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Introduction to the Lorentz transformation

Video transcript

- [Voiceover] So we've already been able to explore a lot with our little thought experiment where I am floating in space. I'm at the center of my frame of reference, and right at time equals zero in my frame of reference, a friend comes in a spaceship passing me by with a velocity v, I'll say the magnitude is v, and it's going in the positive x direction. We're just going to focus, and we have been focusing, just on the x dimension for simplicity. And we've thought about reconciling space and time in my frame of reference relative to her frame of reference. The conundrum that we've faced in previous videos is how do we reconcile that the speed of light is always going to be the same in every frame of reference? And to reconcile them, we had to essentially come up with the idea of spacetime. I should say it even faster, spacetime. Spacetime, let me write it out, spacetime. And the first time I heard about spacetime, I assumed that people were just talking about space and time as independent things and just plotting your point in space and time. But when people talk about spacetime, they're really talking about this continuum of one thing, and we're just talking about different directions in spacetime. They could have called this something else. They could have called this spime, or tace, or stace, or a lot of different things. But this is spacetime, and it's this idea that it's this continuum. And when we started to make spacetime diagrams, we realized in order for the speed of light to be absolute, that time and space weren't as independent of each other as we thought, and they weren't as absolute as we thought. And we constructed these Minkowsky spacetime diagrams for each of our frames of reference. So my frame of reference, the spacetime diagram, is here in white, and for my friend's frame of reference, her spacetime diagram is here in this blue color. The angle formed between these axes, between the x and the x prime axis and the ct and the ct prime axis, this angle alpha here, this is going to be dependent on her relative velocity in my frame of reference. So if her velocity is v, or the magnitude of her velocity is v in my frame of reference, this angle we've already seen is going to be the inverse tangent or the arctangent of the ratio between her relative velocity and the speed of light. So this is going to be equal to the inverse tangent of v over c. So the faster she goes, these two things are going to start squeezing together, and if somehow she were to approach the speed of light, they would both approach a 45 degree angle and actually start to coincide if they were actually able to approach the speed of light. And that's all interesting already, this idea that space and time are not as independent, that it's all a continuum called spacetime, but some of you have probably said, well, I want to deal with some more tangible numbers here. For example, if this event right over here in spacetime, we can think about it from my frame of reference, and we can think about it from her frame of reference. In my frame of reference, I would view the coordinates here. This coordinate would be x, and this coordinate right over here would be ct. We had a whole video on why we think of time in terms of ct. The units here literally would be meters. We could think of it as light-meters if we like. So that would be the coordinates in my frame of reference. Well, what would be the coordinates in her frame of reference? Well, we've already thought about how to read these Minkowski spacetime diagrams. To find her x prime coordinate, we would just go parallel to the ct prime axis. So that would be the x prime coordinate in her frame of reference. And to figure out the ct prime coordinate, we would just go parallel to the x prime axis. So this would be the ct prime coordinate. Now how do you actually go in between, transform, from x to x prime and from ct to ct prime? And to do that, we're going to introduce in this video the Lorentz transformations. And what they do is they allow us to do exactly what we just needed to do. They'll allow us to go from x, ct to x prime and ct prime. And to help us think about it, I'm going to introduce some variables, and hopefully it will show the symmetry of the Lorentz transformations. You might see them written in other ways in other sources, and we'll reconcile all of those in the future. But the Lorentz transformations, we'll start with what we call the Lorentz factor because this shows up a lot in the transformation. So I'll just define this ahead of time. So the Lorentz factor, denoted by the Greek letter gamma, lowercase gamma, it is equal to one over the square root of one minus v squared over c squared. Now sometimes you might even see it written like, well, I'll write it another way. Sometimes you might see it written as gamma, let me do it in that same color, same reddish color, gamma is equal to one over the square root of one minus beta squared. You might say, well, what is beta? Well, beta is another variable that shows up a lot when we're thinking about special relativity. And beta is just the ratio between the relative velocity, her relative velocity in my frame of reference, and the speed of light. It shows up a lot, even this angle alpha here, we could have said this is the inverse tangent of beta. This becomes a little bit simpler when you write the Lorentz factor. And when we look at the actual transformation between the coordinates, we'll see that beta becomes useful again, at least the way I like to write it. So if we want to think about what x prime is going to be, so we can write x prime is going to be equal to the Lorentz factor, let me do it in that red color, is going to be equal to the Lorentz factor times x minus, and now we're going to say beta times ct, and ct prime is going to be equal to the Lorentz factor gamma, let me do that same color again, switching colors is sometimes difficult, gamma times, and we'll see it's just the other way around. It is going to be c, let me do it in that green color, ct minus, you might even guess what I'm about to write based on the symmetry that we see here, ct minus beta times x. And I really want you to appreciate this because it really shows that space and time are really just different directions in the spacetime and there's a nice symmetry to them right over here. Notice we have an x and we have an x. We have a ct and we have a ct. So when we're thinking about x prime, it's x minus beta times ct, and we're scaling it by the Lorentz factor, and then when we're thinking about time, well we do it the other way around. We're still scaled by the Lorentz factor, but now it is ct minus beta times x. Now this might all seem like Greek to you, and we actually are using Greek letters, but in the next video, I'll actually use some sample numbers here, and you'll see that evaluating these is just a little bit of straightforward algebra.