So now we come to the twin paradox which is probably the most famous paradox in the special theory of relativity. In reality, it's a pseudo-paradox as we've mentioned before because you can explain it very logically and consistently within the confines of the special theory of relativity. But this seems counterintuitive to our common sense. And even counterintuitive to some of the principles of relativity because one of the principles is it seems like things should be relative, right? If you have a symmetrical situation, then what happens to one person should happen to the other if you reverse the analysis. And so let's set it up here, you can see we've got a space time diagram, we'll get to that in a minute. But so over here we'll have Alice traveling this time. So she's in spaceship going with velocity v. Bob is observing and Alice heads over to this star which is three light years from Bob. And Bob's frame of reference where velocity is going to be 0.6 times the speed of light and the gamma factor then is 1.25. And just from our previous analysis with things like time dilation, we can see that from Bob's perspective here. Let's just start with Bob and figure out okay, what's going on with Alice. Well and the velocity of 0.6c as she travels away from him, that means if the star is three light years away. It will take her five years to get there, five years times 0.6 times the speed of light gives us the three light years or other way around, you could, let's say that three light years times. Well I'm going to say five times 0.6 because it's written we'll just leave it at that. So what happens here is this is her world line, Alice's world line going up to five years and traveling three light years there. So that's the speed of 0.6 times the speed of light. And then she turns around at that point and comes back again, all right? So another five years, according to Bob's clocks, another five years and then she's back at ten years, in terms of what he is thinking. And then let's think about what he sees Alice's clocks do. Well, Alice's clocks will be time-dialated. They will be running slowly as he observes them, according to his lattice of clocks. Remember, both Alice and Bob have their imaginary lattice of clocks, all synchronized. And so we know that, let me just call it Alice's clock, t sub A. We'll call it delta t sub A, as a certain period of time in Alice's clock. It's going to be one over gamma, delta t sub B because Bob is observing Alice moving. And so if you do the math which really isn't too difficult here because if Bob says sees five years for Alice to get out there, well five years divided by 1.25. So this is going to be one over 1.25 times Bob measures five years in his frame of reference for Alice to get there and you get 4 years. So he sees Alice's clocks tick off four years, when she gets there they take a picture. So his lattice of clocks will read five years, her clock, at that point, will read four years. And then on the return trip, the same thing happens, that now she's coming back still at a velocity of 0.6c. Bob will see his clocks tick off five years ago, because that's how long it takes to go three light years back again. And at the same time, he will see Alice's clocks run more slowly than his and so he'll see Alice's clock run for four years. So we'll have essentially four years for Alice here, another four years here. Meanwhile Bob has five years and five years. So from Bob's perspective it seems like Alice should come back having aged eight years, while he has aged ten years. You could say that's somewhat paradoxical, but not within the confines of the of the theory of relativity, the special theory here, because we know about time dilation. Where the paradox comes in, as we'll see in a minute, is Alice is understanding of this. In fact, let's think about that right now, because you could say well let's just reverse this situation, because in Alice's frame of reference, she sees Bob moving away from her, right? And we could draw a similar diagram like this for Alice except Bob would travel over here this way in a negative direction and then travel back to meet Alice again as it were. So, whether Bob is on Earth of planet or just his own spaceship there, Alice travels away to the star comes back again okay? Or Alice could see Bob from her frame of reference Bob travels that way, goes that way. The star comes to her and then back again the star moves away and Bob comes this way. And if you do that analysis from Alice's perspective she sees her clock ticks off four years and then you could say okay so this is Bob's analysis and then if we got Alice's analysis shall say, well, from my prospective, your clock's Bob are taking more slowly than mine. My clock's take of four years, we both agree on that because that's what my clock read when I got there. But I see your clock's ticking more slowly during my trip out there. If I do comparisons along the way, I see them actually going one over gamma delta tA. And if you that actually, [COUGH] if you do delta tA equals four so this is one over 1.25 and this is four. We get 3.2. Something like that, four divided, yeah, okay. 3.2 years, so she sees his clocks tick off 3.2 years there. And so this is where the real paradox comes in. I mean, part of the paradox is just in the idea of time dilation and all that, that there might be differences in times when they get back. But the real paradox is Bob can understand just from regular time valuation when Alice comes back that her clocks run more slowly. Yet Alice though can do the same analysis and say well, if I observe Bob's clocks going 3.2 years out and 3.2 years back shouldn't that be 6.4 years total? My clocks run eight years while Bob's clocks run 6.4 years. I should expect Bob to be younger when I get back. So, what's going on here, you can't have both. Either one is older and one is younger or they're the same age. And so we have to figure out what's going on with this in terms of the special theory of relativity. Can we explain this? So that we can understand that when they get back, Alice actually, the answer is, Alice has aged eight years, and Bob has aged ten years. And yet this part of the analysis, also for Alice, is correct. Now, one other thing to mention here, because often when you redevelop the twin paradox, and just sometimes when you think about it a little bit. You'll often come to the point that well there's a real problem here because for Alice to get to the star she has to change direction. And the change direction means she has to decelerate zero and then accelerate again or sort of going a loop. So how were she does it, she's accelerating and as we've mentioned before, the special theory of relativity only works for unusual frames of reference when you have constant velocity motion. And so it seemed that that just throws the whole thing out the window, that we cannot analyze this situation using the special theory of relativity. And sometimes you'll hear it said that way. The twin paradox, because it involves acceleration, therefore you have to use something like the general theory of relativity which involves acceleration to really understand it. And in actual fact you don't. The difference in times as we'll see when they get back is due to relativistic effects from the special theory of relativity. And here's how we can handle the acceleration. Essentially, we'll make the acceleration, the time of deceleration and acceleration again, very, very small compared to the travel time. So in other words, essentially be a quite an instantaneous turnaround there, if I decelerate rapidly, and then accelerate rapidly, and turn around and come back again. And you can make that possible if it's not possible within our numbers here you just lengthen the trip that needs to be made. And then you can make the time of acceleration really as small as you want. Now of course there are things about you have to factor in. What level can a human body take a sort on amount of deceleration and acceleration. In fact next week we'll talk a little bit more about that. But we'll this assume for now, yes we can decelerate, and accelerate rapidly enough so that the time of acceleration is negligible compared to the time when it's at constant when Alice is at constant velocity there. But as we'll see the key point about that deceleration and acceleration and actually as you can see form the diagram here as well. Is that Alice changes her frame of reference? And that's what we have to take into account. And as long as we take that into account then we can analyse this from the perspective of the special theory of relativity. And understand that yes, Alice will see Bob's clocks when she's traveling along here on the outbound trip. She will observe Bob's clocks ticking away. And the total time of 3.2 years while hers do four years, meanwhile Bob sees his clocks five years and hers at four years, on the inbound trip as well. She'll see Bob's clocks 3.2 years while hers do four years and Bob will observe his clocks for five years. So that's what we want to understand. And what we're going to do in part two here of the twin paradox is break this diagram down. Expand it out to the outbound trip and the inbound trip. And do a combined, two combined space time diagrams actually, to see if we can understand a little bit more diagrammatically, visually what's going on with the twin paradox.