WEBVTT

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Prof: Okay.

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The subject is special
relativity.

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And right at the end of last
class, I had written down this

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factor, gamma.

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And gamma is the key thing,
which tells you how

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relativistic you are.

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Gamma = 1 over the square root
of [1 - (V^(2) /

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c^(2))].

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And we talked about this factor
a little bit.

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If V over c is
equal to zero or approaches

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zero, then gamma,
obviously, is 1.

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And when gamma is 1,
that's the Newtonian

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case--then, everything is just
like Newton's law said.

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Okay.

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On the other hand,
as V over c goes

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to 1--that is to say,
as the velocity approaches the

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speed of light,
this gamma factor goes to

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infinity, because 1 minus 1 in
the denominator--that's zero in

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the denominator,
so the thing has to go to

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infinity.

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And then, all these bizarre
relativistic effects start

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taking place.

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And the one we talked about in
particular came about from an

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example of how this gamma is
used--namely,

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that the relativistic mass is
equal to gamma times the rest

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mass, which is the Newtonian
mass.

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And, obviously,
if gamma = 1,

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then the Newtonian mass is
equal to the--then,

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the mass is equal to the
Newtonian mass,

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and you're in Newton's laws,
and everything is fine.

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When the velocity approaches
the speed of light,

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then this total relativistic
mass goes to infinity--the

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consequence of which is that you
can no longer accelerate,

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regardless of how much--so,
no more

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acceleration--regardless of how
much force is applied,

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because force equals mass times
acceleration.

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And if the mass is infinite,
then any amount of force will

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not give you an acceleration.

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An acceleration is a change in
velocity, and so,

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the consequence of this is that
you can't go faster than the

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speed of light.

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It's also another side
consequence of

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this--sorry--there was?

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Oh, excellent, yes ask it.

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Student: [Inaudible.].

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Prof: V--okay.

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V is the velocity that
something is traveling.

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There's no escape velocity
here, at the moment.

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There are all kinds of
different Vs floating

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around, so it's important to
keep them straight.

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Yeah, can't--you can't go
faster than the speed of light.

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A side comment from this is
that photons,

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particles of light,
which obviously,

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by definition,
do go at the speed of light,

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have to have zero rest
mass--because otherwise they'd

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have--they'd end up having
infinite mass and infinite

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energy which isn't--which isn't
physical.

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So, photons which go at the
speed of light,

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for which gamma is therefore
infinite,

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have to have M--this
little M_0

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here, equal to zero.

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So, you have zero times
infinity, and that can equal a

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finite number,
otherwise they'd have infinite

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energy.

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Yes?

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Student: I was just
wondering if we're talking about

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velocity as a factor or,
like, the speed of light?

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Prof: At the moment what
I'm talking about is velocity as

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a speed.

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So, I'm talking about the
magnitude of the velocity.

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And you can tell,
actually, that that's the case,

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because it comes in as the
velocity squared.

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So, even if it's a vector,
when you square it,

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that gives you a scalar
quantity.

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Okay.

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So, let me go on from here and
talk about an intermediate case.

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We've talked about V
equals zero, we've talked about

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V equals c.

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Let me talk about an
intermediate case.

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And the particular intermediate
case--and this will show up on

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your problem set – this is the
thing we didn't get to on

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Thursday.

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This is what's called the
post-Newtonian approximation.

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This is when you're just a tiny
little bit relativistic.

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So, V over c, or
more properly,

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V^(2) / c^(2) is
small, but not zero.

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And here, a little mathematical
concept that may have flashed

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before your eyes when you were
in eleventh grade,

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or something,
comes in, which is the series

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expansion.

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We won't do the mathematics of
this, but here it is.

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If you take 1 plus epsilon
[ε]--and I should say,

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epsilon is a--used by
mathematicians to mean something

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small.

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Famous mathematician named
Erdos used to refer to children

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as "epsilons."

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That seems to be carrying it a
little bit too far.

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But, this is how mathematicians
think.

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So, epsilon,
in this case,

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is something that's much,
much less than 1.

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And if you take 1 plus
epsilon--so that's a number

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that's slightly greater than
1--and you take it to the

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nth power,
you can then expand this as a

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series.

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And the series goes like this:
1 + n ε,

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plus a bunch of other terms,
each of which is multiplied by

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a higher power of epsilon.

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So, there's a term in ε^(2)
and there's a term in ε^(3),

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and so forth.

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And this is an infinite series.

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But the thing is,
if epsilon's small,

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then ε^(2) is even smaller.

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So, if epsilon's already small
and nε is pretty small,

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then ε^(2) is far smaller than
that,

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ε^(3) yet smaller,
and these are negligible.

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And so, the approximation is,
(1 + ε)^(n) is approximately

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equal to 1 + n ε,
if epsilon is much,

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much less than 1.

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So, here's what we're going to
do.

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We're going to use this series
expansion, and we're going to

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generate expressions for some
things--in particular,

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the mass, but some other
things, too--where the 1 is the

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Newtonian term.

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And the thing represented by
this n ε,

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which is much smaller,
is a correction to the

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Newtonian term,
and is the first sign that

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things are becoming
relativistic.

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So this is, in this context,
going to be a Newtonian term,

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and this is the post-Newtonian
approximation.

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Okay.

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Oh, so, let's just do an
example.

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Let's think about the motion of
the Earth--the Earth's orbit

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around the Sun.

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Earth's orbit,
you may recall,

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from the last section of the
class, how fast the Earth moves.

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It's about 3 x 10^(4) meters
per second.

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Thirty kilometers a second.

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The speed of light is 3 x
10^(8) meters per second.

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And so, V^(2) /
c^(2) = [(3 x 10^(4)) /

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(3 x 10^(8))]^(2) = 10^(-8).

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And so, that quantity,
which is going to turn out to

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be the post-Newtonian effect,
is one part in 10^(8) for the

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Earth's orbit.

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And so, pretty small.

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Considering that we think that
9 = 10, something in the eighth

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decimal place isn't going to do
a lot of damage.

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So, it's not a particularly
relativistic situation.

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All right, so,
now I want to go back and I

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want--that's just an aside.

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I want to go back and apply
this approximation to gamma.

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Gamma is equal to 1 / (1 -
V^(2) / c^(2)),

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the whole thing to the ½
power.

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That's (1 - V^(2) /
c^(2))^(-1/2).

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I got to get a better pen,
sorry about that.

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And now, this can be expanded.

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This can be expanded in just
this kind of way.

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The first term is 1.

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Epsilon is - V^(2) /
c^(2).

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n is - ½.

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Minus times a minus is a
positive, so this is plus 1/2

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V^(2) /
c^(2)--plus additional

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terms,
the first of which is

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(V^(2) / c^(2))
^(2),

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which in the case of the Earth
would be 10^(16).

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So, it would be vastly smaller
still, so, we ignore that.

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And so that--and so,
you can substitute this for

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gamma in situations where
V is kind of small.

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For example,
let's go back to this mass

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equation.

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1 + 1/2 V^(2) /
c^(2).

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This is the total mass.

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Now--oh, let's separate these
terms out.

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M_0 --that's
the Newtonian term.

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And now, here's a
post-Newtonian approximation.

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1/2 M_0
V^(2) divided by

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c^(2).

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Okay, high school physics
experts, do you recognize this

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term?

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Kinetic energy.

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Absolutely right.

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This is from high school
physics.

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The kinetic energy--that is to
say the energy in the motion of

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this object.

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So what is this whole term?

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This whole term is the kinetic
energy.

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So, M is equal to the
Newtonian rest mass,

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plus the kinetic energy divided
by c^(2).

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Now, you'll recall that
E / c^(2) =

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M.

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So, this is the mass equivalent
of the kinetic energy.

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So, given this nice
relativistic idea that mass and

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energy are interchangeable,
well, here it is.

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Here is the kinetic energy
expressed in terms of mass.

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And so, this little equation
M = M_0

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γ has a variety of
consequences.

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When V /
c--V^(2) / C^(2),

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really, goes to zero,
then it's just Newton.

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It just says,
mass equals mass,

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because gamma is equal to 1.

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And that expresses an important
Newtonian concept,

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that the mass of something is
an intrinsic property of that

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object,
which doesn't change--which is

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how things are in Newton,
but not how things are

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relativistically.

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In the other extreme,
where V^(2) /

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c^(2) approaches 1,
this equation expresses the

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fact that light--the speed of
light is a speed limit.

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And in the post-Newtonian case
where V^(2) /

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c^(2) is small,
but not zero,

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this same expression expresses
the fact that E =

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mc^(2).

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So, all three of these concepts
come out of the same equation.

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It's this kind of thing that
makes physicists say things

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like, the theory of relativity
is incredibly beautiful.

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It's hard to know--I realize,
it's hard to know what that

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means, you know.

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What makes a piece of
mathematics beautiful?

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It's when you get a situation
like this, where a simple

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mathematical concept sort of
spawns all kinds of new and

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unexpected ideas,
depending on how you look at

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it--and that they all kind of
come together as one.

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I have this vivid memory in
graduate school,

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I was sitting in a class,
you know, and the professor was

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doing the thing that professors
do.

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He was writing down all kinds
of miscellaneous information

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really, really fast,
from relativity theory and

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nuclear physics.

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All sorts of stuff was going up
on the board.

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I was doing the thing that
students do, where you

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desperately try and write it all
down,

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so that you can then go back
and figure out what the hell he

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was talking about later.

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And suddenly,
in the midst of this rather

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typical class,
I realized what was happening:

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that in about twenty minutes,
he was going to put all this

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stuff together and prove
Chandrasekhar's limit.

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We talked about Chandrasekhar's
limit a week or so ago.

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That's the limit whereby a
white dwarf can't be bigger then

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1.4 times the mass of the Sun or
it continues to collapse.

13:44.419 --> 13:47.879
And, I suddenly saw where all
this was going and I sort of

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wrote down in my notebook,
"Chandra," and underlined it,

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put my pencil down.

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And then, I just watched for
twenty minutes.

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It was great.

13:55.929 --> 13:59.459
It was--the only thing I can
compare it to is listening to a

13:59.455 --> 14:02.495
great piece of music,
because it kind of unfolds in

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time, and you see where it's
going, and it's just a great

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feeling.

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So, don't be condescending to
the physics majors when they're

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working like hell on those
problem sets late on a Thursday

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night,
because they have access to

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realms of aesthetic experience
that you can only imagine.

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But perhaps--it's true,
it's true, I promise.

14:24.509 --> 14:28.079
But you have to work hard to
get there and ask questions.

14:28.080 --> 14:31.620
So, at this point,
let's do the question thing.

14:31.620 --> 14:37.660
Here's what I'd like to do:
talk to the people around you.

14:37.659 --> 14:40.439
Introduce yourself to the
people around you--groups of

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two, three or four will do fine.

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And come up,
in consultation with your

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neighbors, with a question to
ask.

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Now, this could either be a
question of something that you

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guys don't understand about what
I've said,

14:52.529 --> 14:55.919
or something where the
question, if answered,

14:55.918 --> 15:00.228
would deepen your understanding
beyond what I've said.

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So, take a couple minutes.

15:01.920 --> 15:03.130
Talk to your neighbors.

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Make a friend.

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Think of a question,
and in a couple minutes' time,

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we'll regroup and I'll answer
some of the questions.

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If your particular question
doesn't end up being answered,

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you know, hand it in at the
end,

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so that I know what you were
thinking of--and so,

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definitely write them down.

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And we'll try this out.

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Let's see how this goes.

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So, introduce yourself to your
neighbors and come up with a

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question.

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And when you have it,
put your hand up and ask it.

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We'll come around and ask.

15:33.910 --> 15:34.680
All right.

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I've now heard the same
question twice,

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so let me answer it,
and then we'll ask for other

15:40.965 --> 15:41.765
things.

15:41.769 --> 15:45.169
The same question,
which has been asked in a

15:45.170 --> 15:48.730
couple of different ways,
is, what is mass?

15:48.730 --> 15:52.220
That's an excellent question,
because it's awfully hard to

15:52.217 --> 15:55.887
parse what's going on with this
M_0 and this

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other kind of M,
and stuff, if you don't know

15:59.187 --> 16:00.597
what the concept means.

16:00.600 --> 16:02.120
So, that's a really good
question.

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What's mass?

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Mass is defined--you can think
of it this way.

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What is mass?

16:09.930 --> 16:14.150


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Mass is defined as from this
equation, in a certain sense.

16:21.510 --> 16:23.250
F = ma.

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It tells you,
for a given object,

16:25.744 --> 16:29.024
how hard it is to move--or,
more precisely,

16:29.017 --> 16:31.587
how hard it is to accelerate.

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So, higher mass requires
greater force to accelerate by a

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certain amount.

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It's a property of an object,
and it tells you how much it

16:52.037 --> 16:54.397
resists being pushed around.

16:54.399 --> 16:59.279
It's sometimes referred to as
"inertial mass" for this reason.

16:59.280 --> 17:02.520


17:02.519 --> 17:06.429
It's an expression of the
object's inertia--how much it

17:06.425 --> 17:08.445
resists being accelerated.

17:08.450 --> 17:12.570
And so, in Newtonian physics,
this is purely a property of

17:12.565 --> 17:14.005
the object itself.

17:14.009 --> 17:16.359
You can say,
if you've got a basketball,

17:16.359 --> 17:18.529
or if you've got a car,
or something,

17:18.527 --> 17:21.597
that the car has more mass than
the basketball.

17:21.600 --> 17:22.750
Why?

17:22.750 --> 17:25.200
Because if I go--do I have a
basketball?

17:25.200 --> 17:25.950
I don't.

17:25.950 --> 17:29.360
All right, car,
a car has--the lectern has more

17:29.363 --> 17:32.633
mass than the little piece of
cardboard here,

17:32.627 --> 17:37.077
because if I apply force to the
cardboard I can move it.

17:37.079 --> 17:39.519
And if I apply the same amount
of force to the lectern,

17:39.519 --> 17:40.829
I move it a whole lot less.

17:40.829 --> 17:44.089
And so, what is the difference
between those two things since

17:44.090 --> 17:45.830
the force applied is the same?

17:45.829 --> 17:48.519
The difference is that this one
has a whole lot more mass.

17:48.519 --> 17:52.219
And in Newtonian physics,
the mass is a property of the

17:52.222 --> 17:52.842
object.

17:52.839 --> 17:56.679
In Einsteinian physics,
there is something that's the

17:56.679 --> 18:00.149
property of the object--that's
its rest mass.

18:00.150 --> 18:03.440
But the inertial mass,
the amount by which it resists

18:03.439 --> 18:06.219
being pushed,
is also a property--not just of

18:06.223 --> 18:08.883
the rest mass,
but also of its motion.

18:08.880 --> 18:12.280
So, if this were moving at a
large fraction of the speed of

18:12.282 --> 18:15.042
light and I applied some kind of
force to it,

18:15.039 --> 18:17.849
it wouldn't move nearly as much
as it does--or,

18:17.846 --> 18:21.446
it wouldn't change its motion
nearly as much as it does when

18:21.445 --> 18:22.905
it's standing still.

18:22.910 --> 18:29.660
And so mass--the inertial mass
of the thing varies depending on

18:29.657 --> 18:32.157
how fast it's moving.

18:32.160 --> 18:37.700
So, that's an important
question for understanding this.

18:37.700 --> 18:40.550
Let's see--yes?

18:40.549 --> 18:42.419
Student: Let's say that
you're on Earth.

18:42.420 --> 18:44.440
If you were to push an object
that's standing still versus an

18:44.439 --> 18:45.989
object in motion,
you're going to have the same

18:45.987 --> 18:46.557
affect on it?

18:46.559 --> 18:47.929
Prof: No,
you're not going to have the

18:47.934 --> 18:50.634
same effect on it,
because if the object in motion

18:50.627 --> 18:54.027
is going to have a little extra
piece to its mass.

18:54.029 --> 18:55.299
Student:
Okay, so it has a very small

18:55.299 --> 18:55.859
effect [inaudible].

18:55.860 --> 18:57.130
Prof: A very,
very small effect.

18:57.130 --> 18:59.660
In fact, if you try this on the
Earth itself,

18:59.660 --> 19:02.020
which is moving at 30
kilometers a second,

19:02.019 --> 19:04.779
that effect is going to be 1
part in 10^(8).

19:04.779 --> 19:10.069
So, that's what we calculated
before, where we--where was it?

19:10.070 --> 19:14.600


19:14.600 --> 19:16.500
Yeah, here it is.

19:16.500 --> 19:18.370
That's this purple part here.

19:18.369 --> 19:20.829
You calculate V^(2)
/ c^(2),

19:20.829 --> 19:21.699
that's 10^(-8).

19:21.700 --> 19:25.220
So, it's a difference if
something moving even as fast as

19:25.219 --> 19:28.429
30 kilometers a second,
which is pretty fast--this

19:28.425 --> 19:30.935
difference is only going to
be--actually,

19:30.941 --> 19:32.641
1/2 of 1 part in 10^(8).

19:32.640 --> 19:35.450
So, it's a hard thing to see.

19:35.450 --> 19:39.070
Now, the place you can see this
stuff is in particle

19:39.070 --> 19:41.200
accelerators,
because there,

19:41.200 --> 19:44.010
you can take sub-atomic
particles and accelerate them to

19:44.005 --> 19:45.785
very close to the speed of
light.

19:45.789 --> 19:50.389
And then you see these very
dramatic effects.

19:50.390 --> 19:52.570
Yes?

19:52.569 --> 19:54.709
Student: So,
if something's going at the

19:54.706 --> 19:56.656
speed of light,
so that the force you would

19:56.656 --> 19:58.696
need to stop it is infinite
[inaudible]?

19:58.700 --> 20:01.260
Prof: Yes,
yes, that's correct.

20:01.259 --> 20:03.639
You cannot stop something
that's going at the speed of

20:03.644 --> 20:06.124
light, because the force you'd
need to accelerate it,

20:06.119 --> 20:09.209
which could either be a--making
it faster or slower,

20:09.208 --> 20:10.418
would be infinite.

20:10.420 --> 20:16.060
But, that's why photons don't
have any rest mass at all.

20:16.059 --> 20:19.809
That statement that you made
only applies to something which

20:19.806 --> 20:22.596
has non-zero--greater than zero
rest mass.

20:22.599 --> 20:24.009
Student: And they never
go?

20:24.009 --> 20:25.869
Prof: And they never go
to the speed of light,

20:25.867 --> 20:27.257
because you can't get them that
fast.

20:27.259 --> 20:30.039
You'd need an infinite amount
of force to get it to go that

20:30.035 --> 20:30.365
fast.

20:30.370 --> 20:32.280
Photons are a different story.

20:32.279 --> 20:34.219
And, of course,
if you stop a photon,

20:34.218 --> 20:35.078
it disappears.

20:35.079 --> 20:38.269
Because if it's going less than
the speed of light,

20:38.274 --> 20:41.534
then it's got no energy,
because it's a finite gamma

20:41.532 --> 20:43.132
times zero rest mass.

20:43.130 --> 20:45.580
Yes?

20:45.579 --> 20:46.589
Student: What about all
the weird, like,

20:46.590 --> 20:47.360
time stuff you always hear
about?

20:47.359 --> 20:49.229
Prof: The weird time
stuff you always hear about.

20:49.230 --> 20:51.310
Excellent.

20:51.310 --> 20:52.650
Yeah, okay.

20:52.650 --> 20:53.920
A couple of other Lorentz
transformations.

20:53.920 --> 20:57.650
Time is equal to gamma times
time zero.

20:57.650 --> 20:59.990
This is time dilation.

20:59.990 --> 21:05.100
The faster you go,
the slower your clock works.

21:05.099 --> 21:07.329
This is the origin of all that
nice science fiction,

21:07.328 --> 21:08.768
where you get on a rocket ship.

21:08.769 --> 21:10.339
You go close to the speed of
light.

21:10.339 --> 21:11.069
You go somewhere.

21:11.069 --> 21:11.799
You turn around.

21:11.799 --> 21:12.719
You come back.

21:12.715 --> 21:16.375
Your clocks are running slow,
so, it only takes a year in

21:16.379 --> 21:17.229
your time.

21:17.230 --> 21:19.260
And you come back and
everybody--you know,

21:19.262 --> 21:20.752
a hundred years have passed.

21:20.750 --> 21:22.090
All your friends are dead,
and so forth.

21:22.089 --> 21:23.729
So, that's a whole science
fiction thing.

21:23.730 --> 21:26.100
There's also length contraction.

21:26.100 --> 21:27.550
That looks like this.

21:27.550 --> 21:31.660


21:31.660 --> 21:36.420
Take the pen. Put it in motion.

21:36.420 --> 21:39.130
Make that motion at some
substantial fraction,

21:39.134 --> 21:41.854
at the speed of light,
and it gets shorter.

21:41.850 --> 21:45.480
Amazing.

21:45.480 --> 21:50.460
We'll talk more about this in a
little while.

21:50.460 --> 21:53.750
Yeah, these are--these things
with the gammas in them,

21:53.752 --> 21:57.482
they're collectively known as
the Lorentz transformations.

21:57.480 --> 22:00.960
That's just a name.

22:00.960 --> 22:11.080
And these are the ways in which
basic properties--space,

22:11.083 --> 22:16.973
time, mass--change with
velocity.

22:16.973 --> 22:18.633
Yeah?

22:18.630 --> 22:21.090
Student: So with the
equation, M =

22:21.094 --> 22:22.934
M_0 times
gamma.

22:22.930 --> 22:26.040
M_0 is the
intrinsic mass?

22:26.040 --> 22:26.790
Prof: Yes.

22:26.789 --> 22:28.939
No, no, no,
M_0--yes.

22:28.940 --> 22:30.870
Let me think.

22:30.869 --> 22:32.419
M_0 is the
intrinsic mass.

22:32.420 --> 22:34.230
M is the inertial mass.

22:34.230 --> 22:35.320
Student: Okay.

22:35.319 --> 22:36.869
Prof: That's a way of
thinking about it.

22:36.869 --> 22:39.879
Student: So,
if something was traveling at a

22:39.880 --> 22:42.170
very small speed,
wouldn't it mean that

22:42.169 --> 22:45.179
V^(2) / c^(2) was
close to zero?

22:45.180 --> 22:46.150
Prof: Yes.

22:46.150 --> 22:48.120
Student: Which means the
inertial mass would be close to

22:48.117 --> 22:48.337
zero?

22:48.340 --> 22:50.100
Prof: No.

22:50.099 --> 22:53.769
If it goes--so,
you were right up to the very

22:53.769 --> 22:54.269
end.

22:54.270 --> 22:55.810
Remember how this works.

22:55.809 --> 23:00.119
γ = 1 / [(1 - V^(2) /
c^(2))^(1/2)]

23:00.116 --> 23:04.946
If V / c goes to
zero, gamma goes to 1.

23:04.950 --> 23:10.020
And in that case,
the inertial mass is equal to

23:10.016 --> 23:12.326
the intrinsic mass.

23:12.329 --> 23:15.859
And, if it's going faster,
then the inertial mass is

23:15.863 --> 23:18.153
bigger than the intrinsic mass.

23:18.150 --> 23:21.120
The intrinsic mass is usually
referred to as the rest mass.

23:21.119 --> 23:22.599
Student: Well,
then how did you get

23:22.598 --> 23:23.678
V^(2) / c^(2)?

23:23.680 --> 23:24.770
Prof: Ah.

23:24.769 --> 23:28.669
That's the second term of the
series expansion.

23:28.670 --> 23:33.800
1 / [(1 - V^(2) /
c^(2))^(1/2)]

23:33.798 --> 23:39.978
expands to be 1 + 1/2
V^(2) / c^(2) plus

23:39.975 --> 23:41.835
other terms.

23:41.840 --> 23:43.550
Here's the first term.

23:43.549 --> 23:45.689
That's the rest mass,
because we're going to multiply

23:45.688 --> 23:46.468
this by M.

23:46.470 --> 23:53.950
And this is then the kinetic
energy divided by c^(2).

23:53.950 --> 23:54.950
Student: So the
[inaudible].

23:54.951 --> 23:55.951
Prof: It's the second
term..

23:55.953 --> 23:57.523
Student: [inaudible]
inertial mass can never be

23:57.515 --> 23:57.655
zero.

23:57.662 --> 23:58.932
It'll just go closer to the
front?

23:58.930 --> 24:00.820
Prof: Yeah,
the rest mass doesn't change.

24:00.819 --> 24:02.789
If V^(2) / c^(2)
goes to zero,

24:02.793 --> 24:06.053
what happens is,
the kinetic energy goes to

24:06.049 --> 24:10.259
zero, which is exactly what you
want it to do.

24:10.255 --> 24:10.905
Yes?

24:10.910 --> 24:14.440
Student: How does--how
is momentum conserved when

24:14.444 --> 24:16.184
something's relativistic?

24:16.180 --> 24:18.120
Prof: How is momentum
conserved when something's

24:18.120 --> 24:18.660
relativistic?

24:18.660 --> 24:20.090
Excellent.

24:20.089 --> 24:22.809
There is an equation of
relativistic momentum.

24:22.809 --> 24:28.039
And it's that thing that's
conserved, which is different

24:28.041 --> 24:32.011
from M times V,
because V behaves

24:32.008 --> 24:33.958
differently and M behaves
differently.

24:33.960 --> 24:36.840
And you can work out exactly
where all the gammas go in that,

24:36.838 --> 24:39.188
and I don't remember it off the
top of my head.

24:39.190 --> 24:42.680
But there is a quantity that is
conserved, but it doesn't look

24:42.679 --> 24:45.139
quite the same as the Newtonian
quantity.

24:45.140 --> 24:49.780
In the limit where the velocity
is small, it reduces down.

24:49.779 --> 24:52.939
The first term of that series
is M times V.

24:52.940 --> 24:56.500
Yes sir?

24:56.500 --> 24:59.550
Student: When you define
mass by the equation force

24:59.553 --> 25:01.383
equals mass times the
acceleration,

25:01.375 --> 25:03.245
how, then, do you define force?

25:03.250 --> 25:04.190
Prof: Yeah, okay.

25:04.190 --> 25:08.510
So, this gets a little bit
circular, right?

25:08.509 --> 25:09.229
Okay.

25:09.230 --> 25:11.960
Force--you can define it either
way.

25:11.960 --> 25:15.420
Force is the thing that--force
is the ability to do work.

25:15.420 --> 25:16.900
That's the technical definition.

25:16.900 --> 25:19.950


25:19.950 --> 25:26.110
Force: ability to do work.

25:26.109 --> 25:29.039
And then, work also has a
technical definition.

25:29.039 --> 25:32.269
The problem with these things
is you kind of do--if you just

25:32.269 --> 25:34.239
define them all around the
circle,

25:34.240 --> 25:37.000
they do tend to come back and
bite themselves in the tail.

25:37.000 --> 25:38.340
I agree with that.

25:38.339 --> 25:43.989
But force is also related to
energy.

25:43.990 --> 25:47.210
How much energy is required to
exert a certain force?

25:47.210 --> 25:49.550
So, all these things,
interrelate--but it is

25:49.548 --> 25:52.868
definitely true that there's a
circularity to the definitions,

25:52.865 --> 25:55.145
if you follow them all the way
around.

25:55.150 --> 25:58.250
So, if you keep asking that
kind of question,

25:58.246 --> 26:02.676
I'm going to go around until I
get back where I started from.

26:02.680 --> 26:05.270
Yes?

26:05.269 --> 26:07.849
Student: Why is the
speed of light constant in all

26:07.854 --> 26:08.274
frames?

26:08.269 --> 26:10.799
Prof: Why is the speed
of light constant in all frames

26:10.798 --> 26:11.428
of reference?

26:11.430 --> 26:15.700
That's problem two of your
problem set, and I'll come--I'll

26:15.704 --> 26:19.394
say a few things--that's an
insufficient answer,

26:19.390 --> 26:22.630
but I'll come back and say a
little bit more about that in a

26:22.634 --> 26:23.134
minute.

26:23.130 --> 26:25.030
Yes?

26:25.029 --> 26:28.429
Student: We were just a
little bit confused with the

26:28.426 --> 26:30.706
original Lorentz
transformation--how,

26:30.710 --> 26:33.470
in most cases,
that transforms to E =

26:33.472 --> 26:34.502
mc^(2).

26:34.500 --> 26:36.780
Could you show that to us?

26:36.780 --> 26:37.790
Prof: Yeah.

26:37.789 --> 26:40.729
So, let's go back and take
another look at that.

26:40.730 --> 26:45.890


26:45.890 --> 26:47.860
All right, so,
here's what I'm going to do.

26:47.859 --> 26:50.779
First of all,
I'm going to take gamma--this

26:50.779 --> 26:53.489
quantity--and I'm going to
expand it.

26:53.490 --> 26:56.190
This is equal to gamma,
and now I'm going to expand it.

26:56.190 --> 26:59.950
That gives me 1 + 1/2
V^(2) / c^(2).

26:59.950 --> 27:03.620


27:03.619 --> 27:06.689
Let's see what have I done with
the--my--the pieces of paper are

27:06.694 --> 27:08.114
getting out of order,
here.

27:08.110 --> 27:11.380
Here it is.

27:11.380 --> 27:14.230
So, inertial mass is equal to
the rest mass,

27:14.231 --> 27:15.161
times gamma.

27:15.160 --> 27:17.710
So, now, I'm going to
substitute in this approximation

27:17.712 --> 27:18.292
for gamma.

27:18.289 --> 27:21.809
And then, I'm going to multiply
M_0 through

27:21.805 --> 27:22.575
both terms.

27:22.579 --> 27:25.369
So this is--so,
the inertial mass is equal to

27:25.371 --> 27:28.291
the Newtonian mass,
plus a term that looks like

27:28.289 --> 27:28.859
this.

27:28.859 --> 27:36.019
One half M V
squared is, in Newtonian terms,

27:36.021 --> 27:38.661
the kinetic energy.

27:38.660 --> 27:42.410
And it appears here,
divided by c^(2).

27:42.410 --> 27:46.850
And so, a way you can write
this is that the inertial mass

27:46.851 --> 27:50.751
is equal to the rest mass,
plus the kinetic energy,

27:50.748 --> 27:52.928
divided by c^(2).

27:52.930 --> 27:57.750
And that is an example of the
fact that energy divided by

27:57.751 --> 28:01.971
c^(2) is another way of
expressing mass.

28:01.970 --> 28:04.760
Student: Then,
how is it not saying that

28:04.755 --> 28:08.025
energy over c^(2)
equals--why is it not from the

28:08.025 --> 28:10.285
energy of c^(2),
literally, algebraically,

28:10.289 --> 28:11.499
whether it be M minus
M_0

28:11.498 --> 28:11.798
[inaudible].

28:11.801 --> 28:14.371
Prof: Oh,
which M are you talking

28:14.365 --> 28:14.835
about?

28:14.840 --> 28:15.480
Student: Yes.

28:15.480 --> 28:19.200
Prof: Yeah,
its M_0 here,

28:19.203 --> 28:21.533
and it's M over here.

28:21.529 --> 28:23.749
And so, somebody
asked--actually you're now the

28:23.754 --> 28:26.704
second person who's asked this
question, so I better answer it

28:26.704 --> 28:27.434
explicitly.

28:27.430 --> 28:30.510
I answered the question,
"What is mass?"

28:30.510 --> 28:32.570
What's energy?

28:32.569 --> 28:37.089
E = mc^(2)
squared.

28:37.090 --> 28:37.920
So, what do I mean by this?

28:37.920 --> 28:42.280
M--this is the inertial
mass.

28:42.280 --> 28:45.350


28:45.349 --> 28:48.629
So, that's M,
which is equal to,

28:48.625 --> 28:51.895
in this case,
M_0 plus

28:51.900 --> 28:55.090
kinetic energy over
c^(2).

28:55.090 --> 28:56.780
And notice the difference here.

28:56.780 --> 28:58.450
This is one kind of energy.

28:58.450 --> 29:01.430
This is kinetic energy.

29:01.430 --> 29:06.290
That's the energy in--that's
the energy contained in the

29:06.285 --> 29:07.075
motion.

29:07.079 --> 29:14.939
Another way you could write
this is the rest energy over

29:14.944 --> 29:17.094
c^(2).

29:17.089 --> 29:20.069
So E,
here, is the total energy.

29:20.069 --> 29:23.629
And there are two kinds of
energies being expressed here.

29:23.630 --> 29:26.200
There's a rest energy,
divided by c^(2),

29:26.196 --> 29:29.316
and there's a kinetic energy,
divided by c^(2).

29:29.319 --> 29:31.429
But in the Newtonian case,
you think of them as two

29:31.426 --> 29:33.066
completely different
things--whereas,

29:33.069 --> 29:37.919
in the relativistic case,
they're two manifestations of

29:37.916 --> 29:41.416
the same thing,
and they add to make the

29:41.415 --> 29:42.665
inertial mass.

29:42.672 --> 29:43.482
Yes?

29:43.480 --> 29:45.300
Student: At what
velocity are a fractions of

29:45.297 --> 29:47.547
c-- would something need
to be traveling on Earth for us

29:47.551 --> 29:49.661
to--for there to be a
perceptible relativistic effect?

29:49.660 --> 29:50.650
Prof: Okay.

29:50.650 --> 29:53.950
So, how fast do you have to go
to have a perceptible

29:53.947 --> 29:55.367
relativistic effect?

29:55.369 --> 29:57.719
That depends how good your
instruments are.

29:57.720 --> 30:01.930
If you've got something that
can perceive--that can measure a

30:01.925 --> 30:05.985
velocity or an energy or a mass
to eight decimal places,

30:05.990 --> 30:08.090
then it doesn't have to go so
fast.

30:08.089 --> 30:10.939
If you have something that's
fairly crude,

30:10.937 --> 30:13.157
then it has to go much faster.

30:13.160 --> 30:16.140
But I should say--special
relativity, these Lorentz

30:16.141 --> 30:18.881
transformations,
have been measured upside down

30:18.883 --> 30:21.033
and backwards in the laboratory.

30:21.029 --> 30:23.429
This work to many,
many decimal places.

30:23.430 --> 30:25.410
Here's an example.

30:25.410 --> 30:28.330
If you take sub-atomic
particles that are unstable,

30:28.333 --> 30:31.553
that decay--they have a half
life, so, half of them will

30:31.549 --> 30:33.829
decay in ten seconds,
or something.

30:33.829 --> 30:36.429
And you take those particles,
and you accelerate them to a

30:36.430 --> 30:38.210
large fraction of the speed of
light.

30:38.210 --> 30:42.560
It takes much longer for them
to decay, because they're going

30:42.562 --> 30:45.612
so fast and their time clocks
slow down.

30:45.609 --> 30:49.319
One manifestation of this is,
there's a kind of unstable

30:49.319 --> 30:53.029
particle that's created at the
top of the atmosphere.

30:53.030 --> 30:54.410
These things are called "muons."

30:54.410 --> 30:56.530
What happens is,
cosmic rays come in.

30:56.529 --> 30:57.879
They blast the top of the
atmosphere.

30:57.880 --> 30:59.280
They make these muon things.

30:59.279 --> 31:01.999
The muons propagate close to
the speed of light,

31:01.998 --> 31:03.558
and they arrive at Earth.

31:03.559 --> 31:05.679
And you can measure,
if you have a little muon

31:05.676 --> 31:08.546
detector--never mind what the
heck a muon is--but imagine that

31:08.546 --> 31:11.036
you have something that could
detect it on Earth,

31:11.039 --> 31:13.569
and you see a whole bunch of
these things caused by cosmic

31:13.569 --> 31:13.879
rays.

31:13.880 --> 31:16.600
But, if you ask,
how long did it take them to

31:16.595 --> 31:19.795
get from the top of the
atmosphere down to where your

31:19.804 --> 31:22.864
muon detector is,
it's much, much longer than

31:22.857 --> 31:24.027
their decay time.

31:24.029 --> 31:26.029
So, you really shouldn't see
any of them at all.

31:26.030 --> 31:27.450
But you do see them.

31:27.450 --> 31:29.770
And the reason is because
they're going at close to the

31:29.769 --> 31:30.499
speed of light.

31:30.500 --> 31:35.960
And so this is--these effects
are observed and measured to

31:35.956 --> 31:42.076
great accuracy in the laboratory
and in astrophysical situations.

31:42.084 --> 31:42.854
Yes?

31:42.849 --> 31:45.359
Student: I know it's
practically impossible,

31:45.364 --> 31:48.234
but what would happen here if
we--if you were to go faster

31:48.231 --> 31:49.641
than the speed of light?

31:49.640 --> 31:50.700
Prof: Faster than the
speed of light?

31:50.700 --> 31:53.570
Well, you'd have super infinite
mass.

31:53.570 --> 31:56.300
This would not be good.

31:56.300 --> 31:59.630
Well, okay.

31:59.630 --> 32:04.260
So, let us imagine what the
equations would do.

32:04.259 --> 32:06.799
What would happen is,
as you go faster than the speed

32:06.798 --> 32:09.728
of light, if you try and slow
down toward the speed of light,

32:09.727 --> 32:11.287
you'd have the same problem.

32:11.289 --> 32:13.679
That is to say,
it would take you more and more

32:13.678 --> 32:16.428
effort to slow down to close to
the speed of light.

32:16.430 --> 32:18.890
And the consequence of this is
that if you're faster than the

32:18.885 --> 32:20.555
speed of light,
you can't slow down to the

32:20.563 --> 32:22.203
speed of light,
or slower than that.

32:22.200 --> 32:24.670
There are hypothetical
particles--and this doesn't

32:24.673 --> 32:27.253
exist in the real world--but,
there are hypothetical

32:27.247 --> 32:28.607
particles that do this.

32:28.610 --> 32:29.970
These are called tachyons.

32:29.970 --> 32:34.630
They have the odd effect that
they go backwards in

32:34.634 --> 32:39.304
time--because again,
if you believe the equations,

32:39.298 --> 32:40.248
right?

32:40.250 --> 32:41.910
Time dilation.

32:41.910 --> 32:47.790
T = γ
T_0.

32:47.790 --> 32:49.090
Now, look at gamma.

32:49.089 --> 32:52.059
Square root of 1 - V^(2)
/ c^(2).

32:52.060 --> 32:53.940
So, this is awkward.

32:53.940 --> 32:57.210
If V is bigger than
c, this is a square root

32:57.210 --> 32:58.530
of a negative number.

32:58.530 --> 33:02.840
That becomes imaginary. Right?

33:02.839 --> 33:05.309
Square root of a negative
number--hard to do.

33:05.310 --> 33:06.550
So, this becomes imaginary.

33:06.549 --> 33:09.059
So, your time axis becomes
imaginary, and all sorts of very

33:09.055 --> 33:10.605
bad things start to happen to
you.

33:10.610 --> 33:13.550
Yeah, yes?

33:13.549 --> 33:15.989
Student: So,
as your mass increases with

33:15.992 --> 33:19.182
velocity, do you just--does the
particle or the entity have a

33:19.177 --> 33:20.927
greater gravitational effect?

33:20.930 --> 33:23.320
Prof: Does it have a
greater gravitational effect?

33:23.320 --> 33:25.050
Excellent question.

33:25.049 --> 33:28.439
That's the next lecture,
because that's general

33:28.441 --> 33:29.401
relativity.

33:29.400 --> 33:33.750
And what you're doing is you're
assuming that Newtonian gravity

33:33.748 --> 33:34.658
is correct.

33:34.660 --> 33:38.910
Newtonian gravity--gravity is a
force proportional to the mass.

33:38.910 --> 33:42.320
Turns out--and this is one of
the things that led Einstein to

33:42.322 --> 33:45.002
his theory of general
relativity--turns out that

33:44.995 --> 33:48.175
relativistically,
it's--that gravity should not

33:48.178 --> 33:50.508
be thought of as a force,
at all.

33:50.509 --> 33:54.399
And there's a whole different
way of thinking about it,

33:54.400 --> 33:58.080
and your question has to be,
in a kind of Zen sense,

33:58.075 --> 33:58.935
unasked.

33:58.940 --> 34:00.760
We'll get to that.

34:00.763 --> 34:02.893
We will get to that.

34:02.890 --> 34:05.550
But the point is that,
in relativistic theory,

34:05.550 --> 34:07.560
gravity is not a force,
anymore.

34:07.559 --> 34:11.539
And so, the question doesn't
arise in quite the same way.

34:11.543 --> 34:11.973
Yes?

34:11.969 --> 34:15.169
Student: [Inaudible.].

34:15.165 --> 34:17.475
Prof: Sorry?

34:17.480 --> 34:21.070
Student: [Inaudible].

34:21.068 --> 34:26.838
Prof: E =
M_0 /

34:26.836 --> 34:30.166
c^(2) + K.E.

34:30.168 --> 34:31.448
Yeah.

34:31.449 --> 34:34.029
You have to put--you have to
get the c squareds in the

34:34.028 --> 34:36.348
right place, and that's
basically a unit conversion.

34:36.349 --> 34:38.219
Student: What's
K.E. / c^(2)?

34:38.219 --> 34:39.889
Prof: No,
E -.

34:39.894 --> 34:41.884
Student: Oh,
E.

34:41.880 --> 34:45.970
Prof: - is equal to
c^(2)

34:45.972 --> 34:49.852
M_0 +
K.E.

34:49.849 --> 34:52.379
Basically, what I've done is
I've multiplied this equation by

34:52.384 --> 34:53.614
c^(2) on both sides.

34:53.610 --> 35:02.030
Yeah. Yes?

35:02.030 --> 35:06.490
Student: I don't know if
this is really a worthwhile

35:06.488 --> 35:09.178
question, but,
what is--what are the

35:09.178 --> 35:12.328
implications for,
like, what time is?

35:12.329 --> 35:13.689
Prof: Well,
this is the--.

35:13.694 --> 35:16.214
Student: If you have
one meaning of what time is?

35:16.210 --> 35:16.800
Prof: Right.

35:16.799 --> 35:18.289
So, what are the implications
for what time is?

35:18.289 --> 35:21.999
This is the big stumbling
block, from a philosophical

35:21.999 --> 35:24.709
point of view,
to all of this stuff.

35:24.710 --> 35:27.060
Turns out, time isn't absolute.

35:27.059 --> 35:29.759
That is to say,
one has the feeling,

35:29.761 --> 35:33.081
as we go through our everyday
life, that,

35:33.079 --> 35:36.069
you know, my watch and your
watch are kind of measuring more

35:36.065 --> 35:38.835
or less the same thing,
to the extent that they're

35:38.843 --> 35:39.953
accurate to do so.

35:39.949 --> 35:42.959
So, if we have identical,
perfectly accurate watches,

35:42.959 --> 35:45.679
and we go about our daily life
and come back,

35:45.679 --> 35:48.189
they're going to read the same
time at the end if they read the

35:48.193 --> 35:50.023
same time at the
beginning--because time moves

35:50.017 --> 35:51.757
the same way for me as it does
for you.

35:51.760 --> 35:53.330
This turns out to be false.

35:53.329 --> 35:58.779
And so, time is not an absolute
quantity.

35:58.780 --> 36:00.410
Neither is space,
for that matter,

36:00.409 --> 36:01.889
because length changes also.

36:01.890 --> 36:05.100
This is quite disturbing.

36:05.099 --> 36:07.889
And the reason it's so
disturbing to us is our brains

36:07.894 --> 36:10.694
have evolved in a situation
where we're always moving

36:10.688 --> 36:12.568
really,
really slowly compared to the

36:12.568 --> 36:13.278
speed of light.

36:13.280 --> 36:15.600
And therefore,
you don't have to worry about

36:15.600 --> 36:18.300
this stuff in everyday life,
down to some factor of

36:18.298 --> 36:18.998
10^(-12).

36:19.000 --> 36:22.420
But, nevertheless,
it turns out to be true that

36:22.424 --> 36:25.704
time does not tick off in an
absolute way.

36:25.699 --> 36:28.129
And this is why people get so
freaked out by this

36:28.132 --> 36:31.022
stuff--because it seems counter
to everyday experience.

36:31.020 --> 36:33.410
But, it can be measured.

36:33.410 --> 36:35.850
And it turns out to be true.

36:35.849 --> 36:39.599
So, the Newtonian case,
what we're used to and the way

36:39.601 --> 36:42.361
our brains work,
turns out to be only an

36:42.362 --> 36:45.692
approximation of the way things
really are.

36:45.690 --> 36:49.230
There are other things that are
conserved, no matter how fast

36:49.229 --> 36:49.759
you go.

36:49.760 --> 36:57.050
And we can get to that at some
point--but time isn't one of

36:57.051 --> 36:57.681
them.

36:57.679 --> 36:58.559
Yes?

36:58.559 --> 37:01.519
Student: Why is the
speed of light this magic

37:01.518 --> 37:02.038
number?

37:02.039 --> 37:06.639
Prof: Why is the speed
of light this magic number?

37:06.640 --> 37:07.190
Okay.

37:07.190 --> 37:15.050
So, the fact that there is a
magic number comes out of this

37:15.046 --> 37:17.616
kind of equation.

37:17.619 --> 37:22.019
Because, when this quantity is
equal to 1, bad things start to

37:22.022 --> 37:24.912
happen, and all the equations
blow up.

37:24.909 --> 37:29.609
The fact that it is the speed
of light that happens to be

37:29.612 --> 37:33.892
that, in that equation,
is because light consists of

37:33.894 --> 37:36.754
particles with zero rest mass.

37:36.750 --> 37:40.750
And a particle at zero rest
mass has to have this thing go

37:40.745 --> 37:43.615
to infinity or it doesn't exist
at all.

37:43.619 --> 37:46.509
Student: [Inaudible].

37:46.513 --> 37:50.753
Prof: Why is the world
this way?

37:50.750 --> 37:52.020
Student: Well
[inaudible].

37:52.024 --> 37:54.014
Prof: That I--no,
seriously, that's what you're

37:54.013 --> 37:55.813
asking, and it's a good
question, and I can't answer it.

37:55.809 --> 37:59.119
You know, because this is where
physics turns into--seriously,

37:59.121 --> 38:01.511
this is where physics turns
into theology.

38:01.510 --> 38:06.910
You can't--all you can say from
science is that this is the way

38:06.911 --> 38:07.871
it works.

38:07.869 --> 38:10.909
You can't answer the "why"
question.

38:10.909 --> 38:12.829
You got to talk to my
colleagues in some other

38:12.827 --> 38:14.017
department about that one.

38:14.019 --> 38:16.879
I'm sorry, because it's the
question one would really like

38:16.876 --> 38:18.426
to know the answer to,
right?

38:18.429 --> 38:23.329
But that one I can't cope with.

38:23.329 --> 38:24.119
Yes?

38:24.119 --> 38:26.159
Student: Thinking about
black holes, when the light

38:26.158 --> 38:27.908
enters the black hole across the
event horizon,

38:27.909 --> 38:31.189
I've heard that it often times
orbits or may orbit Jupiter?

38:31.190 --> 38:35.200
How is that possible if the
light doesn't have mass?

38:35.199 --> 38:37.419
Prof: Okay,
so, the question is,

38:37.421 --> 38:40.111
can light go into orbit around
a black hole?

38:40.110 --> 38:42.530
Basically.

38:42.530 --> 38:43.310
And the answer is, yes it can.

38:43.309 --> 38:46.119
And then, the next part of the
question was,

38:46.123 --> 38:49.333
how does that work if it
doesn't have any mass?

38:49.329 --> 38:51.739
You're asking the same question
he did over there,

38:51.743 --> 38:53.913
because what keeps it in orbit
is gravity.

38:53.909 --> 38:57.899
And if you reject the view that
gravity is a force,

38:57.899 --> 39:01.569
then that question doesn't
become important.

39:01.570 --> 39:05.710
And we'll--as I say,
we'll talk about that later.

39:05.710 --> 39:08.990
So, that, again,
is a question that pops up

39:08.985 --> 39:12.725
because you're thinking of
gravity as a force.

39:12.730 --> 39:15.470
Student: How does--with
that equation,

39:15.474 --> 39:19.094
not to say "why," if it is,
why it is, but where--how is it

39:19.091 --> 39:19.841
derived?

39:19.840 --> 39:21.500
Prof: How is it derived?

39:21.498 --> 39:21.818
Okay.

39:21.820 --> 39:27.520
So, let me go on with the
presentation in the last couple

39:27.515 --> 39:30.765
of minutes of the class,
here.

39:30.769 --> 39:31.749
Here's--so, let me ask your
question in a different way.

39:31.750 --> 39:37.170
Here's what--why did Einstein
think this up?

39:37.170 --> 39:39.620
What a crazy-ass thing to do,
right?

39:39.619 --> 39:42.359
Why not--there's little
discrepancies when things are

39:42.364 --> 39:45.324
moving at the--at high speeds,
but who the heck cares?

39:45.320 --> 39:46.940
It's hard to measure things at
high speeds.

39:46.940 --> 39:49.870
Why not just stick with Newton,
which has done very well for

39:49.869 --> 39:51.209
two-and-a-half centuries?

39:51.210 --> 39:52.440
Here was the problem.

39:52.440 --> 39:55.610
In the late nineteenth century,
there were a whole series of

39:55.613 --> 39:59.723
experiments, all of which,
one way or another,

39:59.719 --> 40:04.099
had the same
consequence--namely,

40:04.098 --> 40:11.898
that light--the speed of light
is the same in all--for all

40:11.897 --> 40:13.947
observers.

40:13.950 --> 40:17.360


40:17.360 --> 40:17.860
Okay?

40:17.857 --> 40:20.837
That's extraordinarily weird.

40:20.840 --> 40:24.000
It's not obvious immediately
why that's so weird.

40:24.000 --> 40:26.080
But let me try and explain it.

40:26.079 --> 40:28.869
Here's a guy,
and let's say he's a baseball

40:28.874 --> 40:31.274
pitcher, and he's throwing a
ball.

40:31.269 --> 40:33.569
So, here's the ball and it's
moving this way at some

40:33.567 --> 40:35.817
velocity, which we'll call
V_1.

40:35.820 --> 40:38.050
And you're down here,
or over here somewhere,

40:38.050 --> 40:39.420
it doesn't really matter.

40:39.420 --> 40:42.150
You're down here,
and you've got a radar gun,

40:42.146 --> 40:45.116
and you measure the velocity of
that baseball.

40:45.120 --> 40:46.840
How fast is it going?

40:46.840 --> 40:48.510
Well, if he throws it at
V_1,

40:48.514 --> 40:50.564
and you're at rest with respect
to him, then you measure

40:50.559 --> 40:51.489
V_1.

40:51.489 --> 40:56.499
Now, supposing we put our
pitcher on top of a moving

40:56.495 --> 40:58.355
train, all right?

40:58.360 --> 41:02.060
And the train is moving at some
other velocity,

41:02.059 --> 41:03.909
V_2.

41:03.909 --> 41:06.319
So, the guy's on top of the
train.

41:06.320 --> 41:07.660
He throws it 100 miles an hour.

41:07.659 --> 41:09.769
The train's moving at 100 miles
an hour.

41:09.769 --> 41:12.779
You're standing next to the
track with your radar gun.

41:12.780 --> 41:15.260
How fast do you measure that
baseball to move?

41:15.260 --> 41:19.520
Well, you measure a total
velocity, which is the sum of

41:19.516 --> 41:23.766
the velocity of the baseball
with respect to the train,

41:23.772 --> 41:26.372
and the train with respect to
you.

41:26.373 --> 41:27.243
Okay?

41:27.240 --> 41:31.800
That seems pretty clear.

41:31.800 --> 41:34.520
But now, supposing this
guy--instead of throwing a

41:34.521 --> 41:36.411
baseball, he's got a flashlight.

41:36.409 --> 41:39.819
So he's got some light source,
and the light is moving at the

41:39.815 --> 41:42.305
speed of light,
because that's the speed that

41:42.311 --> 41:43.221
light moves.

41:43.219 --> 41:47.109
And you've got some device,
down here, which measures how

41:47.113 --> 41:49.203
fast that light moves along.

41:49.199 --> 41:51.779
And he's standing at rest with
you.

41:51.780 --> 41:54.680
And you measure the speed of
light, and it comes out to be

41:54.681 --> 41:57.431
c--not surprisingly,
the same speed of light.

41:57.429 --> 41:58.909
Now, let's put the guy on the
train.

41:58.909 --> 42:02.449
Here he is, moving at
V_2,

42:02.445 --> 42:03.955
or any V.

42:03.960 --> 42:07.530
You would expect that the speed
of light that this guy measures

42:07.532 --> 42:10.412
down here would be c +
V_2.

42:10.413 --> 42:10.993
Right?

42:10.989 --> 42:13.849
Because, it would the speed of
light with respect--from the

42:13.850 --> 42:16.760
flashlight, with respect to the
train, plus the speed of the

42:16.760 --> 42:18.290
train, with respect to you.

42:18.289 --> 42:20.279
You would expect it to be
c plus

42:20.277 --> 42:21.477
V_2.

42:21.480 --> 42:24.860
But it isn't. It's c.

42:24.860 --> 42:29.330
V_total =
c, not c +

42:29.332 --> 42:31.392
V_2.

42:31.390 --> 42:34.060
Very weird.

42:34.059 --> 42:38.379
The speed of light is the same
no matter how fast you're moving

42:38.378 --> 42:40.118
relative to the source.

42:40.119 --> 42:43.529
You could be moving at 99% of
the speed of light,

42:43.527 --> 42:47.147
toward a light source,
and it would not be coming at

42:47.148 --> 42:50.058
you at 1.99 times the speed of
light.

42:50.059 --> 42:52.629
It would be coming towards you
at the speed of light.

42:52.630 --> 42:55.060
You could have a friend who's
standing still,

42:55.062 --> 42:58.272
and you would both measure the
same speed of that light,

42:58.269 --> 43:00.889
even though you're moving
really fast compared to your

43:00.894 --> 43:01.344
friend.

43:01.340 --> 43:02.910
Very strange.

43:02.909 --> 43:05.679
And there are a whole series of
experiments, the most famous of

43:05.679 --> 43:08.359
which is something called the
Michelson-Morley experiment,

43:08.360 --> 43:12.050
which, one way or another
demonstrated that this was the

43:12.050 --> 43:12.520
case.

43:12.520 --> 43:14.420
What to do?

43:14.420 --> 43:20.130
And, what Einstein did was,
he said, hmmm.

43:20.130 --> 43:22.970
Okay, various things were tried.

43:22.970 --> 43:24.130
The experiments were wrong.

43:24.130 --> 43:26.500
The equations were slightly
screwy, who knows.

43:26.500 --> 43:30.280
But Einstein took--Einstein's
great genius was to take this

43:30.280 --> 43:33.280
seriously, and to say,
okay something is really

43:33.279 --> 43:35.299
screwed up with velocities.

43:35.300 --> 43:36.730
What is velocity?

43:36.730 --> 43:45.370
Velocity is space over
time--miles per hour.

43:45.369 --> 43:50.039
Therefore--so,
space and time are messed up

43:50.039 --> 43:54.819
when you get close to the speed
of light.

43:54.820 --> 43:58.410
Now, of course,
it also has to be true that

43:58.410 --> 44:03.030
when you're going slowly,
the original Newtonian result

44:03.026 --> 44:04.476
is recovered.

44:04.480 --> 44:07.310
So, this is your problem set.

44:07.309 --> 44:10.709
I have written down the
relativistic formula for the

44:10.712 --> 44:12.382
addition of velocities.

44:12.380 --> 44:16.390
And your problem set is going
to be to use that series

44:16.391 --> 44:20.401
expansion to demonstrate that in
the Newtonian limits,

44:20.403 --> 44:21.693
you get this.

44:21.690 --> 44:27.190
And in the limit where one or
the other of those velocities is

44:27.186 --> 44:30.336
the speed of light,
you get this.

44:30.340 --> 44:31.940
So, you'll see.

44:31.940 --> 44:33.740
The algebra's actually not so
bad.

44:33.739 --> 44:37.459
And that's what prompted
Einstein to go on this whole

44:37.461 --> 44:41.471
rampage, generating all of these
gammas, and so forth.

44:41.469 --> 44:45.769
It was the experimental
evidence that there's a serious

44:45.769 --> 44:47.839
problem with velocities.

44:47.840 --> 44:49.160
Okay, we got to stop.

44:49.160 --> 44:50.000
More next time.