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Professor Charles Bailyn: Last time, I wrote down the Schwarzschild metric, which was the sort of culmination of the theoretical part of the — of this section of the course. And I realized, as I was talking to some of you afterwards about some of the implications of this, that the way I — one of the ways in which I wrote it down might have been a little confusing. So, let me write it down again and I'll say a couple things.

And what I'm going to do is I'm going to write it down with a little r instead of a capital R and I'll tell you why in just a minute. Obviously, it doesn't matter. I've just changed the notation slightly. And the reason I want to do this is because I want to emphasize that this r, the little r, what I've now written down as a little r in this equation, is not the radius of the object; r is just a coordinate. It can be anything. It's not the radius of the object for which the mass is giving you the Schwarzschild radius. The Schwarzschild radius is a very specific radius.

rs = 2GM / c2 so that's a specific number. The little r, here, is just a coordinate and so is φ and θ, which combine into this thing which I've called Omega [Ω] and T.

And so if you have an object of mass M and radius — I'll give — I'll call that R0 just to make sure that you understand that this is the radius of the object. The radius of the object doesn't come into this metric. The mass does, because the mass turns into the Schwarzschild radius and that's part of the equation. But the radius of the object doesn't matter.

And so, if you then ask, well, what is a black hole? A black hole is something in which the radius of the object is less than the Schwarzschild radius. Because if that's true, then there is some coordinate r, some little r where little r = RS. And then, if you go up into these equations here very — these very strange things happen when little r is equal to RS. Yes, question?

Student: Yesterday, the big group had [Inaudible]

Professor Charles Bailyn: Oh you're right, you're right, it has to be the other way around, I'm sorry.

Student: [Inaudible]

Professor Charles Bailyn: This has to be squared. You're right on both counts, good. RS / r. Yes, it better be that way because RS / r in the normal part of space is less than 1. So, thank you very much, sorry about that. Okay, I think we're all right. Are we all right? Yeah, good.

Okay, so there is some r where r is equal to RS, and then these very bizarre things start to happen. If R0, the radius of the actual object, is bigger than its Schwarzschild radius, then that isn't true. There's no r where r is equal to RS — then no r equal to RS. Now, you might think that somewhere inside the object, there would be a little r, which is equal to the Schwarzschild radius. But that isn't — that turns out not to be right either, because the relevant M, the relevant mass-

Student: Can you lower the top slide?

Professor Charles Bailyn: Oh sure.

Student: The whole — the slide not the -

Professor Charles Bailyn: Oh yeah, okay. So, yeah. Let me put this back. The relevant mass is the mass inside r. So if you're — if you've got some mass M, like this, and you're outside, if you're here, then the whole of the mass counts. But if you're sitting inside this thing, then the only mass that counts is this mass here. And so, for r — for coordinates for positions inside the object, then the relevant mass that you put into the equation is less than the mass of the whole object.

And what I'm going to do is I'm going to write it down with a little r instead of a capital R and I'll tell you why in just a minute. Obviously, it doesn't matter. I've just changed the notation slightly. And the reason I want to do this is because I want to emphasize that this r, the little r, what I've now written down as a little r in this equation, is not the radius of the object; r is just a coordinate. It can be anything. It's not the radius of the object for which the mass is giving you the Schwarzschild radius. The Schwarzschild radius is a very specific radius.

rs = 2GM / c2 so that's a specific number. The little r, here, is just a coordinate and so is φ and θ, which combine into this thing which I've called Omega [Ω] and T.

And so if you have an object of mass M and radius — I'll give — I'll call that R0 just to make sure that you understand that this is the radius of the object. The radius of the object doesn't come into this metric. The mass does, because the mass turns into the Schwarzschild radius and that's part of the equation. But the radius of the object doesn't matter.

And so, if you then ask, well, what is a black hole? A black hole is something in which the radius of the object is less than the Schwarzschild radius. Because if that's true, then there is some coordinate r, some little r where little r = RS. And then, if you go up into these equations here very — these very strange things happen when little r is equal to RS. Yes, question?

Student: Yesterday, the big group had [Inaudible]

Professor Charles Bailyn: Oh you're right, you're right, it has to be the other way around, I'm sorry.

Student: [Inaudible]

Professor Charles Bailyn: This has to be squared. You're right on both counts, good. RS / r. Yes, it better be that way because RS / r in the normal part of space is less than 1. So, thank you very much, sorry about that. Okay, I think we're all right. Are we all right? Yeah, good.

Okay, so there is some r where r is equal to RS, and then these very bizarre things start to happen. If R0, the radius of the actual object, is bigger than its Schwarzschild radius, then that isn't true. There's no r where r is equal to RS — then no r equal to RS. Now, you might think that somewhere inside the object, there would be a little r, which is equal to the Schwarzschild radius. But that isn't — that turns out not to be right either, because the relevant M, the relevant mass-

Student: Can you lower the top slide?

Professor Charles Bailyn: Oh sure.

Student: The whole — the slide not the -

Professor Charles Bailyn: Oh yeah, okay. So, yeah. Let me put this back. The relevant mass is the mass inside r. So if you're — if you've got some mass M, like this, and you're outside, if you're here, then the whole of the mass counts. But if you're sitting inside this thing, then the only mass that counts is this mass here. And so, for r — for coordinates for positions inside the object, then the relevant mass that you put into the equation is less than the mass of the whole object.

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