Adam Brown: evaluation

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…Absolutely. People in the late 18th century wrote this formula down. I think both Michell and Laplace had this. And they said that if you had an object that was that massive and compact, light would not be able to escape. That reason is not particularly compelling by modern standards but it turns out to be particularly correct, including crazily this factor of 2, which is correct for completely coincidental reasons. But let me give you a more compelling argument that something funny is going to happen around this radius. And to do that, let’s think about trying to extract energy from objects by lowering the objects down towards a central mass. Let’s start off perhaps with the Earth. Here it is. We’re going to start off a long way away from the Earth, with a brick of mass m. I’m going to take this brick, attach it to a pulley system, and then slowly lower the brick down towards the surface of the Earth and deposit it with zero velocity on the surface of the Earth down there. In doing so, I can extract energy from the brick. I’ve extracted energy from the brick because there’s a force pulling the brick down. That force I’m doing through a certain distance, and that gives you an energy. We know, at least in Newtonian physics, what the formula for the amount of energy you can extract from the brick is: G times the mass of the Earth times the mass of the brick, divided by r, the radius away. It’s an energy, not force, so it’s an inverse distance law, not an inverse square distance law. That’s the energy you can extract from the brick by lowering it down to a distance r away from the Earth. Of course, if you try to lower it beyond the surface of the Earth, this formula changes. But let’s just put it on the surface of the Earth. So this is the amount of energy I have extracted from the brick, out here a long way away. You can ask, what fraction of the rest mass energy of the brick have I extracted? This is a question that you would only naturally ask once you’ve invented special relativity and know that the rest mass energy is given by mc2. We can straightforwardly calculate, at least in this approximation, that the fraction of the energy that you’ve extracted is that divided by the rest mass energy you started with. The mass of the brick, of course, is going to cancel, but not the mass of the Earth. This is going to be given by G times the mass of the Earth, divided by c2 times the radius away from the Earth at which you stop, the radius of the Earth. So what fraction have I got out of it? If you lower down to the Earth’s surface, the answer is you haven’t really extracted that much from the brick. You’ve extracted a fraction 7x10-10 of the original rest mass energy 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. 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.…

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