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Current time:0:00Total duration:9:14

Video transcript

so let's revisit a scenario that we've seen in several videos especially the last video where we tried to find this neutral frame of reference let's say we're in spaceship a we are in an inertial frame of reference and let's say right at time equal zero in our frame of reference spaceship B is exactly where we are but it's traveling in the positive x-direction at eight ten to the speed of light we've already seen that we can overlay the space-time diagrams for each of these frames of reference if we make this if we do our axes our you know the X is one direction of space-time that we associate with the X direction of space and then the CT this vertical axis this is another direction of space-time that we perceive as the passage of time well you can you can then overlay B's frame of reference and you kind of have these tilt these these skewed these skewed axes and you could look at a change in time so for example if you were to compare the event right when B passes us up to let's say some amount of time later some amount of so let's say this this right over there is a change in CT in our frame of reference these two events are happening at the exact same place they're just separated in our frame of reference we would perceive them as being separated by time but if you want to say well how much time seems to separate these events in B's frame of reference well then you would want to go parallel to the X prime axis reviewing B's frame of reference as a prime frame of reference and see where we intersect the CT prime the CT prime axis so this is our change in CT while this is our change in CT this looks like our change in CT prime and it looks longer but we have to remember that we haven't scaled these things and actually the scales change depending on the relative the relative velocities but we can actually verify algebraically that our change in CT prime is going to be larger than our change in CT and we just have to look at the Lorentz transformations to realize that so our change in CT prime is going to be equal to the Lorentz factor times our change in CT - beta times our change in X we've seen that multiple times before well our change in X our change in X is zero it looks stationary in our frame of reference so that term is zero so our change in CT prime is going to be equal to the Lorentz factor times I have to use parentheses anymore times our change in CT and our Lorentz factor is going to be greater than one I can actually calculate that let's do it so the Lorentz factor the Lorentz factor here so gamma is going to be equal to one over the square root of 1 minus well be the B's relative velocity so 0.8 C over the speed of light squared well what is this going to simplify to the C's cancel out 0.8 squared is 0.6 for one - that is 0.36 this is going to be and then you take the square root of that it's going to be 1 over 0.6 which is equal to 1 over 6/10 which is the same thing as 10 over 6 which is equal to the same thing as 5 over 3 5/3 which is equal to 1 and 2/3 so you can see our change in CT prime is going to be 1 and 2/3 times the change in CT now you might want to just say well does this do these two look like 1 and 2/3 and it might look at a little bit like that the way I've drawn it but you can't just look at it purely on you can't just take a ruler for this length and a ruler for this length because the scales are different and I haven't marked off the scales with this at least helps us visualize but let's think of let's think about let's take a round let's imagine the change in CT prime between right where the spaceships pass by and a little bit later now I'll do this in a different color because this is actually a different event in space-time then the one that we were focusing on right now and we're going to be viewing it from B's frame of reference so that's our change in CT prime but what is it going to be our change in CT well to think about it we could go parallel to the x axis the x axis right over there so you go parallel parallel to the x axis and you get right over there and so our change in CT looks like it's more and once again we can algebraically verify it by really doing the same thing our change in CT is going to be equal to gamma times our change in CT prime minus beta times Delta X prime and if you were to actually if we actually did have a change in X prime here the beta the velocity has a different direction so it'd be the negative but our change in X prime from B's point of view these two events are happening in the same place so our change in X prime is zero so you have change in CT is equal to gamma times change let's again I don't need my parentheses times change in CT prime it's going to be the same gamma because remember we're taking V over C and so whether it's either V or negative V when you square it it gets the same value so once again gamma is going to be one in two thirds so it's going to be one and two thirds so it seems a little bit strange you know I have some passage of time in my frame of reference where it looks you know something that looks stationary two events that look like they're happening in the same place but one after another it looks like their change in time it takes longer for those two events to happen in the CT prime for in the in the moving frame of reference but and then if we have some event happen with that look stationary and B's frame of reference and they're separated by change in CT prime it looks like the change in CT between those two events is even larger so it looks like this somewhat bizarre phenomenon and to help us reconcile these and to visualize a little bit better actually even to to be able to put CT and CT prime on on us on the same scale we can look for this neutral frame of reference which we did in the in and actually the last video I made I don't know if it's the last video you've seen or we could say look if a and B are traveling with a relative velocity of 0.8 times the speed of light relative to each other B is traveling 0.8 C in the positive x-direction relative to a stationary a or a is traveling 0.8 C in the negative x-direction relative to a stationary B you can find a frame of reference where a and B were in that frame of reference to a stationary observer in that frame of reference a and B are both traveling outwards at half the speed of light and we figured that out when we did those videos on a neutral frame of reference and what's neat about that is if when you if you make with what looks like a Minkowski space-time diagram in that neutral frame of reference then the C T and C T prime a and B's frame of reference is get equally skewed to the you know if you think about the time axis to the left on the right and since they're equally skewed the time-dilation relative to this rest frame is the scaling is going to be the same so you could put both of these on both of these on the same scale and you can see that this this this neutral frame of reference the C T prime prime frame of reference I drew them over here that if you look at either of a or B's frame of reference they're going to be in between a and B frame of reference but here it's the neutral one it's the one where we're drawing the time and or we're drawing the two axes being perpendicular to each other and now if you look at these two events so if you look at this first event where you have our delta CT so you look at this first event where you have a delta in CT between right when the spaceships pass and right over there you see that if you were to look at that if you look at the CT prime coordinate for that event well you go parallel to the X prime axis you get right over there so this is that is your change in CT prime and likewise if you have that other event that I drew in that blue green color right over here and this is your change in CT prime this is your change in CT prime well then if you think of it from a z-- frame of reference well we just followed the x axis not the X prime we go parallel to it we end up right over there now what's really interesting about this what's really interesting about this is the is from a z-- frame of reference the yellow event happens before the blue event but in B's frame of reference the blue event happens before the yellow event so really really really fascinating things going on but I really like about this diagram is that is that a and B's frame of reference are going to have the same same scale since they're both equally skewed to the left and the right if we think about the time axis and this type of diagram is called a lodal diagram lodal lodal diagram which is really a variation of a Minkowski diagram but it lets us really appreciate the symmetry between these frames of reference