Frontiers and Controversies in Astrophysics: Lecture 10 Transcript
February 20, 2007
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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 - (
V
2
/
c
2
)]. 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
V
s 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
M
0
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.