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Current time:0:00Total duration:5:06

Lorentz transformation for change in coordinates

Video transcript

we spent it we spent all several videos now getting familiar with the Lawrence transformations what I want to do now instead of thinking of what X Prime and CT prime is in terms of X and CT I'm going to think about what is the change in X Prime and the change in CT prime going to be in terms of change in X and change in CT and we'll see it's just going to involve some some fairly straightforward algebraic manipulation so let's think about a change in X prime is going to be X prime final minus X prime X prime initial well X prime final let me just pick a suitable color for that X prime final is going to be gamma times X final minus beta times CT final all I did is I use this formula up here if I want to figure out my final x prime well I'm just going to think of my final X and my final CT so that's that and from that I am going to subtract the initial X prime well X prime initial is just going to be I get another color here X prime initial is just going to be Lorentz factor gamma times X initial minus beta times CT initial so now let's see we can factor out the gamma so this is going to be equal to and I'll do it in my color for gamma if we factor out the gamma we're going to get gamma times we're going to have X final let me do this in a so we have let me do that in white actually we're going to have X final and then if we distribute this negative sign minus X initial and then let's see if we distribute this negative sign well I don't want to skip too many steps here so that's that and then we're going to have negative we're going to have negative beta CT final negative beta CT final and then we have plus we distribute this negative plus beta C T initial plus beta C T initial and so what can we do it here well this that's just going to be change in X so this piece right here is just change in X so I can rewrite this as being equal to gamma gamma times change in X change in X and now let me subtract let me just subtract let me take let me take out factor out a negative beta so I'll say minus beta times minus beta times well then you're going to have CT final see T final minus CT initial minus CT initial and well what's C T final minus CT initial actually I'm not going to skip any I think I'm skipping too many steps already well that's just going to be change in CT so we get this is going all going to be equal to gamma our Lorentz factor times change in X minus beta minus beta minus beta times change in CT times change in CT and since C isn't changing you could also view to see times change in T either way so there you have it so these notice it takes almost the exact same form X prime is equal to gamma times X minus beta CT and change in X prime is going to be gamma times change in X minus beta times change in CT and I'm not going to do it in this video but you can make the exact same algebraic argument for your change in CT prime as you'll see and I encourage you to do this on your own change in CT prime which you could also view as since C isn't changing C times Delta T prime these are equivalent is going to be equal to is going to be equal to exactly what you would imagine is going to be it's going to be gamma times change in CT change in CT minus beta minus beta times change in X and encourage you right after this video actually do this one too just hey Delta X prime is going to be X X prime final minus X prime initial and then do what I just did here just little bit of algebraic manipulation and you can make the same exact argument over here to get to the result that I just wrote down you say well our change in CT prime is going to be CT prime final minus CT prime initial and you can substitute with this do a little bit of algebraic manipulation and you'll get that right over there and the whole reason I'm doing this is well now we can think in terms of change in the coordinates which will allow us to think about what velocities would be in the different frames of reference which is going to be pretty neat