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Professor Charles Bailyn: Okay. The subject is special relativity. And right at the end of last class, I had written down this factor, gamma. And gamma is the key thing, which tells you how relativistic you are. Gamma = 1 over the square root of [1 - (V2 / c2)]. And we talked about this factor a little bit. If V over c is equal to zero or approaches zero, then gamma, obviously, is 1. And when gamma is 1, that's the Newtonian case — then, everything is just like Newton's law said.

Okay. On the other hand, as V over c goes to 1 — that is to say, as the velocity approaches the speed of light, this gamma factor goes to infinity, because 1 minus 1 in the denominator — that's zero in the denominator, so the thing has to go to infinity. And then, all these bizarre relativistic effects start taking place. And the one we talked about in particular came about from an example of how this gamma is used — namely, that the relativistic mass is equal to gamma times the rest mass, which is the Newtonian mass. And, obviously, if gamma = 1, then the Newtonian mass is equal to the — then, the mass is equal to the Newtonian mass, and you're in Newton's laws, and everything is fine. When the velocity approaches the speed of light, then this total relativistic mass goes to infinity — the consequence of which is that you can no longer accelerate, regardless of how much — so, no more acceleration — regardless of how much force is applied, because force equals mass times acceleration. And if the mass is infinite, then any amount of force will not give you an acceleration. An acceleration is a change in velocity, and so, the consequence of this is that you can't go faster than the speed of light. It's also another side consequence of this — sorry — there was? Oh, excellent, yes ask it.

Student: [Inaudible.]

Professor Charles Bailyn: V — okay. V is the velocity that something is traveling. There's no escape velocity here, at the moment. There are all kinds of different Vs floating around, so it's important to keep them straight. Yeah, can't — you can't go faster than the speed of light.

A side comment from this is that photons, particles of light, which obviously, by definition, do go at the speed of light, have to have zero rest mass — because otherwise they'd have — they'd end up having infinite mass and infinite energy which isn't — which isn't physical. So, photons which go at the speed of light, for which gamma is therefore infinite, have to have M — this little M0 here, equal to zero. So, you have zero times infinity, and that can equal a finite number, otherwise they'd have infinite energy. Yes?

Student: I was just wondering if we're talking about velocity as a factor or, like, the speed of light?

Professor Charles Bailyn: At the moment what I'm talking about is velocity as a speed. So, I'm talking about the magnitude of the velocity. And you can tell, actually, that that's the case, because it comes in as the velocity squared. So, even if it's a vector, when you square it, that gives you a scalar quantity.

Okay. So, let me go on from here and talk about an intermediate case. We've talked about V equals zero, we've talked about V equals c. Let me talk about an intermediate case. And the particular intermediate case — and this will show up on your problem set – this is the thing we didn't get to on Thursday. This is what's called the post-Newtonian approximation. This is when you're just a tiny little bit relativistic.

Okay. On the other hand, as V over c goes to 1 — that is to say, as the velocity approaches the speed of light, this gamma factor goes to infinity, because 1 minus 1 in the denominator — that's zero in the denominator, so the thing has to go to infinity. And then, all these bizarre relativistic effects start taking place. And the one we talked about in particular came about from an example of how this gamma is used — namely, that the relativistic mass is equal to gamma times the rest mass, which is the Newtonian mass. And, obviously, if gamma = 1, then the Newtonian mass is equal to the — then, the mass is equal to the Newtonian mass, and you're in Newton's laws, and everything is fine. When the velocity approaches the speed of light, then this total relativistic mass goes to infinity — the consequence of which is that you can no longer accelerate, regardless of how much — so, no more acceleration — regardless of how much force is applied, because force equals mass times acceleration. And if the mass is infinite, then any amount of force will not give you an acceleration. An acceleration is a change in velocity, and so, the consequence of this is that you can't go faster than the speed of light. It's also another side consequence of this — sorry — there was? Oh, excellent, yes ask it.

Student: [Inaudible.]

Professor Charles Bailyn: V — okay. V is the velocity that something is traveling. There's no escape velocity here, at the moment. There are all kinds of different Vs floating around, so it's important to keep them straight. Yeah, can't — you can't go faster than the speed of light.

A side comment from this is that photons, particles of light, which obviously, by definition, do go at the speed of light, have to have zero rest mass — because otherwise they'd have — they'd end up having infinite mass and infinite energy which isn't — which isn't physical. So, photons which go at the speed of light, for which gamma is therefore infinite, have to have M — this little M0 here, equal to zero. So, you have zero times infinity, and that can equal a finite number, otherwise they'd have infinite energy. Yes?

Student: I was just wondering if we're talking about velocity as a factor or, like, the speed of light?

Professor Charles Bailyn: At the moment what I'm talking about is velocity as a speed. So, I'm talking about the magnitude of the velocity. And you can tell, actually, that that's the case, because it comes in as the velocity squared. So, even if it's a vector, when you square it, that gives you a scalar quantity.

Okay. So, let me go on from here and talk about an intermediate case. We've talked about V equals zero, we've talked about V equals c. Let me talk about an intermediate case. And the particular intermediate case — and this will show up on your problem set – this is the thing we didn't get to on Thursday. This is what's called the post-Newtonian approximation. This is when you're just a tiny little bit relativistic.

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