Lorentz transformation derivation part 1

so in all of our videos on special relativity so far we've had this little thought experiment where I'm floating in space and right at time equals zero a friend passes by in her spaceship she's travelling in the positive x-direction velocity is equal to V and we draw space-time diagrams for both of us first I draw my space-time diagram in white and then I overlay her space-time diagram and the angle that is formed between the time axis and the position axes right over there that's going to be dictated by how fast V is how fast she is actually traveling and we give the space-time diagram from her frame of reference we see with the little Prime's right over there now one thing that you might have been thinking about throughout this entire series what is well if I perceive her traveling with a velocity of V in the positive x-direction if we took her point of view and that's what I have right over here if she views herself is just floating in space what she will see me as right at time equals zero actually I'd say right at T prime is equal to zero we're saying that T Prime and X Prime at T Prime and T equals zero are coinciding right at that moment she will see me flyby at negative V going in the pot in the negative x-direction once again there is no there's no absolute frame of reference these frames of reference are all absolute hour so these frames of reference are all relative and so you could imagine what will look like is if we drew her space-time diagram where her CT prime axis and X prime axes that they are perpendicular to each other and then based on that my space-time diagram would be at an angle and it's at an angle like this you can kind of see the positive see T axis is in the second quadrant here because I'm traveling with the velocity of negative E but these angles are going to be the same this is going to be alpha and that is going to be let me let me write this is going to be alpha and this is going to be and this right over here is going to be alpha now what I want to do in this video is use this symmetry use these these two ideas to give us a derivation of the Lorentz transformation or the Lorentz transformations and the way we might start and this is actually a reasonable way that the Lorentz transformations were stumbled upon is to say alright we could start with the Galilean transformation where we could say alright the Galilean transformation would be X prime is equal to is going to be equal to X minus V times T V times T now we already know that if you just use the Galilean transformation then the speed of light would not be absolute it would not be the same in every frame of reference and so we have to let go of the constraints that time and space are absolute and so there's going to be some type of scaling factor involved and so we can call that scaling factor gamma so we can say alright let's just let's just postulate that there's going to if we that X prime if we assume the speed of light is absolute it's going to be some scaling factor gamma times X minus VT well you could make the same argument the other way around if you view if you view it from her frame of reference and you're trying to translate into my coordinates you could say well X instead of just using the Galilean transformation that X is going to be equal to X is going to be equal to X Prime and now instead of a V we have a negative V right so if you subtract a negative V and let me just write it that X minus negative V times T prime that would be the Galilean transformation but whatever scaling factor we used here these are there's a symmetry here I shouldn't have to use a different scaling factor if I assume a different kind of if I'm in a different frame of reference so if we assume the absoluteness of the speed of light we're going to have some other scaling factor just like that or we could rewrite this as as X let me do that same color we could rewrite it as X is equal to this scaling factor I'm really having trouble changing colors today I'm going to be equal to that scaling factor x times X prime subtract a negative plus V T Prime and if you ignore the scaling factor right over here this is the Galilean transformation from the primed frame of reference to the non primed frame of reference so an interesting thing is well what is the scaling factor how do we figure out what that scaling factor is going to be and so we can do a little bit of interesting algebra here what we could do is actually let me just write what I just wrote let me write it right below here so we could say that X again changing colors is difficult we could write that X is equal to our scaling factor gamma times X prime times X prime plus V T Prime and now what I'm going to do in order to to have my self an equation that involves all of the interesting variables I'm going to multiply both sides of this equation by okay so I could I one way to think about it I'm going to multiply both sides of this top equation by X so if I multiply the left-hand side by X I'm going to have x times X prime x times X prime and then the right-hand side of the equation I can multiply by X but X is the same thing I'm saying it's the same thing as gamma times all of this business so I'm just going to multiply the left hand sides of the equation and I'm going to multiply the right hand sides of the equation so if I multiply the right hand sides of the equation I am going to get gamma squared times and I'm going to have a big expression here and so just really applying the distributive property twice x times x prime x times X Prime and then x times positive VT so so web set prime doesn't look like a prime x times X prime plus x times positive VT plus x times actually positive VT prime I should say you got to be careful here x times positive VT prime and then I have negative V T times X prime so it's going to be negative V T times X prime times X Prime and then finally I'll have negative VT times positive VT prime so that's going to be I could write that as negative let's write that as VT squared V I started negative V squared and actually let me delete this this parenthesis I don't want to force myself to squeeze for no reason so I'm going to have negative V times V so that's negative V squared x times T times T prime times T Prime and now let me place my parentheses so how can I use all of this craziness here to actually solve for gamma and here we're going to go back to one of the fundamentals postulates one of the assumptions of special relativity and that's the speed of light is absolute you're going to measure it to be the same in any frame of reference and to think about that let's imagine an event that is connected with the origin with a light beam so let's say right at time and T prime is equal to 0 I were to shoot my flashlight and let's say it hits something at some point right so do we look at some event right over there and they're connected by a light beam by photon so let me connect them so let me connect them and so if you say once again this could be me turning on my flashlight and the photon at some future at some forward distance and some forward time could just be at some position or maybe it hits something it triggers some type of reaction who knows what it does but we're going to talk about this event right over there that events in my frame of reference it's coordinates are going to be X and CT and CT and since we know the the speed of light is absolute and the way that we've set up these diagrams the any any path of light is always going to be at a 45 degree or a negative 45 degree angle we know that X is going to be equal to CT X is going to be equal to CT for this particular case I could draw it on this I could draw it on this diagram as well if I like just to show that I can so let me draw that so it would look like this it would look like this and we would once again have X equaling CT how would you read that well to get the x coordinate you go parallel to the CT axis so that would be the x coordinate on this diagram and then the CT coordinate you go parallel to the x axis so just like that but once again X is going to be equal to CT and similarly because the speed of light is going to be absolute in any frame of reference if we look at X prime X prime is going to be the same is going to need to be equal to CT prime CT prime if we look it over here X prime X prime is going to be equal to CT prime once again because light this is going to be at a 45-degree angle so X prime is equal to see T prime they're connected by light events so if you take your change in X divided by the change of time is going to be the speed of light so what we can do is use this information for this particular event if if gamma is going to be true for all transformations it definitely should be true for this particular event I can use this I can use this information to substitute back in and then solve for gamma and that's exactly what I'm going to do in the next video although I encourage you to try it on your own before you watch the next video you