0:09

Don't be too worried here about all of the math you see on the board,

we're just going to do a quick reminder about a few things here,

most of which we covered in the math review.

But, for those of you who are taking more a qualitative approach to the course we'll

use our Festina Lente motto, as it were make haste slowly.

So we're not going to try to rush through this.

For those of you, for

which some of this is old hat, you can fast forward as you wish, of course.

So, Michelson Morley experiment, before we do the quick math review, or

math reminders here, really, the goal of this was to detect

the either wind that pretty much everyone thought should exist.

So we'll come back in a few minutes to actually talk about what that was,

why they were looking for it, and so on and so forth.

Let's just do this quick overview of math reminders.

1:03

Because this really is the first topic that will be more mathematical but

remember it's all algebra.

We'll do it step by step so that, again, for those of you who may be taking a more

qualitative approach that the goal here is to follow along, see where we end up, and

understand the concepts involved and for those of you who like

the more quantitative approach then we can dive into the math a little bit here.

So, just a few math reminders, notation to start off with.

When we have one over a, a just being some number,

representing some number symbol, representing some number,

it could be x or b or you know whatever we want to do there.

So one over a,

we also write that as a to the negative first power, a to the negative one.

One over a squared, a to the negative two power.

Square root of a, right there is a to the one half.

Square root of a times the square root of a gives us a back again.

One over the square root of a then is one over a to the one half equals

a to the negative one half.

And so on and so forth with things like that.

2:19

Typically an integer, one two three four five, that type of thing, but

doesn't have to be necessarily.

That means one plus a times one plus a times one plus a times one plus a,

in times, so if this was five, it'd be five times you'd have to multiply it by.

Clearly that gets sort of messy, and

in a minute we'll see a way to get around that in certain circumstances.

When we're adding fractions together, a over c plus b over d for

example, remember how we used the little mathematical trick.

We can multiply to get a common denominator here so

we can add the two fractions together into one fraction.

We can multiply each term by one in the form of d over d

in the first term so giving us cd in the denominator on the bottom.

And then c over c for the second term, again, multiplying by one, giving us dc or

cd on the bottom.

And so, then, once we have the common denominator there,

we can just add them together.

So we had ad plus bc over cd, and I just did a little quick example here,

if we're doing three fourths plus two fifths.

You can see it works out to twenty three twentieths there.

Another thing that we'll encounter is form here.

If we had a minus b, we can factor out the a from each term.

So this would be a times one is a.

And then a times b over a is just going to be b.

The a is cancelled, so a minus b is the same as a times one minus b over a.

Or the form we'll actually see is this, a squared minus b squared,

we can factor out an a squared from each term, so leaving a one for the first term.

a squared times one is a squared.

And for the second term, a squared times negative b squared over a squared,

the a squares will cancel just leaving us the b squared there, and

before we get to this then, also one other form like this,

which this is actually what we will see here in a little bit.

Square root of a squared minus b squared.

We'll use this little mathematical technique so

factor of the a squared, all under the square root sign.

A squared times one minus b squared over a squared.

Then if you have the square root of two products, if you have the square root of

like xy, you can do that as the square root of x times the square root of y, so

that's what we did here.

We have a squared times one minus b squared over a squared.

So that's the same as squared of a squared times the square root of one minus b

squared over a squared.

And the square of a squared is just a.

4:53

So, most of that we did in our math review.

Something we actually didn't do in our math review that's going to be

helpful this time is something called the binomial expansion.

We're not going to try to prove this or

get in the details why it's called binomial expansion.

We're just going to use it.

It's a type of thing that physicist loves to use all the time

when they can because it enables us to simplify very complicated expressions.

So here how it works If you have one plus a to the n.

So again, this type of situation here.

It's one plus a times one plus a times one plus a times one plus a.

If a here is a very small number, a small positive number.

And if we say a is much less than one.

That's what this symbol means.

Much less than instead of just less than, it's much less than one.

So a is very small compared to one.

Down near zero but not quite zero.

5:44

Then one plus a to the tenth power,

this whole thing we'd have to write out becomes very simple.

We can write it as approximately equal to one plus n times a.

Now we'll just take the next one here, whatever it is, maybe it's a three.

One plus a cubed so this one plus three a.

Or if we have one minus a to the nth power and again, a is much less than one,

becomes, because we have a minus sign here one minus n a.

Or even one plus a to the negative n power.

Again because you're essentially multiplying the exponent

times whatever the a value is assuming again a is much less than one.

It's one minus na.

And this by normal expansion so any time we can do something like that, replace

a multiplication with something simple and remember, it is an approximation,

it's not exactly equal to it but it's close enough usually for

what we want to do as long as a is much less than one.

And in the special theory of relativity, we often have, maybe not often,

but we have a number of cases where a here is v over c,

6:54

Will have like one plus v over c.

Or one plus v squared over c squared.

Where c is the speed of light.

And v is just another velocity.

And so in many cases, if the velocity is less than the speed of light.

Much less than the speed of light.

It often is.

Then we can do this substitution, and it's called the binomial expansion.

Because we're expanding this out in a series of terms,

that's where it comes from.

We're actually only using the first two terms,

there would be terms in the series here.

But the first two terms are usually good enough for what we want to do.

Okay, so that's just relatively brief review,

some reminders of the math we'll be encountering.

Again, we'll try to take a measured pace here and

do many of the steps, so you should be able to follow along.

But if you're rusty with some of this you might want to just jot some of this down,

play around with it a little bit, and hopefully knock some of that rust off.

So let's talk about just the basic idea here of the Michelson-Morley Experiment.

We're going to break this up into several parts.

If we did this in one long video clip,

it be awhile here, so we don't want to do that.

So we'll do it part by part.

8:20

Okay, first part is we just want to understand what's going on here.

What were Michelson, who was Albert Michelson, Edward Morley, 1880's.

Michelson did an earlier version in 1881 of this.

The more classic version, 1887, is more precise, was able to fix up it a little

bit in terms of how exactly he was, how precisely he was able to measure things.

So this is that case for those of you that are interested.

Maybe you were in Ohio,

I was actually at what became Case Western University in Cleveland, Ohio.

9:19

So when you're talking about wave speed and

of course eventually we're going to apply this to the speed of light.

Because that's what Michelson and Morley were dealing with, the speed of light.

So when we think about wave speed, remember we talked about out the movement

of the source so

if you have a light source.

We did examples with a source of waterways a paddle like in our long tank of water.

Paddle moving up and

down generating those waves such that what happens to the wave speed, then?

Nothing happens.

Whether you're moving the paddle, the source of the waves, through the medium,

the water, or the paddle is just standing still,

enjoying the waves, the speed of the waves traveling through the medium,

through the water in this case, is going to be the same.

Now it does affect, remember, the wave length and the frequency of the waves but

does not affect the velocity for the wave, the speed of waves through the water.

Same thing with light.

If you have a flashlight, some sort of light, and

you measure the speed of light coming from that flashlight as it goes by you and

then you bring that flashlight, you know speed it up and a truck or something,

a very fast jet plane, and you watch it moving by and you measure.

The speed of the light from that flashlight going by

it is not the speed of the plane plus the speed of light you measure.

You still measure just the speed of light as it was.

So the movement of the source does not effect the velocity of the waves.

But what does effect the velocity of the waves,

at least in principle is a movement of the medium.

11:04

The movement of the medium does effect the velocity otherwise and again the water

examples we used remember, were such that if you have a wave traveling through water

saying a still pond and you're an observer measuring the speed of the wave traveling

across the water, traveling through the water there you get some value.

If you then take the exact same wave and generate it but generate it in

a moving water situation such as a stream or river or something like that.

The speed of the wave will be the speed of the wave through the water of course,

through the medium but

then also you have the velocity of the medium adding to the velocity of the wave.

And so in that case you will see a difference in the velocity of the wave

from the still situation where it's just traveling through the pond, say, or

a tank that's still.

Versus if the tank is moving.

Or if you're in a situation where the river is moving then the movement of

the whole medium, the movement of the water itself carries the wave along, so

if I'm on the bank observing it, I see the water going by pass, and of course

intuitively like I think it sounds like that or if I'm going up stream, right?

Then the wave is going up stream and

the water is moving other direction and then it's slowing the lake.

So that's the idea of wave speed.

And two key facts about wave speed.

12:27

How does that apply to the ether wind?

Well, remember the luminiferous ether was the idea that if light is a wave,

it has to have some medium.

Something has to be actually doing the waving, the oscillations.

And the idea was it was the ether remember, if you want,

you can spell it with A-E-T-H-E-R here if you want.

But we'll just use E-T-H-E-R, that's actually how it's

used in the multi volume, series, of Einstein's correspondence.

So they chose to use E-T-H-E-R, so

that's why [INAUDIBLE] will go along with them but there are many other books that

use the A-E-T-H-E-R so it's not that big a deal, anyway so side point there.

Movement of the source versus movement of the medium, let's apply that to light

then, okay so now we're talking about movement of the medium is the ether.

If the either is sort of the stationary background for the universe,

the universe is just sort of consistent either and everything in it.

The earth moving around the sun and then rotating on it's axis as well, so

it's revolving around the sun, rotating on it's axis.

Clearly as it moves around it must be moving through the either.

13:52

So depending on my orientation for

the sake of argument say I was moving through the either this way so

if I can feel that I would feel it coming by this way but in another direction.

I wouldn't feel it depending on my orientation.

And so, that's what Michelsen Morley set out to try to test a little bit to

see could we determine how we're flowing, how we're moving through the ether wind.

As the Earth moves around the sun at about 30,000

meters per second I think is the figure.

So we should feel, again we can't feel it but if we send a light beam basically.

So here's our light beam and we're going to send it.

Maybe into the ether wind versus sending a light beam the other direction, maybe

with ether wind, so will assume let's just say, we'll call this to ether wind.

14:53

And again that wind is going to be generated just by the motion

around the sun.

And clearly we're not always going to be directly into it depending how we're

standing and where we are on earth in a given time etc.

But you can take all those effects into account.

So we'll just do the simpler situation,

we'll assume going into the ether wind here,

we're either sending a light beam into the ether wind, or with the ether wind.

And from our basic wave speed fact,

the movement of the medium in this case should affect the velocity of the light.

In other words if I'm an observer here I should be able to detect a difference in

velocity of light depending on which way it's moving in the ether wind.

And so that's what Michelson–Morley set out to do and so in next video clip then

we're actually going to set it up and start doing calculation.

Just to see what the result was and why it was so significant.