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Hello everyone, I'm Norman Wildberger. In this video, we're going to take a look at a new way of thinking about special relativity, which is one of the most fascinating and interesting developments in the history of science. We want to approach this topic from a perspective rooted in our understanding of linear algebra. This is quite a mathematical point of view, and we'll see that it's, in many ways, much
0:38
more elementary and simpler to understand than almost all classical treatments that I've seen. Okay, so the history is, of course, well-known. Albert Einstein, in 1905, working as a Swiss patent clerk, published the special theory of relativity. It was based on, or had connections with, quite a lot of previous work in physics before that—in particular, Maxwell's theory of electromagnetism, so-called Maxwell's equations;
1:13
the experiment of Michelson and Morley that showed that essentially there was no ether, that the velocity of light seemed to be the same both in the direction of the Earth's orbit and transverse to it; work by Fitzgerald and Lorentz on the symmetries of Maxwell's equations, which had also been investigated by the great mathematician Henri Poincaré; and then after Einstein, Hermann Minkowski came along and reformulated some aspects of special
1:45
relativity in a somewhat more geometrical way, involving what we now call space-time. Alright, so this was a major revolution in scientific thought. It really upended people's ideas of how we fit into the universe, and many familiar concepts were turned on their heads and had to be re-examined from a completely new light. In particular, we had such hard-to-understand concepts as the relativity of simultaneity—that it's no longer
2:18
possible to say that two events occur at the same time, as that's an observer-dependent statement; the idea of length contraction, where moving rocketship lengths contract; that time dilates, meaning clocks run slower; and then this remarkable relationship between mass and energy that's really at the heart of the nuclear revolution in the 20th century. Okay, so I guess one could say that, simplistically, Einstein looked at
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Maxwell's equations in the same way that others had done and was able to see clearly how to organize the symmetries of Maxwell's equations, going beyond a Newtonian framework to a new kind of framework. Ultimately, he was able to base the logical structure of the theory essentially on two postulates. Okay, two assumptions that he said, "Let's start with these assumptions and see where they bring us," and they ended up giving us special relativity.
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So the first postulate is that all inertial frames are equivalent with respect to the laws of physics. An inertial frame is one that's not accelerating, but that can mean that it's moving at a constant velocity with respect to some other inertial frame. So he had this idea, perhaps best thought of in terms of things moving around in space—one thing drifting around in space and then some other object drifting around with some velocity with respect to the first—that it's
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impossible to say that one of these is stationary and the other one's moving. The laws of physics don't really distinguish between these two inertial frames. That's his first postulate. And the second one is that the speed of light, at least in empty space, is independent of the state of motion of its source. So even if you have a rocket ship going very fast in that direction and it shoots a beam of light ahead of it, that beam of light will still be
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measured as having the speed of light by all observers, whether they're on the rocket ship, going transverse, or going in opposite directions. From these two postulates, he was able to derive all of these very interesting and surprising conclusions. So right from the start, when Einstein proposed his theory, there was a lot of difficulty and reluctance in accepting it—not only among the general community but even among his fellow
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physicists. It took many decades for this to slowly be accepted into the mainstream of physics, only after considerable evidence had been found to support it. Some of the reasons for that are significant challenges in learning special relativity. One very obvious one is that it's reasonably mathematical. It does involve mathematics, and if you're not willing to come to grips with the mathematics, it's very hard to get anything
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but a superficial understanding of it. Another reason is that the effects are mostly visible only at high velocities, so they're divorced from everyday experience and intuition. Special relativity generally doesn't play any role at all in our ordinary lives, so our biology has not been evolutionarily trained to deal with it—to give us mental constructs or physical constructs that allow us to understand what's going on. So that's a big reason why it's difficult.
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There are highly counterintuitive insights into space and time—the idea that space and time really ought to be thought of in one configuration rather than space here and time ticking away completely independently is very hard for many people to get their heads around. Alright, it seems as if space and time are completely different kinds of things. It's also difficult to unravel certain paradoxes or apparent paradoxes that come up—notably, for example,
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the twin paradox. These really are quite challenging to explain completely clearly, and even I think a lot of textbooks don't do such a good job when it comes to that. In my opinion, also, there's an additional difficulty, which is that there are unacknowledged and unjustifiable assumptions built into many standard treatments. When you look at a classical book on special relativity—and there are dozens of these that have been written over
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the years—some of them, many of them in fact, have built-in certain assumptions that are part of the story. And these assumptions, when you look more carefully at them, are not entirely obvious, and it's not clear why they're justifiable if they are at all. I think this renders the whole subject a little bit logically dubious. If you're learning the subject for the first time, you may not be explicitly aware of these things, but it's sort of there underneath the surface—that
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it's a little bit wobbly in terms of what exactly we're talking about at various stages. So in particular, what I'm thinking about is the sameness of clocks and rulers. This is a very common situation that one runs into, that treatments assume that there is such a thing as a clock which can be reproduced, and we can put one version of it in one rocket ship going that way and another version of the same clock in another rocket ship going at a very high velocity
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in a completely different direction. The same thing with rulers, that there's some kind of standardized ruler that we've just handed out to all the participants in our global experiment. But if you think about it, there's actually really no good reason why we should be able to assume that. If we've manufactured clocks or rulers on one rocket ship, how are we actually going to get those clocks and rulers onto a nearby rocket ship? Well, we could try to sort of throw them from one
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rocket ship to another as the other one is going by—little trajectory, a little bit of a shuffle or something like that. But if the velocities are high, if these rockets are going at half the speed of light, that will invariably involve crashing and destroying the clocks or the rulers even under the most optimistic circumstances. So it really is a built-in assumption that is, in my opinion, against the essential spirit of relativity. Now, we're not going to assume that
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things are as you might expect naively. Okay, so this is a significant problem. Can we really compare measurement tools on two different rocket ships? Alright, this is something that physicists might like to give a little bit more thought to, I think. So more generally, this is obviously a very foundational topic. Okay, we're redefining what we think about in terms of space, time, energy, mass, momentum,
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etc. And so it's really important that the physics community addresses the seriousness of this, right? One really has to have a very broad and careful discussion about what all the words mean. And those of you who watch my various videos will know that I'm a bit of a stickler about these kinds of things, that I think the mathematical community, for example, doesn't spend enough time and effort being very careful about the meanings of the fundamental
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words that it's using and the fundamental concepts. They bandy things like angles and irrational numbers and transcendental functions around, and if you actually look carefully at what those things exactly mean, you find some kind of waffling and imprecision. And so it's careful that physicists don't fall into the same kind of trap. We're laying out really fundamental things here—it's important to have a solid discussion, make sure everybody is on the right page,
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that we know exactly what we're talking about, illustrated with lots of examples, and we ask questions. There needs to be a real vigorous discussion before we really nail things down. So a symptom of this that I think is very popular these days, that weakens, I think, a lot of discussion of modern physics in the popular press, has to do with the origins of the universe and the Big Bang. Okay, so in 2014, there's this story that there was a Big Bang, and,
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you know, in the first trillionth of a second, such things happened. And then in the next trillionth of a second, various other inflationary things happened. And then a millionth of a second later, there was something else that was going on. And I think it's fair to say that a lot of this discussion is very, very waffly and very vague to the point almost of being deceptive. I mean, it seems to me kind of obvious that you may have a notion of a second—what a second is,
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okay? Maybe it's a pendulum; you have a pendulum of leaden lengths of a certain length, and you define this to be a second. Or maybe you have a second defined in terms of how far light goes, for example—maybe, you know, from here to the moon, maybe that's one second. Or maybe you have it in terms of the decay in a cesium atom or something like this. Whatever your definition of a second is, if you want to apply that definition to the first trillionth of a second of the universe,
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then surely you need to make sure that you have a definition of what that word means in the context of the first trillionth of a second of the universe. Does it make any sense if we didn't have any pendulum clocks hanging around at that time? Does it make any sense if we didn't have a fixed distance that light could go to measure the distance? Does it make sense if there weren't any cesium atoms at all back then? You see what I'm saying? So I'm not saying that this is wrong,
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that the Big Bang is wrong; I'm saying that the usual presentations of the Big Bang, as we see in the popular press, are highly suspect, I think, from anyone with a sort of logical, skeptical point of view. And I think those same kinds of problems are coming in here, even with a much more established theory of special relativity. It's going to be something relevant to our discussion of this interesting topic.
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So in this video and in the next one or two, we're going to lay out a new and non-standard but elementary introduction to special relativity, which I claim will be more elementary and understandable than almost all other treatments. And we're going to use basically only some simple linear algebra. And we're also going to let go of many of the usual assumptions. In particular, we are not going to adopt Einstein's postulates. We are not going to assume the constancy of the
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speed of light. We are not going to assume the equivalence of inertial frames or the ability to distribute identical clocks and rulers to all inertial frames. To me, these are assumptions that may or may not be true, but there's no need to build them into our theory. So I'm going to show you a mathematical theory of special relativity, essentially, and it's going to turn out that most of the formulas—perhaps, you know, the vast
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majority of the formulas—are already captured in this mathematical version. But the mathematical version is somewhat more general and broader and maybe is logically prior to the physical theory. So we're going to instead return to a simple high school Newtonian view—that's going to be our starting point. Space is absolute, time is absolute—we're going to go back several centuries. And more surprisingly, I'm going to replace light as the primary communication medium with sound.
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This is going to help me distinguish between what I'm going to show you and the usual point of view. So I know that it can be very challenging for people to relearn a subject if they already know something. One is very reluctant to learn a new version of it—I see this all the time with my fellow mathematicians when it comes to rational trigonometry. They already know trigonometry; they've already spent years sort of mastering all those formulas and the various ins and outs.
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The idea that there's an alternative, even if the alternative is much, much simpler—which it certainly is—and much, much more powerful—which it certainly is—is still hard to accept because of this big potential hurdle of getting over the work that you've already put into it. So what I want to do is I want to very clearly distinguish between this new way of thinking about special relativity from the old one. And one way of doing that is to remove light altogether
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from the picture and replace it with sound. You might be surprised that this is going to work, right? You may think, "Sound doesn't have the properties of light." Yes, but we'll see that the essential mathematics of special relativity works even with sound, if you approach things in the right way. So the speed of sound is much lower than the speed of light, of course—it's about 300 meters per second or a thousand feet a second. So it takes about five seconds to go a mile. Alright,
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that's much, much slower than light, which everybody who's watched a lightning bolt will know. You see the light, and then a few seconds later, you get to hear the sound of the thunder. So in fact, sound has a lot of advantages because it's more something in our world; it's something that we can understand much more clearly. Light is much, much more complicated. And I remind you that for a long time in history, people didn't know that there was
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even a finite speed to light. The ancient Greeks debated this—Aristotle believed that light had infinite speed. So did Descartes, and so did most 16th-century mathematicians and physicists. It was really only in the 1700s that Ole Rømer, who was a Danish astronomer, actually made a calculation involving the moons of Jupiter that showed that light, indeed, yes, had a finite velocity, and he estimated it reasonably closely.
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Okay, so that's what we're going to do. Quite interesting—will that work? Well, we'll see. We're going to see that it's very possible to claim that special relativity is, in fact, a logical consequence of Newtonian mechanics and doesn't essentially depend on Maxwell's equations or any of the later developments of 19th-century physics. It could have been discovered anytime after 1700. And about the only other person I know who has something of this orientation is
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Hermann Bondi, who introduced the k-calculus point of view towards special relativity. Okay, but as I said, we're going to replace light with sound as the primary communication medium. And let me remind you about sound and echolocation, which is how bats can fly around so successfully at night time when it's very hard for them to see. So bats have an acute sense of sonar—they send out these little clicks at 15 or 20,
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50 or 100 a second, and then the sound of those clicks bounces off whatever they're close to—trees, other bats, insects—and their ears pick up these returning sound signals. And that allows their brain to form some kind of image of the world around them, even though it might be in the middle of the night. It's a remarkable thing these creatures can do. So a bat like this can hear the location of a little bug, and it can fly towards it. It's
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like a radar, but the sonar version, and it can just snatch up that bug, and that's basically how it hunts. Of course, sonar is also involved in military things. So if you have a destroyer and it's looking for an enemy submarine, then one way of doing that is to send the pings into the water and hope that it bounces off the metal hull and comes back and alerts you to the presence of something down there. So this idea of bouncing sound off something to identify how far away it
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is, maybe to figure out how fast it's moving—in fact, this can actually be taught to people. So it turns out that blind people can learn to use echolocation for navigation and recognition. This is a lovely thing, and a fellow called Daniel Kish is a pioneer in teaching blind people to do this. You can see this for yourself—if you walk around carefully with your eyes closed and you talk, say,
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towards a wall, you can very quickly sense that there's something different about the way the sound comes back at you if you're pretty close to a wall. And people who have been trained in this—blind people—can recognize chairs and other people, often quite at some distance from them, and can learn to do lots of things that they otherwise couldn't do by using sound. They use little clicks or some other noise-generating thing to listen to the echoes of things.
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So we're going to use sound and echolocation as our primary means of communicating with the world. We're going to pretend we're bats. So welcome to one-dimensional bat world. We're going to assume that we're in a one-dimensional world—maybe you can think of it as a long cave, which is basically a long tunnel so that mostly bats and other creatures can just go back and forth in this cave. And here we are, we're having a little rest. Bats like to hang upside down from
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their perches up on the top of the cave because that way they can take off easily—they just let go and then they can flap. They don't have enough power to get off the ground just by jumping up like birds do generally. So we are just here resting—this is us, we'll call it bat A. And here's our friend, bat B, and she is hunting. She's going through the tunnel like this, and this represents her vector. She's sending out sonar messages to figure out where things are,
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and we are also emitting a constant little click just to check out what's going on around us. Okay, and our friend B senses a little bug over here, a little bug which we'll call C, which is moving in this direction, okay? It does so eventually until it starts sensing that there's a bat coming down the tunnel towards it, at which point it immediately turns around and heads off the other way in somewhat more urgent fashion. But B is a very excellent hunter, and she zips
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into it very quickly, and right about there, she catches that bug and eats it. Alright, so we're watching all of this—or maybe not watching it, but we're listening to all of this—so we have a mental picture in our mind. And I want to express what's just happened in terms of a spacetime diagram. Okay, so here is a two-dimensional representation of what's just happened—sort of like a movie in
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some way where the basic axis here along the cave is reproduced here as the x-axis, and here's our time axis, so that various points here represent events in the world and have both a position and a time coordinate. So, for example, here is us—we're just hanging, so we're just staying in the same spot as time marches on. Our world line just goes vertically up. Here's our friend B, who is moving through the tunnel from left to right, and she's going at a fair clip, which
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is represented by the fact that this is a pretty slanted or almost horizontal line. Then over here is a bug—this is the world line of bug C, which starts over here and is moving first to the left, represented by this line here. And at this point here, the bug realizes that it's in trouble and goes the other way, so here the world line changes and then it heads off in this direction. But it's not as fast as our friend B, so B catches it right there, and so this dot represents what
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we might call an event—let's call it P. That's the event where bat B catches bug C. Okay, and we're going to try to understand what's going on in our one-dimensional bat world in terms of a spacetime diagram like this. It's very instructive so we can plot the motion of things. Alright, so let's return to our static picture. Here we are, we're hanging upside down, contemplating life and the universe. Now, let's say we're scientifically oriented as a bat. You
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know, we haven't had any proper education in science or anything, but we're intelligent bats, and we're thinking, "Hmm, wouldn't it be nice to have a more scientific understanding of the world around us?" Basically, this tunnel that we live in and the various things that go on in it. So one of these days, we have an idea because there's a stalactite hanging not too far away from us. Okay, a stalactite coming down from the top of the cave. And so, while we're clicking along,
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sending out little clicks, these clicks are also bouncing off this stalactite, and these clicks are coming back to us, and we're registering this—the presence of the stalactite. And what we can do, maybe, it occurs to us, is that we can play a little game here. So we can send a click towards the stalactite, and when that click comes back to us, as soon as it comes back, we can send another click, also in the direction of the stalactite. So we get a click, click,
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click, click, just bouncing back and forth, this sound off the stalactite, and that's giving us essentially a clock. And this is an interesting idea—the presence of the stalactite allows us to introduce, first of all, a basic distance measurement. So, for example, we could define the distance between us and the stalactite to be, say, one meter. This doesn't have to have anything to do with the usual meter that we know; it can just be a word that we're going to adopt. Alright,
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here's a stalactite—let's agree that from here to the stalactite is what we'll call one meter. And now here is the spacetime diagram of not just us at 0, but also now of this stalactite, which is now at position 1—we've agreed that this is one meter over. So the stalactite is just right there, that's its world line, and here's us hanging upside down. And this sound that's bouncing back and forth is represented by these red lines. So let's say here is the sound starting here, and it
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bounces off the stalactite and comes back there, and at that point, we send off another signal—it comes back there, we send off another signal, and so on and so forth. This is defining for us kind of a clock, internally—internally, we're going click, click, click. And it makes sense, say, to call these clicks at 2, 4, 6, and so on, so that we can think about the time taken for this sound to go from us to the stalactite as being exactly one unit. So we're going to
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define the time for a sound to go from A to S—here or here, or equivalently from S to A, presumably these are the same—as one second. So we're going to introduce a time measurement; we define a second to be the amount of time it takes for sound to go from us to the stalactite, or from the stalactite to us. We're going to assume that those are the same—why shouldn't they be? So we, namely bat A, now have a clock by sending out these constant clicks to the
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stalactite and just registering when they come back, just sort of counting. We have an internal clock that we can use for timekeeping. Now we also have the beginnings of a kind of spatial understanding of the world around us because we've now established a unit of length. We don't know where anything else is yet, but we can say for sure that we're at position 0 and this stalactite is one unit away, one meter away. So this is a pretty good intellectual
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accomplishment—I think we can be proud of ourselves for having arranged this understanding. But once we're engaged in this kind of thinking, we can go further. Alright, so now I'm going to introduce some very important ideas, okay? We're going to show how to measure not just where we are and where the stalactite is, but more generally, the position of a general event. In particular, let's go back to our friend B, whose world line we'll put like this, okay? So she's going left to
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right, and let's say that at time minus 6, she's passing us. So we've just synchronized our clock in some fashion, arbitrarily decided that this is going to be the zero time point. So that's 0, 1, 2, 3, 4, and so on, and here are negative numbers corresponding to the earlier times. So bat B is heading in this direction, and this represents when bat B is actually at our position—she's just passing us at time T equals minus 6. So we can make a record of that—that B
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is going past us at time minus 6 according to our little clicking watch or clock that we're keeping in our head. And then a little while later, boom, she catches the bug C. Here's the world line of bug C—it started here, it was moving left rather slowly, and then realized that the bat was coming towards it, so it headed off in the other direction. But still, it wasn't fast enough, so it was caught at the point P. That point P is an event—it's someplace in the spacetime diagram,
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so it has an x-coordinate and a T-coordinate. But how could we determine what those are? We're just hanging upside down, okay? We don't have any rulers, we don't have any friends that can go out here and make any measurements, we're just a bat hanging upside down. All we can do is send out these sonar clicks. Nevertheless, we can figure out exactly where and when this event took place, and it's absolutely essential that you understand how this is done.
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Alright, so make sure you understand this box, okay? So how is it working? Well, besides having this constant clock, bats are multifunctional when it comes to dealing with sonar sounds. We can deal with sonar clicks that are heading off in lots of different directions—we can be conscious of the tree over here, and our 14 bats that are flying with us over here, and these 39 bugs that are in
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front of us there. So the bat's brain is perfectly capable of dealing with a fair number of sonar clicks at the same time. In particular, while this clock clicking back and forth is going on, we're sort of keeping track of that. At the same time, we're also still sending out clicks into the general area to see what else is happening. In particular, we can sort of hear the place where bat B goes "snap" and catches that bug. And how can we do that? Well, if we had sent out a click
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right at this point here, then this line here represents the sound that would have emanated, a click that would have emanated from us and would have gotten to the point P just as it was happening. Of course, we might have sent out all kinds of other clicks before and after that, but it's this particular click that bounces off the crash of B and C and then bounces back again. Now, this is a sound wave, so it's going at the same speed as these sound waves, so it's parallel to
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it, okay? So this is starting at some point here and then it's coming back at some point there. Because we have this clock ticking away, we can, if we're good, remember when this click was sent off, and we can note when it comes back. We can, in our minds, remember that that click was sent at time T equals minus 1/2. We could call that the initial time of the event P from our point of
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view. And then when this sound wave comes back to us, we can register that return, we can call that T sub-final of the event P, and that's going to be between 3 and 4—we'll just estimate what that is, say 3 and a half. So we can be confident that this event P has the properties that the initial time that we sent off the click was at minus 1/2, and we received that click back at 3 and a half.
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And now have a look at these two formulas. The first one says that X is TF minus TI over 2. Why should that be? TF minus TI is the total amount of time taken, the total amount of time taken from the start of this sound emission to receiving it when it comes back. The total amount of time. And how fast does the sound go? Well, we've arranged it so that the sound travels one meter every second—that's how we arranged our meter and second units. We agreed that in one second,
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the light goes one meter, so in TF minus TI seconds, the light has gone a total of TF minus TI over two meters. We have to divide by 2 because this represents the total distance from here to P and back again—that's because we're counting this portion and this portion. So the total time, which is the difference between these two times, divided by 2, is also the number of meters that that point P is away from us. In other words, that's the x-coordinate. And
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what about the T-coordinate? When did this event take place? Well, it was emitted at this point, it was received at this point. The light is going at the same speed this way as it is this way, so the time of that event must be halfway between these two markers here—it must be the average of these two time values. In other words, TF + TI over 2. Notice the close proximity of these—TF minus TI over 2 is the x-coordinate, TF plus TI over 2 is the T-coordinate.
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So in our particular situation, we can easily make the computation—TF is three and a half, T sub-I is minus 1/2, so the x-coordinate is the difference divided by 2. So 3 and a half - minus 1/2 is—that's three and a half plus a half, or four, divided by two is two. And the time is three and a half plus minus 1/2 over two, which is three over two or one and a half. Conclusion: this event happened at position x equals two at a time T equals one and a half. So we are now,
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even though we're upside down and we're only emitting little clicks, in a position to find out when and where any event happened in our one-dimensional world. We now have coordinates on our spacetime. So Batman would be proud of us—we've done very well, and now we're in a position to study the world around us quantitatively. Alright, so let's measure some velocities. Alright, so velocity is change in x over change
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in time. Let's have a look at our friend Barbara the bat, who has gone from here to here in terms of the world line. So this means this position is zero, minus six—at time T equals minus six, she was at position zero, namely where we are now. At time T equals minus six by our clock, and then here, where she caught the bug, we just figured out that that's position two and one and a half. Alright, so the velocity of B is determined by taking the difference in
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the x-coordinates, just 2 - 0, and dividing by the difference in the time coordinates, which is 1/2 - -6. Alright, that's 2 over 1 and 1/2 plus 6 will be 7 and 1/2, which is the same as 15 halves. When we divide that, we get 4/15—that's the speed of bat B. The speed of sound—what is the speed of sound? Well, it goes one meter in one second, so its speed is 1. So this is about 1/4, a little bit more than 1/4 of the speed of sound. Alright,
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so here is Barbara's velocity represented by that slope—here is the sound's velocity, sound is faster, it's more horizontal. This one is about 1/4 the speed of this one. Let's have a look at our bug C and see if we can determine what its velocity is. Well, it actually had two sections of its journey—from here to here, while it was still a happy-go-lucky bug, and here, where it was in a blind panic trying to get away, okay? So let's have a look at this
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position right here, where it happened to turn around. Let's see if we can figure out what the coordinates of this point are. So how do we do that? Well, we have to think in terms of sound—so sound travels at this kind of slope. So if I go parallel to that, to here, then I can say that that initial position time there will be maybe minus 1 and 1/2, okay? I'll put that there—that's minus 1 and 1/2, that's the initial time for R. And what is the final time?
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So that's when such a signal would come back to us after hitting our R. It comes over like this, and now it's going to come back again, parallel to our sound wave, and it would come out to about there. Let's say T sub-final of R is two and a half. Alright, so what is R? So the position is the difference divided by two—two and a half minus minus one and a half is a total of four, divided by two is two. And its time is the average of these two, so if we average these two,
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we get two and a half plus minus one and a half is just one, we divide by two to get a half. Okay, maybe my drawing wasn't very accurate because, in fact, I've got this thing right underneath P, which means that it's actually going very slowly. It probably should be a little bit less than two; if I'd done this more accurately, that might be more like 1.8 or something like that. If this was actually like this, then its speed there would be change in x divided by change in T would
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actually be zero. If this was the actual correct value, so the velocity from, say, C from P to R would be—if these values are correct—would be 2 - 2 over 1 and 1/2 - 1/2, which would be zero. Let's have a look at S and see if we can figure out what its position and time are. So again, we're going to be going in a direction parallel like this because this is sound, so we're just going to go parallel down here, alright, and maybe around here. So what I'm saying
42:09
is that if we emitted a click at time minus 4, we could expect that click just to reach that event, okay? And then, so it would go like this in red, and now it would come back. Or I guess it would actually probably be at zero, so that would be the initial—that would be the final—for the event S. So initial for event S: minus four, and final for event S: zero. Okay, so what is the position and the time here? The position is the difference divided by two,
42:53
which is another two. And let's make it a little bit nicer—let's suppose that it's 2.2, for the sake of argument, just to have something different. So 2.2—so maybe I should make this, to make that honest, let's make this minus 4.4. How's that? Alright, so this minus this over two would be 2.2, and then the corresponding time would be the average of these two, which would be minus 2.2. Alright, in that case, what is the velocity of C from S to R? Let's write
43:38
that—equals delta x over delta T. So we're going from S to R, so 2 - 2.2 all over a change in time, 1/2 - -2.2 for a total of, that would be, I guess, minus 0.2 up there in the numerator. 1/2 is 0.5, so 0.5 plus 2.2, that's 2.7. Then, what would that be? Well, 20 over 27, so minus 20 over 27. If it really was over there, that's just the way this kind of thing works—it's actually relatively simple. So the main thing to understand is how we calculate the position and the time for any
44:46
event. It's always referenced to our position—here we are, hanging upside down in the same position all the time, emitting these clicks which are always moving at the same speed, represented by lines either in this direction if they're going this way, or this kind of slope if they're going this way. So any event has an initial time and a final time where we emit the click and we receive the click. By taking the difference divided by two, we get the x-coordinate, and by taking the
45:18
average of the two, we get the T-coordinate. So we can compute velocities, we can do mechanics here. Alright, this is all very good, but where's the relativity? Well, to bring in relativity, we need someone else. In our next video, we're going to introduce our friend Barbara in a little bit more of a serious role—not just hunting bugs, but we're going to tell her about our new approach to coordinate-izing our little spacetime. And she's going to become very enthusiastic,
45:52
and she's going to want to play along with us, she's going to want to try this out herself. And that's where relativity is going to start, and we're going to see, in a way that's really not much more complicated than this, using just basic linear algebra, pretty much as we've been talking about in the earlier videos of the first part of this course, we're going to see special relativity just fall out from the interaction of bats A and
46:22
B. I hope you'll join me for that. I'm Norman Wildberger—thanks for listening.
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