On to part three of The Twin paradox and in this part we will do the quantitive analysis Bob's perspective versus Alice's perspective. Before we get to that though, I want to go over one more point here that I did not emphasize probably quite enough in the earlier video clips. Because of you may will be thinking, well it still seems to symmetrical to me. In other words, Alice travels to the star and back again. That's from Bob's perspective. From Alice's perspective, she sees the star come to her. Bob go the other direction. Turn around as it were and come back to him. How do you tell the difference between those two situations? Isn't her, on her observations, the same? Shouldn't they be the same as Bob's observations? In actual fact, this is where the acceleration does come in, and that is, only Alice undergoes acceleration. In other words, when you accelerate, you actually feel something, unlike constant velocity motion. So, when she gets to the star and she has to decelerate, she'll feel that deceleration. Same thing accelerating back towards Bob. Whereas Bob does not undergo any acceleration or deceleration, does not feel anything. In fact, only Bob between the two of them can, where you can have a diagram where he has the same reference frame. He does not change his reference frame at all because he's stationary as it were. Whereas Alice does change her reference frame as we've seen. So, that's where the asymmetry comes in. And certainly acceleration is involved there and deceleration. But what we've argued is that by confining it just to a very small region, the effect is such that we can still do the analysis using the special theory of relativity, as we'll see right here. And so we saw in the last video clip with the the diagrams, but it's certainly is true that the acceleration, deceleration is what makes Bob's situation asymmetrical different from Alice's situation. Okay, so let's see if we can do the quantitative analysis then. It's really using concepts that we've used a number of times before. Time dilation, length contraction, leading clocks lag, the three big ones there or the relativity of simultaneity of synchronized clocks. So, let's see how we put this together. And Bob's analysis is pretty straightforward as we've said before. He observes Alice travel to the star. A constant velocity, the star and his form of reference is 3 light years away, she's travelling at 0.6 times the speed of light and again, we're using units such as c is 1, 1 light year per year and 3 divided by 0.6 is just 5 years then, okay? And then Alice's clocks, Alice is moving at velocity V with respect to him. So this is the outbound trip to the star ticking more slowly because it's a moving clock, time dilation, and so 1 over gamma factor, gamma being 1.25 for velocity 0.6c. So 5 divided by 1.25 is 4 years. So he will see her clocks tick off 4 years while his clocks tick off 5 years for the outbound trip. And then she turns around, comes right back again. So, he sees the same thing happen on the inbound trip. She covers three light years. It takes her five years at 0.6c. Her clocks tick off another four years. His clocks tick off five years. And when she gets back, he says, you're only eight years older than you were when you left. Now I've aged ten years, in this case. So again, just using concepts of time dilation there, it makes sense that Alice would be younger. The hard part again comes when what about Alice, in terms of her and Alice, what does she think. So let's think about this a minute, Alice in her frame of reference stationary. She's going to see the star coming towards her and Bob was sitting backwards in a distance. And remember that means there's a length contraction effect. So, in Bob's frame of reference, it's three light-years to the star. And her frame of reference, okay, so distance to the star, Or really we just say the distance the star travels to get to her, right, from her frame of reference, her perspective, is going to be three light years over gamma. Three lights years divided by gamma, which is 1.25, and yeah, I had 3.1 there, 3 light years divided by 1.25. When you do that you get 2.4 light years, ly for light years there, okay. Now, she sees the star coming toward her at 0.6 times the speed of light. So 2.4 light years divided by 0.6c. And again c 1 light year per year. Then you do 2.4 divided by .6, you get 4 years. Okay, so that's how much time ticks off her clock. And again, it's consistent with Bob's observation. Although his analysis of how he gets that 4 years is different. He sees his clocks took off five years while she travels at distance of three light years and he sees time dilation, observes her clocks being time dilated, so it's four years. Where she sees the distance of the star coming toward her contracted to 2.4 light years the shorter distance and at that velocity, it takes four years for the star to reach her. So both of them agree, Alice's clock tick off four years early. Alice's clock and her spaceship ticks off four years from the time she leaves till the star, gets to the star or the star gets to her from her perspective. Okay, now what about her observation of Bob's clocks. Okay, so she observes. Bob's. Now, we have to be careful here because really Bob has at least two clocks involved that we're going to interested in. There's going to be a clock at the star and a clock back on let's say his home planet, if he's on a planet, here on Earth, let's say. And so, got one here, one there, they're both synchronized in his frame of reference, they're part of his lattice of clocks. And so, in general, though, she went wherever they are in his lattice of clocks, when Alice observes one of his clocks, she'll observe the clock ticking more slowly than hers, okay? So, she observes Bob's clocks, To essentially read. Well let me, instead of saying read, let's say to, we'll just say to tick, to tick how many years are they going to tick? Well again it's the gamma factor, her clocks tick for four years. So it's four over gamma, she sees the time dilation in his clocks, and when you do that you get 4 divided by 1.25 is 3.2 years, okay? And again here is where the paradox comes in because Alice says, okay on my outbound trip my clock definitely ticks for four years. We both take a photograph there we both agreed my clock ticks for four years but 3.2 years I see Bob's clock ticking more slowly than mine by the gamma factor, and yet, wait a minute, what's this 5 here? Bob is saying his clocks actually took off 5 years not 3.2 years. We've got to be able to take a photograph of that point and get them to agree if the clock is, both at the same point in space and take a photograph, have to agree in terms of their analysis. And the other part of that too is, if you just say okay same thing for the return trip here, distance star moves away, Bob comes back again, 2.4 light years contracted, 0.6c. 4 years, another 4 years on Alice's clocks but she observes Bob's clocks that took more slowly for 3.2 years. And so it would seem like she has four years on the outbound trip, another four years on her clock, so eight years. But Bob's clocks 3.2 plus 3.2, 6.4. And yet Bob's analysis says no. My clock tick off ten years. So,how can we understand that? And we showed, diagrammatically last time, that the effect of her changing her frame of reference, that when she turns around, she was in a frame of reference with going this way and lines of simultaneity, parallel like this when she changes that frame of reference. Now the lines are simultaneity or different, into different frame of reference. And so, there's a jump, it's not an instantaneous jump but the jump occurs during her deceleration and acceleration phase. Bob's clocks tick ahead from 3.2 All the way to 6.8. And so we want to understand that now, quantitatively. Last time we did diagrammatically. We to understand that quantitatively using some of our old friends here, if they're friends to you may be old enemies at this point, but using our special theory of relativity concepts and equations we've developed. So, first thing to note here is, again, let's think about Bob's two clocks. He's got his Earth clock, and he's got his Star clock. And Alice can observe both of those. And let's think about this a minute, that Alice is observing on the outbound trip early as the star come toward her, what's happening? Well, if both clocks, when they start off were at 0, both of their clocks here, as the star comes toward her, we have a leading clock lag effect. In that case, the leading clock in the star Bob frame of reference, compared to Alice, is the clock on Bob's planet, okay? And that means is the star clock here is going to be ahead of the clock back at the planet. So when Alice starts off on her trip or really when the trip starts coming toward her at velocity point 6C. What that means is if Bob's clock here is zero, Alice observes this clock to be ahead of Bob's clock on the planet. Again leading clocks lag, so I've got two clocks that are moving this direction so, compared to Alice, so Bob's clock is going to be the one that's lagging. And the star clock belonging to Bob's lattice is going to be the one ahead. And remember that leading clock leg factor is Dv over C squared. Where that D is the distance in the frame of reference, Bob's frame of reference this case between him and the star and so if we just do that that's going to be three light years and the velocity is .6 C. And divided by c squared, but remember c squared is 1, c is 1, so c squared is 1 in the units we're using. So, we just get 3 times 0.6 and we get 1.8 years. In other words, when Alice starts seeing the star coming toward her from her frame of reference and both Bob and Alice when they start off both their clocks are at 0 at that point, and then starts coming towards them. Bob goes away. That means Bob's clock, which is reading 0 at that point, the star clock will be 1.8 years ahead of Bob's clock at his planet, his home planet as it were. Okay, so let's see what that does them, so as Alice gets to the star real again as the star gets to Alice the star clock is now there, okay? Alice sees Bob's clocks tick off 3.2 years, all right? Running slower than hers but because the star clocks started out 1.8 years ahead compared to the clock here, Bob's clock here. It means Alice's C start at 1.8 years down there, that's coming toward her, tick off another 3.2 years and what's 3.2 times 1.8, it's 5 years. So, when the star clock gets to Alice, she sees, she observes Bob's star clock To read. 1.8 because it starts out at 1.8 at the beginning of the journey over to Alice, plus another 3.2 years ticked off and that's five years. All right, and so they agree, all right? Both of them agree, yes, my, Bob says, yes my star clock there, my clocks are synchronised as far as I am concerned, and I see Alice travel, takes her five years to get there, therefore my star clock reads five years. From Alice's perspective, she sees Bob's clocks, his lattice clocks, tick off 3.2 years. And the star clock starts out 1.8 years ahead of Bob's planet clock as the star starts moving toward her with velocity of v equals 0.6c. And so she gets 5 years too. Yes, Alice's yes I understand why your clock reads five years but it's not because you think they're synchronizes because from my perspective that clock was ahead already ahead 1.8 years and then your clock runs slower for 3.2 years. But they both agreed the photographic evidence would say five years for that. So, now what happens though as the outbound trip, now what happens is on the inbound trip, it's really, now we do have a mirror image type of thing because from Alice's perspective now, okay? So, star came to her, okay? Now the star moves away again and Bob's coming back this way. As soon as that happens, we really have a change in frame of reference but more particularly from this analysis, we have a change in the leading clock's lag factor. Because now the star clock, as it moves away, is the leading clock, okay and it's Bob's clock coming toward her that is ahead. And it's going to be ahead by this same factor, the 1.8 years, because again it's 3 light years, Bob's distance between the star and his home planet at 0.6 c gives us the 1.8. So, here's what's going on, star comes to Alice. She sees Bob's clocks tick off 3.2 years. Plus the star clock was already ahead by 1.8 years and so when the star clock gets to Alice five years that what she sees and of course Bob sees that as well. Then the star clock starts moving away and Bob starts coming this way as the star clock starts to move away it's still five years it can't change you know and we just looked at it it was five years. So, it's going to start moving away again. And here comes Bob's clock now, but this time as they're moving this way, the star clock is going to lag Bob's home planet clock. And the difference there again is this 1.8 years lag time factor between the two as Alice observes it. And therefore as it moves, as the whole system moves back again, the star and Bob's home planet back towards Alice, she will again see his clock tick off 3.2 years because her clock ticked off four years, same analysis as before. His clock tick off 3.2 years. But his home clock coming back, now is ahead by 1.8 years. And again, this is the change in frame of reference here. So really what happens here is, Alice gets to the planet and figures out five years. And so now maybe she's instantaneously at the planet right there. Now, everything is synced up, she's actually in Bob's frame of reference, if they're stationary, with respect to the others. She's stationary for an instant, or two, at the star, before she heads back in. So, she says, hey, yeah, star clock is at five, my clock is at four, everything matches up. Then, she starts heading back again, or, from her perspective, Bob starts heading toward her, and, at that instant, then, Bob's clock, coming back is the clock that's ahead of the star clock. And that's where you get this going up here to another 1.85 + 1.8 gets us to 6.8. Okay, this is a 1.8 leading clock lag factor here. And then another 3.2 years tick off. Okay, so this deceleration-acceleration can be accounted for within the special theory of relativity by the effect it has on leading clocks lag. And essentially, that's the change in frame of reference that as the star comes toward Alice, it's the one that is ahead because the leading clock is Bob, in that case moving this way and it's behind. But then when it goes to the other direction, all of a sudden, this is the lighting clock and Bob's clock is the leading clock by that 1.8 years. Put it all together, and we see the numbers work out. In other words that Bob Alice's is fairly straightforward, he sees ten years tick off. Alice has to take into account not only the length contraction effect, and not only the time dilation effect, she sees Bob's clock moving more slowly, but also the leading clock lag effect as she switches frames of reference there. And put it all together though and it makes sense, and yes they've done experiments like this. I've mentioned this before, where they actually took two very precise atomic clocks, took them up on a plane, commercial airliner or the equivalent, 500, 600 miles an hour in terms of the speed, flew them around. And they had to, because of gravitational effects, they had to take into account general theory of relativity effects of well. But they took into account the special theory of relativity effects and they did show this difference, that the moving clock, the one that travel on the airplane and came back again compared to the clock that was at rest actually age more slowly. And so one of the things we'll do next week actually is look at some of these affects and see how time travel might be possible, all of these special kind of time travel. So, that's between paradox as I mentioned before the most famous paradox, in the special theory of relativity, but really, again, a pseudo paradox. It seems paradoxical, but, within the theory itself, it all is consistent, it works out with the principles we've established before.