Adam Brown: evaluation

Transcript context

…than it would have been in Newtonian physics. This is the general relativity correction and it’s making the gravitational field stronger. You have to accelerate harder to not fall into the black hole. In particular, once r is equal to 2GM/c2, what’s called the Schwarzschild radius, you have to accelerate infinitely. The proper acceleration required to not move in r goes to infinity. In fact, if we now convert this Earth to a black hole, this is a very significant radius over here, 2GM/c2. It’s called the event horizon. It’s called the event horizon because if you want to remain static outside the event horizon, further away from the event horizon, you just need to accelerate with some finite velocity in order to remain static. You need to have a finite gravitational field. But the gravitational field, as you approach the event horizon, becomes infinite. So once you’re at or beyond the event horizon, it is impossible to remain static. You will inevitably get sucked into the black hole no matter how hard you fire your rocket. Now, this is just a static formula. You might imagine, “Okay, it’s impossible to remain static closer than that, but maybe I could avoid falling into the black hole by orbiting really, really fast. If I orbit really fast, I have a huge centrifugal force that pushes me away from the black hole, and I can stay out of the black hole that way.” That actually doesn’t work. The reason it doesn’t work is somewhat instructive for the way gravitational attraction happens in general relativity. Of course, if you think about the International Space Station, why doesn’t it fall towards the Earth? It is precisely the fact that it’s orbiting. The fact that it’s orbiting gives it a centrifugal force that shoots the astronauts away from the Earth and precisely balances the gravitational field on the astronauts, which is why they feel weightless there. So orbital angular momentum, if you’re a long way away from the black hole, helps you stay away from the black hole, stops you falling in. There is this kind of sci-fi notion that black holes just suck in everything around them. Not true. You are perfectly able to orbit around a black hole if you’re a long way away from it, just like you would orbit around any central mass. You are not inevitably falling into the black hole. You can orbit just fine. But orbiting stops helping when you get too close to the black hole. We said that the event horizon is 2GM/c2. In fact, once you’re already within 3GM/c2, orbiting is counterproductive if you’re trying to stay away from the black hole. That’s because there are two effects of orbiting. One effect helps you stay away from the black hole. That’s the centrifugal effect. Orbital angular momentum pushes you away from the black hole due to the centrifugal effect. It’s not too hard to write down the version of this formula that applies when you have angular momentum, and you would see that pushing you away from the black hole. But there’s another effect which drags you towards the black hole, and that is the fact that in general relativity, all energy gravitates, not just rest mass energy. Kinetic energy also gravitates. e. But there’s another effect which drags you towards the black hole, and that is the fact that in general relativity, all energy gravitates, not just rest mass energy. Kinetic energy also gravitates. So the effect of orbiting is that you have an additional pull down towards the black hole from the coupling between the gravitational attraction between the mass of the black hole and your orbital angular energy. When you’re far away from the black hole, the centrifugal force is the more important term. When you’re close to the black hole, that coupling is the more important term. In fact, once you get within 3GM, orbital angular momentum stops helping and starts hurting. There are no ballistic orbits that go within 3GM and manage to escape again. So that’s formula number one. It tells you what the gravitational field is a distance r away from a black hole. In particular, it shows you that once you get to this critical radius, the gravitational field becomes infinite. If you cross that, you must proceed to the center of the black hole, no matter how hard you fire a rocket. That’s called the event horizon. At the event horizon, you are not yet dead. You are, however, doomed if you cross the event horizon. You will never be able to escape, not if you convert yourself to light and try to shoot yourself out, not if you fire your rocket infinitely hard. The other place, of course, is r=0, which is where you actually die. That’s at the singularity, and we’ll describe that a little bit in a moment. In Newtonian physics, the gravitational force only becomes infinite there. In general relativity, it becomes infinite already at the event horizon if you try to resist the force of gravity. That’s formula number one. Now let’s do formula number two. All three formulas I’m going to write down are heavily related to each other. They’re really going to be reformulations of each other. Formula number two asks about gravitational time dilation. Let’s again imagine that you’re sitting here, Dwarkesh sitting here some radius r away from the black hole. I’m sitting out here, way off at infinity, just watching you. We’re static relative to each other. There’s no relative motion. You’re just suspended here by your pulley system. The question is, how fast does your watch go relative to mine? Of course, as far as you’re concerned, your watch is ticking at one second per second. As far as I’m concerned, my watch is ticking at one second per second. But if I look at you, I see your watch as running slow. If you look at me, you see my watch as running fast. The second formula makes that quantitative. How much slower does your wristwatch—which is closer to the black hole—run than mine? It says that the time interval, as measured by your wristwatch, is given by the time interval as measured by my wristwatch a long way away, times this exact same square root factor that’s showing up all over the place: the square root of 1-2GM/(rc2). This factor here is less than one. So if I think one second has passed, you think less than one second has passed. are root factor that’s showing up all over the place: the square root of 1-2GM/(rc2). This factor here is less than one. So if I think one second has passed, you think less than one second has passed. In other words, if I slowly lower you down towards the black hole—you hang out some finite distance away from the black hole for what feels to you like a year, and then I raise you back up a long way away from the black hole—you will return to a world that has aged a lot more than you have. This formula makes that precise. I observe your wristwatch to be running slow. You observe my wristwatch to be running fast. Time passes slower down here than it does up here. This is a fact that has by now been extremely well observed experimentally. In the 1950s, in the Harvard physics department, they put two atomic clocks at two different heights in the building, and noticed the one that was higher was running faster than the one that was lower. This is an effect that is now considerably within the precision of, for example, GPS. It just has to subtract that effect, otherwise everything would drift all over the place. GPS clocks that are sitting on the Earth’s surface are running slow compared to the atomic clocks that are in orbit sending out the signal. You have to account for that difference and subtract it off in order to get an accurate read. This is known as gravitational time dilation. Notice it’s quite different from the relativistic time dilation you see in special relativity, which is caused by two objects being in motion relative to each other. Here, we’re not in motion relative to each other. We’re both static. We’re fixed. This is caused by us being at a different place in the gravitational potential, you deeper in the gravitational potential than me. So those are two different sources of time dilation, and they stack. Let’s say instead of being static here, you’re in orbit. You’re far enough away that you can orbit the black hole. How slow do I see you as moving? There are now two contributions, both of which make you look slow relative to me. One contribution is the gravitational time dilation given by this formula. A second contribution is the good old special relativity correction where moving observers look like they’re going slow, and we’ll have both of those effects. So you’ll look like you’re going even slower than you would have done otherwise as you go around the black hole.…

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