Let's now, look at the key things we did here in week 5. So, we do a little summary, not quite in the order we did them but, hitting all the high points, clearly one of the big things of the week was the Lorentz transformation. It was sort of the theme of space time switches. In other words switching between one frame of reference to another frame of reference and having some nice equations such that we don't have to start from first principles. All the time in doing that so the idea was of course that if you have two frames of references, this is A and B, maybe Alice and Bob again, so if you have Bob doing a measurement in his system he sees an event as x, y, and z space coordinates, in a certain time on his lattice of clocks. Then can we transform that into the coordinates in Alice's system. And here were the equations that we came up with, yet we won't go into all the details because we spent some time actually exploring them a little bit to see what the implications were. But for x sub a, and remember a and b here we're doing b as the frame of reference, moving to the right, positive x direction of speed v. So another way we've written these in the past is with a being the lag frame of reference and b being r for rocket, the rocket frame of reference and sometimes that's easier to visualize and remember which way it goes, as one will remember. it's the rocket's going to the right for these equations here. You may remember, if we have a situation where the rocket is going to the left, then we have minus v and minus v over c squared in those cases. So again advice is, memorize the basic situation, make sure you understand which direction it's going and what's the rocket frame, what's the lab frame or what's the rest frame, what's the moving frame, where here we're just doing a and b for the rest frame and the moving frame in this case. So this would be Alice watching Bob moving and he see's an event. Inserting coordinates that involve x b and t b and therefore, Alice's coordinate where that event occurs x sub a is given by this, t sub a on her clocks is given by this. Remember the y in z directions, we often don't deal with those, but if we had to they are not transformed. There's no difference in them because the things like the length contraction effects, time dilation effects only apply in the direction of motion. So we don't have to worry about those, typically, at least in terms of the more complicated transformation equations. So we explored those equations a little bit, and then we used them to revisit The idea of leading clocks lag and we found that the quantitative amount by which they lag is d v over c squared, we v again is the velocity between the two frames of reference. d is the distance between the two clocks in the frame that's moving so again if you have a rocket ship with two clocks one at either end and we see it moving past us, then the leading clock is going to lag as we've argued qualitatively before this week. The leading clock will lag the trailing clock as we watch it go by and now we've discovered the exact amount it will lag by. In this distance d is the distance in the frame of reference, in this case, of the space ship. So its not a contracted length or anything like that. We've set the spaceship down, measure the distance between the clocks at rest, in that frame of reference, say it was d and then as it was moving past us the lag time between the two clocks would be d times the velocity between the frames of reference divided by c squared, and note that in general this is a very very small number and that's why we don't see it in our ordinary everyday experience even with velocities up near the speed of light. Point 9 c here, we still have this c squared down here. So this does not get significant until both d and v are fairly large. We start seeing that effect. But, what was nice about it and one of the reasons we did it here, was that it helped solve our star tours puzzle where we explored length contraction, time dilatation a little bit more with Alice and Bob Bob taking a trip to a nearby star. And we looked at it from Alice's perspective and Bob's perspective and the different things they would see on their clocks. And what lengths they would measure from Earth to the star or the start moving toward Bob and so on and so forth. And we hit sort of a wall, a puzzle there. And discovered that it was answered very nicely, by remembering this leading clocks lag concept, which really is a special case of the whole idea of the relativity of simultaneity that clocks in one frame of reference will be synchronized, they're Bob's clocks. Clocks in Alice's frame of reference will be synchronized. Yet if one looks at the other's clocks, they're not synchronized. And they're not synchronized by this factor between any two different clocks as you observe them. And that all goes back to remember Einsteins' principle that he derived from his two postulate of the constancy of the speed of light. The speed of light is the same to all observers. And that's where we got this relativity of simultaneity effects, as well as the time dilation and the length of contraction of x. So leading clocks lag, we discovered by this amount, d v over c squared there. And then another thing we looked at was combining velocities. We've done this before in our Galilean transformation case of our everyday experience case, when If you have a car moving along and you shoot a basketball out from that car at a certain velocity with respect to the car, the velocities just add. If you have an observer observing what the velocity of the ball is. It is like running forward and throwing something versus just throwing something. Well because the speed of light is an ultimate speed limit, which we also talked about this week, then you can't have things if you have something going up 0.9 c and you try to add something at 0.7 c, so if you have Bob in a spaceship going at 0.9 c by Alice and then he shoots out an escape pod going up 0.7 c, whatever it is, with respect to him, Alice does not see it as 0.9 plus 0.7 c, 1.6 c, it will never be faster, she will never observe it faster, the escape pod going faster in her frame of reference than the speed of light, it will always be less than the speed of light. And so another thing we were able to do using Lorentz transformation is derive a formula for figuring out what Alice would see in that case. And I've used it in the form a and b here again. But Alice could be the lab frame. b is the rocket frame for Bob. And so the u b is the velocity of the object as measured in the moving frame, so again imagine the escape pod shooting out from Bob's space ship and if it's going at u of b, with respect to Bob in his spaceship traveling away from him, then Alice will measure it using this formula, this speed. So it's u b, u sub b plus the velocity between them, and that would be normally the Galilean transformation in terms of just adding velocities, the normal addition of velocities, but you've got this factor on the bottom. And again, note that unless u sub b and v are close to the speed of light, this thing is typically very small and therefore, we don't see that effect. But at high velocities, we will see this effect. And we demonstrated also that no matter what u sub b is, and what v is, if they're both 0.9c for example, then even so, u sub a will still be less than the speed of light. So we did some examples like that. And we talked about the Ultimate Speed Limit, in terms of what happens to gamma, v, if we could actually have an object like a light clock moving at the speed of light, gamma becomes infinite then, which is clearly a problem, so that is an indication that something is going wrong there. And we also looked at this idea of time freezing, If you could move at the speed of light using our light clock example. That time seems to slow down, but we need to be very careful how we talk about this, because in many cases, when you hear people talking about the special theory of relativity by Einstein, they say well, right evidently, the physicists tell us when things move very close to the speed of light, time freezes. Remember it's not, if we could actually travel in a spaceship with our light clock near the speed of light, 0.999 c say, so a very large gamma factor, we'd seen nothing. Out of the ordinary, inside our spaceship that is. Just our like clock will be functioning very nicely, ticking away normally. It'd be a person outside our spaceship observing our clock as it goes by that would see a really strong time dilation effect such that we can never see. So time would never freeze but if we could remember, we did an example with combining perpendicular velocity, and a more qualitative example with our light clock here just to show that, if we're actually moving at the speed of light, the light beam hitting up to the upper mirror of the light clock would never get there, because the mirror would always be out of reach if If the light clock would move at speed c. So brief summary of the, especially, some of the key equations we looked at this week, and derived this week, and used this week, and the concepts behind them.