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Professor Charles Bailyn: Okay, let's start — let's see, we started talking about black holes last time, and there's now going to be a problem set that is going to be available later today, due next week. And I should also point that I've put together this website on black holes. Actually, some of the help sheets already send you to that website. And I'll put a link on the classes server as a whole, but here it is as an actual URL, cmi.yale.edu/bh for black holes. And that website kind of serves as a sort of online textbook for this part of the course. And so, you have something written down to look at for this whole section of the course. And, as you'll see, many of the things we'll be talking about are actually discussed on that website, as well. So, you can find out more information there.

Okay, so I — last time, I defined a black hole. This is simply something where the escape velocity is faster than the speed of light. Or, alternatively, and this is — amounts to the exact same thing, the radius of the object is less than the Schwarzschild radius, which is defined for an object of any given mass. And this isn't particularly extraordinary or interesting, as long as the speed of light isn't particularly extraordinary or interesting. And one of the things that happens when you start talking about relativity is that it turns out the speed of light is a very important quantity. So this — these are interesting because the speed of light is interesting.

And, we'll talk in a minute about how all this arises, but one of the interesting things about the speed of light is that you can't go any faster. It's the speed limit, and no physical process can make you go faster than the speed of light. So, c is fastest velocity possible. And the consequence of that is, supposing you have one of these black hole things — so, here's some object, and it's got some radius. And its radius and its mass are such that the escape velocity, here, is greater than the speed of light. You could imagine that around it is a kind of imaginary sphere. Let's put it in dots. And this is the sphere where the escape velocity is equal to the speed of light. And you remember what the formula for the escape velocity — Vesc is equal to 2GM / R, the square root of 2GM / R.

And so, if on the surface of this object, the escape velocity is greater than that of light, if you keep moving out, R will keep getting bigger. And so, eventually you'll come to a point in space where the escape velocity is equal to the speed of light, and this is called the event horizon. And the reason it's called that is because, if nothing can go faster than the speed of light, what it means is that any event that takes place inside this imaginary sphere can't radiate any information about what's going on to the outside, because this — the escape velocity's greater than the speed of light. Light can't escape, and because nothing else can go faster than the speed of light, nothing else can escape. And so, no information of any kind can come — comes from inside the event horizon to the outside.

And so, in a certain philosophical sense, this event horizon sort of constitutes an edge of the Universe, because nothing that happens inside can affect what happens outside in any way. You can't see in. You can't detect anything that goes on inside. No event inside this event horizon can tell us what's — about itself or — and we — there's no — in principle, there's no way we can find out anything about what happens inside an event horizon.

Okay, so I — last time, I defined a black hole. This is simply something where the escape velocity is faster than the speed of light. Or, alternatively, and this is — amounts to the exact same thing, the radius of the object is less than the Schwarzschild radius, which is defined for an object of any given mass. And this isn't particularly extraordinary or interesting, as long as the speed of light isn't particularly extraordinary or interesting. And one of the things that happens when you start talking about relativity is that it turns out the speed of light is a very important quantity. So this — these are interesting because the speed of light is interesting.

And, we'll talk in a minute about how all this arises, but one of the interesting things about the speed of light is that you can't go any faster. It's the speed limit, and no physical process can make you go faster than the speed of light. So, c is fastest velocity possible. And the consequence of that is, supposing you have one of these black hole things — so, here's some object, and it's got some radius. And its radius and its mass are such that the escape velocity, here, is greater than the speed of light. You could imagine that around it is a kind of imaginary sphere. Let's put it in dots. And this is the sphere where the escape velocity is equal to the speed of light. And you remember what the formula for the escape velocity — Vesc is equal to 2GM / R, the square root of 2GM / R.

And so, if on the surface of this object, the escape velocity is greater than that of light, if you keep moving out, R will keep getting bigger. And so, eventually you'll come to a point in space where the escape velocity is equal to the speed of light, and this is called the event horizon. And the reason it's called that is because, if nothing can go faster than the speed of light, what it means is that any event that takes place inside this imaginary sphere can't radiate any information about what's going on to the outside, because this — the escape velocity's greater than the speed of light. Light can't escape, and because nothing else can go faster than the speed of light, nothing else can escape. And so, no information of any kind can come — comes from inside the event horizon to the outside.

And so, in a certain philosophical sense, this event horizon sort of constitutes an edge of the Universe, because nothing that happens inside can affect what happens outside in any way. You can't see in. You can't detect anything that goes on inside. No event inside this event horizon can tell us what's — about itself or — and we — there's no — in principle, there's no way we can find out anything about what happens inside an event horizon.

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