Galilean transformation and contradictions with light

things are starting to get interesting in the first video we set up a spacetime diagram for my frame of reference that started to plot things paths in that space-time diagram and then we thought about our friend Sally who write at time equals zero is it x equals zero but she's passing me up at a relative velocity of half the speed of light in the positive x-direction so after one second she has gone one and a half times ten to the eighth meters and the positive x-direction after two seconds she's gone three times ten to the eighth and then we assumed that she was part of a train of spaceships so 3 times 10 to the 8th meters in front of her was another spaceship and it's moving with the same relative velocity relative to me or in my frame of reference so at time equals zero is 3 times 10 to the 8th meters in front of me but then at time equals one if we take this point in space-time in my frame of reference that ship is now it now has so T let me write this if we look at my space-time coordinates T is equal to one second and X is equal to 4.5 times 10 to the 8th meters that's from that's if you look at my space-time coordinates but what about Sally space-time coordinates well to figure out Sally's space-time coordinates we go parallel to the X prime axis and see where we intersect the T prime axis but that's still T is equal to one second this is T is equal to two seconds this is T is equal to three seconds so T or I should say T prime is equal to one second T prime is two seconds T prime is three seconds so T prime is equal to one second still and so in general we can say that T prime is going to be equal to T but what is X prime going to be equal to so X prime well to figure out X prime you go horizontal to the T prime axis and see where you intersect the X prime axis and you see that it's 3.3 times 10 to the 8th 3 times 10 to the 8th meters and hopefully this makes intuitive sense if it doesn't pause the video and really think about it because in her frame of reference that spaceship looks stationary because it's moving with the exact same relative velocity to me it's going to continue to stay 3 times 10 to the 8th meters in front of her which is exactly what we see so that's why it's X prime coordinates stay three times 10 to the 8th meters from my point of view it's getting further and further away from me at the relative velocity at one point five times ten to the eighth meters per second so how do we how do we translate between our between our x-coordinates let me do this in a different color between our x coordinates and our x prime coordinates well you see for these examples we are we see X prime is going to be less than X and that should also make sense because the especially for this case it's stationary from a from the S prime point of view but it's X continuously increasing as time passes on from my frame of reference from Sal's frame of reference from the s frame of reference so if we start with X we should subtract something and the difference between the two the discrepancy between the two is going to be the relative velocity times time and so for this particular example we saw that three times ten to the eighth meters was equal to four point five times ten to the eighth minus the relative velocity 1.5 times 10 to the eighth meters per second times time which was one second and I didn't write the unit's but if you write the unit's it all works out and you get exactly this so hopefully that's starting to get you comfortable with having these two coordinate planes on or two space-time diagrams over on top of each other and the reason why the blue one is distorted is because it's on top they're moving with a relative velocity relative to what I'm considering to be a stationary a frame of reference which is mine obviously there's no such thing as an absolute stationary frame of reference and we'll talk more about that in the future but what I now want to focus on is that photon that I emitted at time equal zero because we saw it moves with the speed of light in my frame of reference after one second it has moved through its x-coordinate is three times 10 to the 8th meters after after 2 seconds after 2 seconds the photon is at 6 times 10 to the eight meters let's see what that photon looks like from the s prime frame of reference from Sally's frame of reference well from Sally's frame of reference let's think about that photon after two seconds so the photons right over there so T prime is equal to two seconds two seconds but what is X prime what is X prime going to be equal to well X Prime we go parallel to the T prime axis is three times 10 to the 8th meters 3 times 10 to the 8th meters so in her frame of reference it took that photon of light two seconds to go 3 times 10 to the 8 meters or it looks like the velocity of that photon is one and a half times 10 to the 8th meters per second in the positive x-direction and this should hopefully make sense from a Newtonian point of view or Galilean point of view this is actually these are called Galilean transformations because if if I'm in a car and there's another car and you see this on the highway all the time if I'm in a car going 60 miles per hour there's another car going 65 miles per hour from my point of view it looks like it's only moving forward at five miles per hour so that photon will look slower to Sally similarly if we assume this Newtonian this Galilean world if she if she had a flashlight if she had a flashlight right over here and right at time equals zero she turned it on and that first Photon we would have Pat we were to plot it on her frame of reference was going to go this P it should be good it should go the frame the speed of light in her in her in her frame of reference so it starts here at the origin and then after one second in the S Prime in the S prime coordinates it should have gone three times ten to the eighth meters after two seconds it should have gone six times 10 to the eighth meters and so it's path on her space-time diagram should look should look like that that's her photon that first photon that was emitted from it so you might be noticing something interesting that photon from my point of view is going faster than the speed of light after one second its x-coordinate is 4.5 times 10 to the 8th meters it going 4.5 times 10 to the 8th meters per second it's going faster than the speed of light it's going faster than my photon and that might make intuitive sense except it's not what we actually observe in nature at any time we try to make a prediction it's not what's observed in nature it means that our understanding of the universe is not complete because it turns out that regardless of what which inertial reference frame we are in the speed of light regardless of the speed or the relative velocity of the source of that light is always going 3 times 10 to the 8th meters per second so we know from observations of the universe that Sally when she looked at my photon she wouldn't see it going half the speed of light she would see it going 3 times 10 to the 8 meters per second and we know from observations of the universe that Sally's photon I would not observe it as moving at four and a half times 10 to the 8th meters per second that it would actually still be moving at 3 times 10 to the 8th meters per second so something has got to give this is breaking down our classical our Newtonian our Galilean views of the world it's very exciting we need to think of some other way to conceptualize things some other way to visualize these these space-time diagrams for the different frames of reference