Adam Brown: prediction

Transcript context

…of the brick, doing useful work a long way away. First observation: this is small. In other words, the gravitational binding energy of something on the Earth’s surface is quite small in natural units. That’s why we didn’t really notice general relativity on the Earth’s surface until we did very sensitive experiments, because general relativity is in some sense a Taylor expansion in this number, where the relativistic effects, where the first order term is just Newtonian, and then the next order terms will give you the GR corrections to the Newtonian answer. Observation number two, and this is something of a digression, is that by essentially sheer coincidence, this number here is very close to the chemical binding energy of rocket fuel. So if you take a rocket fuel like an oxygen-hydrogen mix, the chemical energy binding the rocket together, which is the energy that you’re going to extract when you burn it to make your rocket go, divided by the mc2 of the oxygen and hydrogen you’re going to mix together, is given by 1.5x10-10. First observation: these two are close to each other, even though they came from completely different calculations. This was a gravitational calculation that was something to do with the Earth. This is a chemical property of hydrogen and oxygen. This is also very small. The reason it’s very small is that almost all of the energy in hydrogen and oxygen is not stored in the chemical binding energy of these things going together. The vast majority of it is stored in just the rest mass energy of the protons and the neutrons, which chemical burning doesn’t affect at all. The second largest amount is stored in the nuclear binding energy of the protons and the neutrons to each other, given by the strong force and the weak force, which again chemical reactions don’t touch at all. This is a small number because chemical bonds are very weak compared to the rest mass of the things we’re considering. These two small numbers are almost exactly equal to each other, which is why we can use chemical rockets to get to space, but it’s hard. In particular, this number is a few times bigger than this number, which means that your payload fraction is quite small when trying to use chemical rockets to get to space, because most of your fuel cannot get to orbit. You have to pay a rocket factor that’s going to tell you that most of what’s sitting there on the launch pad is going to have to be burnt up before you get to space, in order to get a small fraction of the rocket up to space. In other words, we can use chemical rockets to get to space in a way that would be totally impossible if we tried to do it from the surface of the sun, but it’s hard. Okay, that’s the fraction on the Earth. But this formula tells you that if you have an object that’s heavier or more compact, the fraction of energy that you extract by lowering the object down to the surface is going to be larger. For example, if you lower it not down to the Earth’s surface but down to the sun’s surface, this would be larger. It’d be a million times larger, because the sun is a few million times the mass of the Earth, but then it’s also bigger, so that takes it away a little bit. to the sun’s surface, this would be larger. It’d be a million times larger, because the sun is a few million times the mass of the Earth, but then it’s also bigger, so that takes it away a little bit. You end up with 2x10-6, the famous redshift from the sun’s surface. You can escalate from there. You can imagine cramming a sun-like mass into an Earth-like radius to make this formula even bigger. Sun mass, Earth radius. That’s pretty much exactly what happens in a white dwarf like Sirius B. And this would get even bigger again. A larger fraction of the mass of the object you’d be extracting by lowering it down to the surface. But it really feels like something has to give before we make an object that is too massive and too compact. In particular, if you look at this formula, what happens for r less than or equal to GM over c2? If this object were so compact and so heavy that it had a radius less than the mass of the object divided by c2, it sure looks like you could get more than a hundred percent. The fraction would be bigger than one. You could get more than a hundred percent of the mass of your brick back by lowering it down to the surface of this object. And that feels wrong. That feels, in fact, more wrong than what’s going on here, because now you’ve got all this energy a long way away. You could perhaps use it to make a whole new brick. You’ve got all this more than mc2 out there. Lower that one down, and it feels like we’ve figured out a way to make a huge amount of energy where there was no energy before. This argument is pretty suggestive that something has to go wrong by the time you get down to that radius. Indeed, when you do the calculation—this is a Newtonian calculation, so it’s only suggestive —in full general relativity, indeed something does go wrong. The thing that goes wrong is that you form a black hole. You can imagine two ways that you could avoid this conclusion. One would be somehow that gravity becomes very weak when you get close to a massive object, weaker than the Newtonian law would predict. That’s sort of what saves you if you try and repeat this same trick in electromagnetism, lowering a charge down towards another charge and trying to extract the electrostatic energy between them. What happens is, essentially due to quantum effects, when one gets too close to the other, they start to fuzz out. The energy going like inverse r gets softened, and you can’t extract more energy because they stop attracting each other so hard. So that’s one possibility, the force gets weaker than Newtonian law would predict as you approach the other object. That’s actually the opposite of how general relativity resolves this. General relativity resolves this paradox by the force getting stronger than Newtonian law would predict. In particular, the force gets so strong when you try to get within this radius, that in fact you cannot slowly lower the brick down towards the surface because you’ve formed a black hole. The gravitational force becomes infinite at a finite distance away—not at r=0, but at some finite value of r—and the brick simply gets ripped out of your hand and you’re unable to extract any more energy out of it. nal force becomes infinite at a finite distance away—not at r=0, but at some finite value of r—and the brick simply gets ripped out of your hand and you’re unable to extract any more energy out of it. That’s the resolution that general relativity provides to this paradox. In particular, you will find that you’ve formed a black hole. So far, everything we’ve written down on the board is Newtonian. It’s just Newtonian, and you start plugging in the speed of light, and you start getting confused. To actually answer some of these questions that we’re asking, you need to go to general relativity, the theory that correctly unifies the speed of light with gravity. This was first done in the context of black holes by Schwarzschild, who wrote down the Schwarzschild metric that describes the gravitational field around a central mass, including potentially around a black hole. Let me write down some of the formulas that emerge. In fact, I think I’m going to write down three formulas, three direct consequences of the Schwarzschild metric. They’re going to give us intuition for what it’s like outside and indeed inside a black hole. The first formula I’m going to write down is the formula for the gravitational field that you would experience if you were trying to remain static outside a central mass. So let’s just talk about static observers. I can discuss how these will get upgraded for observers who are moving around. But for now, I’m just going to imagine that you’re trying to sit here at some fixed radius r away from the black hole. The reason you don’t fall in, maybe I’m lowering you down on a pulley. You’re just sitting here holding the pulley. The question is, how strong a force do you need to stop you falling down? You’re abseiling down very slowly. You’re static. What is the local force of gravity that you experience? Or you can imagine that you’re sitting here, and the reason you’re static is that you’re firing a rocket very hard. The question is, how much acceleration do you locally feel? So by whatever mechanism, you’re remaining static. What is the local force of gravity that you feel? In Newtonian physics, you know what the answer to that question would be. The force of gravity is GM/r2, which is Newton’s famous inverse-square law. But this gets a correction from general relativity. The correction is 1/√(1-2GM/(c2r)), this same 2GM/c2 that we find all over the place. What this tells you: first of all, if you’re a very long way away from the black hole, this here is essentially one. r is very big, and you get Newton’s force law back again. For the Earth, this is very small. As we discussed, it’s down by a factor of 10-10, and then you take the square root. So you don’t really notice it, but you can Taylor-expand this at large r, and you find that you get corrections. You get an inverse-square law plus an inverse-cube law correction plus an inverse-fourth law correction. You find that gravity at short distances is stronger 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.…

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