Introduction to special relativity and Minkowski spacetime diagrams

- [Voiceover] In the last videos, we constructed a little bit of a conundrum for us. We had the situation where I'm drifting through space, and right at time equal zero, one of my friends, she passes me by in a spaceship going half the speed of light in the positive x direction, relative to me. And at time equal zero, is exactly where she is at my position. And then she keeps traveling, and so I draw the path that she is taking. So after one second, she would be here, after two seconds, she would be there, and we set up our scales so that on our time axis, one second is the same length as on our space axis, or on our path axis, three times ten to the eighth meters. And we did that so that at least in my frame of reference, the speed of light would be at a 45 degree angle, or would have a slope of one. But the conundrum we hit was is that the speed of light would be perceived, based on this model I set up, would be perceived differently based on which frame of reference you're in. The photon that I emit from my flashlight, right at time equals zero, well sure I'm going to see that as moving at the speed of light in the positive x direction. But my friend, since she's already moving in that positive x direction at half of the speed of light, if we assumed a Newtonian world, she would see that photon going at half the speed of light. And likewise, if she emitted a photon from her flashlight, to her, it would look like it's going at the speed of light, but to me, it would look like it's going faster than the speed of light. And the reason why that is a conundrum is we know from observation of the universe around us, and this is profound, this is a little mind-blowing, this is counterintuitive to our everyday experience, but we know from observation that the speed of light is absolute, that it doesn't matter your inertial frame of reference. As long as you're in an inertial frame of reference, it doesn't matter your relative velocity relative to another inertial frame of reference. You will always measure the speed of light at three times ten to the eighth meters per second. But that's at a direct contradiction with what our model just set up. So it really forces us to question all of our assumptions. So what are the assumptions that we made in this Newtonian world? Well, we assumed that time is absolute. Time is absolute. And what do I mean by that? Well, we assumed that one second passing for me is going to be one second passing for my friend. That's just our everyday notion. If I'm in a car and you're not in a car, and we both have watches, and we synchronize our watches, they seem to say synchronized. But maybe we need to loosen this assumption. We also assumed that space is absolute. Space is absolute. And what do I mean by that? Well in our everyday experience, regardless of what our relative frames of reference are, we seem to agree that, well if that meter stick over there, that's riding on that train, that is a meter stick, whether I'm sitting on the train or whether I'm not. But maybe things start to break down a little bit as we think about higher velocities. And maybe they break down even at the slower velocities, but we just don't notice them because it's a very small error. And maybe there's something even more interesting. In our everyday experience, we just assume that time is somehow something very different than space, that you can move in the time direction without moving in the space direction, or you can move in the space direction without moving in the time direction, and that they're independent regardless of which inertial frame of reference you're in. But maybe they aren't separate things. In fact, maybe they're all one continuum of spacetime, and I don't mean space dash time, I mean spacetime. I mean the word spacetime, where space and time really aren't different things. It's just one continuum, spacetime. And I keep saying it fast like that, because it's not space dash time, like thinking about the different dimensions, or thinking about two different things. We're renaming one thing called spacetime. So these are the things that we're going to start to hold into question. So what if we loosen these, and we assume the thing that we have observed in the universe, that the speed of light is absolute, regardless of your inertial frame of reference. So let's word from there, do a little thought experiment, and then think about what type of a model we can construct, in which case the speed of light is absolute. And to do that, we're still going to use our little model here, and we're going to focus on my friend's frame of reference, my friend who's on a spaceship, who right at time equal zero, is passing me by at half the speed of light. And we assumed that she's in this train of spaceships. So we assumed already that she's, there are these spaceships that are all moving at half the speed of light in the positive x direction, relative to me, and they're all three times ten to the eighth meters apart. So they're like that. So let's say, and I did these axes in blue, because we're assuming this is my friend's frame of reference, so you could say this is the S prime frame of reference, this is Sally's frame of reference from a couple of videos ago. Let's say a second before she gets to me, and I'm also assuming, the way I've drawn these axes, and we're going to modify them in the future where we're actually going to use the same units for space and time. Right now, I'm still sticking to what we've classically done where we use seconds for time and meters for our path or our space, but in the future, we'll actually use meters for both. But we'll get to that, I don't want to do too many things at once. But the way I've drawn it along my space axis, three times ten to the eighth meters is going to be the exact same length as one second, and I'm doing that so that the path of light on my diagram can be a 45 degree angle. So this is three times ten to the eighth meters right over there. So let's say a second before Sally gets to me, she releases a photon from her spaceship in the direction of the spaceship that is in front of her. How far is it in front of her? It's three times ten to the eighth meters in front of her. And let's say on the back of that spaceship, there is a mirror. So she's pointing her flashlight at that mirror. So what's going to happen from her frame of reference after one second? So for one second, from her frame of reference, she is stationary, so this is Sally still here. She is going to be still at x prime equals zero in her frame of reference, and that ship is still going to be three times ten to the eighth meters in front of her. They're all moving, relative to me, at the same velocity, but relative to each other, they seem to be stationary. And so what would be the path of that photon? Well that photon will have gone from Sally's flashlight from her headlamp, or whatever it is, to that spaceship in front of her would've just reached that mirror on the back of that spaceship. And so we can draw the path of light. It will be, so let me, so that path of light will look like that on this diagram. And then right at this moment, right at t prime equals zero seconds, the photon will be reflected back to Sally. Well how long will it take to get back to Sally? Well Sally's gonna receive the reflection of that photon after one second, because that's how long it's going to take it to go three times ten to the eighth meters. So the path of that very first photon, the path of that very first photon is going to look like that. All right, well hopefully this is pretty straightforward here. This is what would happen from Sally's frame of reference. A second before she reaches me, at t prime equals negative one seconds, emits a photon at t equals zero seconds, and gets to the spaceship that's three times ten to the eighth meters in front of her, essentially one light-second in front of her, and then a second later, it's reflecting back, a second later, she gets the reflection. And so that's what this diagram is describing. But now let's draw it projected on top of my frame of reference. And this is when things are going to get really, really, really, really interesting. So I've drawn my frame of reference here, and I've intentionally not marked off the seconds or the meters on my frame of reference, because once again, I'm not going to assume that a second in my frame of reference is a second in hers, or a meter in my frame of reference is a meter in hers. I have drawn her t prime axis at the same angle as I did before, because for every second we move into the future, she's going to move half a light-second in distance, in the positive x direction. So this slope right here, one way to think about it, the way I've drawn it, this is a slope of two. For every unit she moves in the x direction, she will move two in the time direction. And what we're going to do again is assume that on her axis, I haven't drawn the x prime axis. In fact, this is an exercise to think about where should the x prime axis be. Should it be coincident with the x axis like we assumed before? Or is it going to be in a different place? But we're going to assume that the lengths I draw for one second on, let's say, in the S prime frame of reference, is going to be the same as the length I would draw for three times ten to the eighth meters. And we're also going to assume that the speed of light is absolute, so it's always going to moving at a 45 degree angle with respect to either frame of reference. So that's where things are going to get a little bit whacky, but let's see what's going to happen. So at negative one seconds, we still have Sally turning on her flashlight. She wants to bounce it off of the spaceship in front of her. And so that photon's going to move with the speed of light in either frame of reference. And so let me draw that. Let me draw that, so it's going to look like this. I'm drawing at a 45 degree angle. So actually, I don't know where it gets reflected. It's gonna get reflected where it hits that x prime axis, but I don't know where that is. But we do know that it then gets reflected, and then it gets back to Sally at one second in the future. So the return path of that photon is going to look something is going to look something like this. And that point that it changes direction, I could have, let me, I could have done it like this, whoops, I could've done it like this. But the interesting point is where it changes direction because that's where that spaceship in front of her must be at that point in spacetime. Because now we're going to start thinking of mixing up space and time, but I'm not going to get too much involved in that. Now why is this interesting? Because from Sally's point of view, from Sally's point of view, this point here where the light changes direction, from Sally's frame of reference, is happening simultaneously with when she reaches me. This is happening at t prime is equal to zero for Sally. So anything that is t prime is equal to zero for Sally must be on the x prime axis. So this must sit on the x prime axis. Once again, why do I know that? Because everything on the x prime axis, any event on the x prime axis, let me do it in a different color, I keep doing black. Any event on the x prime axis is going to be at t prime is equal to zero seconds, and from Sally's frame of reference is going to be simultaneous with when she passes me up. So based on that, we know that this is going to be, we know that this point, which is where Sally is, that's going to be the origin from her frame of reference. And we know that this point sits on the x prime axis, so based on that, we can draw the x prime axis. You just need two points to define a line, and so let me try to do it. So let me try to draw. I can do a better job than that. Let me try to draw the x prime axis. It's going to look like, it's going to look like this. That right over there is the x prime axis. Now at this point, you should find this mildly mind-blowing. Actually, even more than mildly mind-blowing. 'Cause it's saying some pretty, pretty crazy things. First of all, let's just make sure we know how to read this. So for any event, and now we're going to start thinking in terms of spacetime, although I'm still using different units for space and time, but we'll address that in the future. If I want to read its coordinates in my frame of reference, well if I want to read its x coordinate, I go parallel to the t axis, and if I wanna read its t coordinate, I go parallel to the x axis. But for Sally's frame of reference, well I essentially do the same thing. If I want its x prime coordinates, I go parallel to the t prime axis, and if I want the t prime coordinates, I go parallel to the x prime axis. But what's really interesting, and I'll go even deeper into this into the next videos, that moment, that moment right over here, that from Sally's frame of reference, it looks like it's simultaneous with her passing me up. It looks like it's happening at t prime equals zero. In fact, it is happening at t prime equals zero. From her frame of reference, it's happening at t prime equals zero. From our frame of reference, it is happening after Sally passes us up. Notice, it is happening at t equals some positive value. It's not happening at t equals zero. So this is starting to get a little bit whacky. One second, and simultaneous, time, and space, and things being simultaneous, we're not going to agree on those, depending on which frame of reference we're in. The thing we will agree on is the speed of light.