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Professor Charles Bailyn: Okay, here's the plan for today. I want to do one last foray into relativity theory. And this is going to be a tricky one, so I hope you're all feeling mentally strong this morning. If not, we — gosh, we should have ordered coffee for everyone. And, in so doing, I want to introduce one key concept, and also answer at least three of the questions that you guys have asked before in a more — in more depth, and also relate the whole thing back to black holes. And then, having done that, we'll have some more questions. And then, having done that, I want to get back to astronomy; that is to say, to things in the sky that actually manifest these relativistic effects. So, that's where we're going today. And along the way, as I said, we'll deal with some of the questions you've been asking in a deeper kind of way. In particular — so, questions.

Watch out for the answers to these questions. Somebody asked, "What's special about special relativity, and what's general about general relativity?" How do they relate? So, we'll come back to that one. Somebody also asked, "Why use the speed of light to convert time into space and vice versa, to get them in the same coordinate system?" So, why use c to convert time to space and vice versa? And then, also, there was the question of, you know, "What is the mathematical formulation of general relativity?" So, how to express general relativity in some kind of equation. And we'll get to the key equation, which is something called a metric, for general relativity, and then we're going to stop. Because to go on from there is fairly heavy calculus and we're just not going to do that. But I want to get at least that far.

Okay, so let's go back to special relativity for a minute. So, special relativity. Flat space-time, no gravity. And you'll recall what happens. As you get close to the speed of light, all sorts of things that you thought were kind of constant and properties of objects, like mass and length and duration, and duration of time, and things like that, all start to get weird and change. So, length, time, mass, all these things, vary with the velocity of the person doing the measuring.

And so, you could ask the question, is there anything that doesn't vary? Is there anything that's an invariant? And the answer is, yes. There are some things that don't vary. So, some things are invariant. And Einstein actually said later in his career that it's actually the invariants that are important, not the things that change. And so, he should have called his theory invariant theory instead of relativity theory. Think of what that would have done to pop philosophy. Instead of saying, "everything is relative," all this stuff, you would have had the exact same theory. You would have called it invariance theory. And the pop philosophy interpretation of this would be, "some things never change." And it would have been a whole different concept in three in the morning dorm room conversations.

Okay, so some things are invariant, what things? Now, let me first give you a little bit of a metaphor and then come back to how this really works in space-time. Supposing you're just looking at an xy-coordinate system and you have two points in a two-dimensional space. So, here's a point and here's a point. Now, if you arrange for some kind of coordinate system — here's a coordinate system. This is x, this is y — and you ask how far apart these points are. Well, you can do that — let's see.

Watch out for the answers to these questions. Somebody asked, "What's special about special relativity, and what's general about general relativity?" How do they relate? So, we'll come back to that one. Somebody also asked, "Why use the speed of light to convert time into space and vice versa, to get them in the same coordinate system?" So, why use c to convert time to space and vice versa? And then, also, there was the question of, you know, "What is the mathematical formulation of general relativity?" So, how to express general relativity in some kind of equation. And we'll get to the key equation, which is something called a metric, for general relativity, and then we're going to stop. Because to go on from there is fairly heavy calculus and we're just not going to do that. But I want to get at least that far.

Okay, so let's go back to special relativity for a minute. So, special relativity. Flat space-time, no gravity. And you'll recall what happens. As you get close to the speed of light, all sorts of things that you thought were kind of constant and properties of objects, like mass and length and duration, and duration of time, and things like that, all start to get weird and change. So, length, time, mass, all these things, vary with the velocity of the person doing the measuring.

And so, you could ask the question, is there anything that doesn't vary? Is there anything that's an invariant? And the answer is, yes. There are some things that don't vary. So, some things are invariant. And Einstein actually said later in his career that it's actually the invariants that are important, not the things that change. And so, he should have called his theory invariant theory instead of relativity theory. Think of what that would have done to pop philosophy. Instead of saying, "everything is relative," all this stuff, you would have had the exact same theory. You would have called it invariance theory. And the pop philosophy interpretation of this would be, "some things never change." And it would have been a whole different concept in three in the morning dorm room conversations.

Okay, so some things are invariant, what things? Now, let me first give you a little bit of a metaphor and then come back to how this really works in space-time. Supposing you're just looking at an xy-coordinate system and you have two points in a two-dimensional space. So, here's a point and here's a point. Now, if you arrange for some kind of coordinate system — here's a coordinate system. This is x, this is y — and you ask how far apart these points are. Well, you can do that — let's see.

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