay. That's important. This equation is not true just in general it's true for the case t sub b equals zero. So Bob's horizontal axis here the xb axis Transforms into Alice's plot as a line with that slope, slope v over c squared. And we could do lines of as well. Again the lines of here would be t sub equals 0, t sub equals 1, t sub equals 2, or really any t sub equals. 3.896, is the line of simultaneity. And again to do that, same idea here, we just say, t sub B = 1, okay? That's the line of simultaneity here, this line of simultaneity for Bob. Equals again our equation, gamma tA minus V over c squared x sub A, and we could solve that again, put it in a form somewhat like that. And we'd find that the lines of simultaneity for the x v axis here on Alice's frame of reference would be parallel to the x sub B axis. So, they'd look something like this. You know, in the lines of same location, it looks something like that. And now, that should look familiar to you, especially. This was our case we did in a previous video clip, where we did it over here on this side, where we had the very skewed axes, both of them at angles that were not. Not vertical or horizontal, and that's why we did it. To show that, now we have both Bob's frame of reference and Alice's frame of reference, so really they're space time diagrams on the same plot here. And so, note that any given point, let's do a red point here. Okay, let's just put, let's do a relatively easy one here as well and do it about right here, okay? Now imagine if we went back to the survey example and we're doing a plot of land. That red dot would indicate an assertive position at that plot of land. And that's constant, it's invariant, it's just that say the location of the well on the plot. On a space time diagram that red dot represents an event, say a flash of flight. You take a photograph at that position and verify yes that's where it is. But as we seen Bob and Alice would have different measuring systems for saying exactly what location and what time that occurred and we can get that now off of this plot. So, this event here, the one reason I choose it because it's very easy and Alice's frame of reference, she said I saw that flash of light. Took a photograph of it and it was at x equals 3, t equals 4. Okay? So at time t equals 4, maybe 4 years, and 3 light years away from my origin, that flash of light occurred. And then the question is. Where does that, where does Bob measure that flash of light in his coordinate system? Well, it's a little trickier than just what we can do on this diagram. We'll talk about that a little bit more in the next video clip. But clearly, it's not at 3, 4, three over and four up, for Bob. Maybe it's at something like, this is like 1, 2. And then one, two, maybe three, so roughly two to three in his diagram. Now, that's very qualitative. It's not going to be exactly like that. But hopefully you see the idea here that by putting both quotient systems really on the same plot. First of all we have to skew Bob's axis here to get to where And but we can still see his lines of simultaneity. They're not going to be just vertical and horizontal. They're going to be parallel to the x b axis for the lines of simultaneity and parallel to the t b axis for the lines of same location. So to Bob Anything that happens along this line here is simultaneous to Bob, or that line. Anything that happens on one of these lines is are things that happen in the same location throughout time. So Bob himself and his cockpit is sitting on this line, or really moving along that world line as he moves through time here. Okay, so we're going to exploit this a little bit later on, and especially when we look at things like the between paradox and the pull in the barn paradox. And we'll also do some more examples of this as we go along.