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Last week we worked through the Twin Paradox. And if you've been thinking a little bit about it, you might be asking, can we put it to some practical use here? And so Alice has been thinking about that. So here is what she would like to do. She would actually like to take a trip to the center of the galaxy and back again so she is going to be in the spaceship, Bob's going to be here on earth observing her. Here's my fanciful drawing of the center of the galaxy, a lot of gas clouds around they actually can't see it in, in visible light we can still using infrared telescopes in the light and. A lot of bright, blue, young stars there, orbiting around actually a, a super massive black hole with the equivalent mass of about 2 million solar masses, even more than 2 million solar masses. So, very interesting region, if we could actually get there, and Alice thinks she can do it. And so, so let's think about this. using our analysis of the twin paradox. We know that, in terms with Bob the observer, it's a fairly straight forward analysis of the situation. Remember Alice, the traveler, is a little more complicated there because we had, some relativity of simultaneity, leading clock flag going on and things like that, but from Bob's perspective fairly straight forward. So let's think about it from his perspective here. He knows that it's actually about 30,000 light years to the galaxy. This is, this is a round figure. It's, it's not easy to actually figure out the exact distance. The best estimates right now are maybe around 27,000 light years or so, so we'll just round it up to 30,000 light years there. So, we know the distance to the galaxy. So if you think about it, Bob is going to see, even if Alice moves pretty close to the speed of light here if she could move the speed of light, it would take her 30,000 years to get there. So maybe we'll just, we'll ignore the acceleration that's needed for the moment. Just assume she's going to constant loss of motion all the way there. 30,000, it takes 30,000 years, traveling
at the velocity of light from Bob's perspective. But remember, if he were to observe her clock that she's carrying with her, and that measures her time and her frame of reference. The Lorentz factor comes in. The time dilation factor. So, we know that the, we'll just do half the trip. Because, we know from Bob's perspective it's symmetrical, so coming back, it's the same, same thing, so we know from Bob's perspective that, she travels not quite the speed of light but very close to it, it's going to take about 30,000 years on his clocks but we also know that as he observes. Her clocks, there's a time dilation factor. He observes her clocks ticking more slowly, and so it's going to be delta T. This perspective, divided by gamma. Well. Let's say that Alice is really in a hurry here, she wants to get there in two years and then take two years back again, she can certainly, you know, there's a slowing down factor as well, we have to account for that, but we're just doing a very rough calculation here, so. We'll assume that she can slow down pretty quickly, and then get back up to speed pretty quickly and come back. Or even do something like, as she gets there, perhaps she can take a path through here, use sort of a slingshot effect around that black hole in the center, as long as she doesn't get too close to it, and shoot her back this way. So however she does it, in terms of the details, we'll assume that we can work something like that, or that she, she could do it. So, 30,000 years, she want, for Bob. She wants to get there in two years. So, if this is two, we have, okay, Alice, she's in a hurry. She wants to get there in two years. Bob says, you know what, even if you go at the speed of light it's going to take you 30,000 years. But her clock, he's going to observe her clock running more slowly. So if her clock is going to run two years on the trip, from zero all the way to two, to get there, what does gamma have to be? Well, obviously very easily. 30,000.
You know, bring the two over here. 30,000 divided by two. This implies that gamma equals 15,000. So, that is the, the Lorentz factor. That Alice needs to get to the center of the galaxy in two years. Assuming constant velocity and motion there. Well, what velocity is that? If you work that out using the Lorentz factor equation, here's what you're going to get. Assuming I got all the digits right here. It's something. Equal to this or very close to it, it's let's see, 1, 2, 3, 4, 5, 6, 7, 8, 7, 7, 7, 7, 7, 8 or something like that. In other words, very, very, close to the speed of light and oh, c here. We put the c at the end. so, 0.99999, essentially times, times the speed of light to get a gamma factor of 15,000. Which will enable Alice to get there according to her clock that she's carrying with her, and what she's aging according too, she'll get there in two years, and then maybe slingshot back in another two years back this way. So she's back in four years, and she gets back and, and she's really excited to tell Bob. Gosh, that was fantastic! All the great stuff. Took some great photographs there. Comes back and then she asks okay where is Bob? Well it's like well Bob who? When you think about it, you know. Alice has aged four years total. People on earth though, time on earth, 30,000 years there, 30,000 years back and it's 60,000 years later. On Earth when she gets back and in fact so you can think of this some what as time travel into the future would actually be possible here with the Twin Paradox analysis or anything similar to that. So its not only at that point its Bob Jr. Bob the third, Bob the fourth, its more like Bob the six hundredth might still be around at that point. So about six hundred generations have passed. So, It'd be very interesting for Alice not only the, to experience the center of the galaxy, what's going on there, but to come back and see what Earth is like 60,000 years later, if anyone is even
still around at that point. another interesting question here is, a couple of interesting questions. One is, what would Alice actually see if she could take this trip? As she speeds along towards the center of the galaxy and of course in, in films like Star Wars and Star Trek you get the idea of you see the stars sort of flowing by you, you know streaks of light. In fact the image we use for this course if you look on Coursera it has sort of that effect. Right? Sort of the stars streaming by on a blue background. An actual fact if you do that analysis, it wouldn't be quite so dramatic unfortunately. It would be more just like a fuzzy glow that you would see. In fact, not only that, it would be very dangerous type of situation for her cause here's what would happen. Remember how we talked earlier in the course about the Doppler effect and train whistles gotta, train whistle coming towards us or if we're moving toward. It, you get the higher pitch and then as it moves away it's a lower pitch. In other words, the frequency of the train whistle, increases as you move toward it. Same thing here with light, there's a Doppler effect. And even a relativistic Doppler effect, but as Alice moves at that velocity and sees stars coming to her the light from those stars would be shifted toward the blue end of the spectrum. And even beyond that into the ultraviolet region, and beyond that into what's known as the extreme ultraviolet region and beyond that to the x-ray region. In other words, all those stars that she's approaching, the light, from those stars coming at her would be x-rays, and so she'd need some pretty good shielding on her spaceship to protect her from, from those x-rays during her, her trip. Now, of course, we can't see x-rays, so in other words, that it hears the visible light from those stars, because of extreme Doppler shift. the frequency shift there would turn the visible light into x-rays, coming at her. And some of you may know that, throughout the universe, there's what's called the cosmic microwave background radiation. It's its sort of a remnant of the big

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