Okay, for the summary this week, week 7, let's review the three major paradoxes that we analyzed. We won't, of course, reproduce all the analysis we did but try to summarize the key concepts here. Get the bigger picture view, so we don't get lost in some of the details. So the pole and the barn paradox, remember the idea here was that we had Bob and Alice. Bob had the barn, Alice had the pole. Alice was running at a high velocity 10 meter pole. Bob's barn, measured by Bob, is only 8 meters. But if Alice is going at a high enough velocity, then Bob sees the pole being length-contracted and says, hey, no problem, the pole will fit in my barn. I can close the doors or take photos to show that at that given instant, the complete pole is within the barn. And so he says, hey, go ahead and try it, Alice. And Alice meanwhile, is saying, my pole, I can see is 10 meters long here, I've measured it. But she sees the barn being length-contracted from 8 meters down to even a lower figure. And therefore, she says, no way the pole is going to fit in the barn. And in our analysis, the conclusion was, they're both correct. And the reason they're both correct is due to the relatively of simultaneity. In other words, that when Bob takes those photographs, showing both ends are at one barn door and the other barn door, Alice sees Bob's clocks not being synchronized. And so when the rear barn door takes the photo of the pole as the pole gets there, in Alice's frame of reference, her pole is still sticking out the other end of the barn. Because Bob's clock here, has not actually taken a photograph yet, his clocks are unsynchronized. And so we did that analysis and showed that, yes, in one sense, Bob is correct. In his frame of reference, the pole definitely fits in the barn. And Alice is correct as well, in her frame of reference, the pole definitely does not fit in the barn. And the reason is that to Bob, his clocks are synchronized, to Alice, Bob's clocks are not synchronized. And then the spaceships on a rope paradox, remember the idea here was, we had a stationary observer. Two spaceships roped between them, stretched top. And a stationary observer in between them would set them off on sort of automatic acceleration such that the distance between the spaceships would always be constant to that stationary observer. And then the question was, what happens to the rope? Does it break or not? It would seem like if that distance is always constant, it shouldn't break. But then Alice's shows, from the stationary observer's point of view, the rope has to break because it's undergoing length-contraction. And if the spaceships remain at a constant distance, the rope cannot stretch that far anymore, so at a certain point, it will break. And then to the spaceship observers though, they say, the rope is going to break because the spaceship in front, to maintain the constant distance to the station observer, the one in front has to accelerate first before the one in back does. Again, it goes back to the relatively of synchronization, the relativity of simultaneity to do the analysis here. So if the one in front, from the spaceship perspective, from their frame of reference, if it accelerates first, then it's going to stretch the rope and break it. So they do not see the spaceships accelerating at the same time, whereas a stationary observer does. And then finally, the twin paradox, of course. Both twins are participants. We had Alice and Bob observe the traveler return from the trip having aged less than the one who stayed at home. Remember, part of the paradox comes from the fact that both of them see the other's clocks running slower than theirs. And so then how can that be? How can Alice observe Bob's clock running more slowly, and Bob observes Alice's clock, observes them running more slowly than his? And yet you get you get this asymmetry in the aging as when Alice comes back, in this case, we had Alice do the traveling and come back. And also, why is it asymmetrical because it would seem like from Alice's perspective, she sees Bob go back the other way and then return? So we spend some time trying to understand that asymmetry. It comes from the fact that there is some acceleration, deceleration and acceleration involved for Alice, or whoever's doing the traveling. And because of that, the key point was, there's a change in reference frames by the traveler. And that accounted for them, the difference in times at the end so that Bob, in this case, who stayed home, aged ten years in our example, and Alice aged only eight years. So that's a brief summary of the key concepts and key points that we've went through with these paradoxes. And next week, we'll continue on. We're going to do a little bit more with actually, sort of, in the spirit of the twin paradox here. We're going to do a little bit more with time travel. And also get into some of the other interesting things with the special theory of relativity in terms of how was received. And even talk a little bit about the general theory of relativity and the concepts of time dilation within a gravitational field and a few other things as well.