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Current time:0:00Total duration:8:21

Introduction to the Lorentz transformation

Video transcript

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 passing 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 and the conundrum that we 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 space-time and I should say it even faster space-time space-time let me write it out space-time and the first time I heard about space-time I assumed that people were just talking about space and time is independent things and just plotting your point in space and time but when people talk about space-time they're really talking about this continuum of one thing and we're just talking about different directions in space-time they could have called this something else I could have called this spine or or or or tastes or stace or I've got of different things but this is space-time and it's this idea there's this continuum and - and when we started to make space-time diagrams we realized in order for light 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 min Scott Minkowski space-time diagrams for each of our frames of reference so my frame of reference the space-time diagram is here in white and for my friend's frame of reference her space-time diagram is here in this blue color and the angle formed between these accesses 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 relative reference so in my frame of if her velocity is V or the magnitude of 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 inverse tangent of V over C so the faster she goes these two things are going to start squeezing are going to start squeezing together and if somehow she would approach the speed of light they would all they would both approach a 45-degree angle and almost 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 the space and time are not as independent that it's all a continuum called space-time but some of you have probably said well I want to I want to deal with some more tangible numbers here for example for example if this event right over here in space-time we can think about it from my frame of reference and we could think about it from her frame of reference in my frame of reference I would view the coordinates here well let's call this 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 unit's 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 we've already thought about how to read these Minkowski space-time 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 this would be the C T 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 Lawrence Lawrence transformations transformations and what they do is they allow us to do exactly what we just needed to do allow us to go from X comma CT to 2 X prime X Prime and see T 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 and 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 fad the Lorentz factor denoted by the Greek letter gamma lowercase gamma it is equal to 1 over the square root of 1 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 that same color same reddish color gamma is equal to 1 over the square root of 1 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 beta is just the ratio beta is just the ratio between the relative velocity her relative velocity in my frame of reference and the speed of light and it shows up a lot even this angle alpha here we could have said this is the inverse tangent the inverse tangent of beta and it also helps you 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 the Lawrence let me do that red color is going to be equal to the Lorentz factor times X times X minus and now we're going to say BAE data times CT times CT and CT Prime and see T prime is going to be equal to the Lorentz factor gamma let me do that same color again switching colors is sometimes difficult to gamma times and we'll see it's just the other way around it is going to be it is going to be C let me do that green color CT CT - you might even guess what I'm about to write based on the what the the symmetry that we see here CT minus beta beta times X and I really want you to appreciate this because this is it really shows that that that space and time are really are really just different directions in the space-time and there's a nice symmetry to them right over here notice we have we have an X and we have an X we have a we have a CT and we have a CT so when we're thinking about X prime its X minus beta times CT and we're and we're scaling it by the Lorentz factor and then when we're thinking about time what 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 do some sample numbers here and you'll see that this is this evaluating these is just a little bit of straightforward algebra