In the last several videos we have been considering that time's a suspect.

We found that for moving clock it runs more slowly than identical clock at rest.

We've also discovered that, according to Einstein's special theory of relativity

based on his two postulates, that length is also suspect, that,

as we have here, length contraction occurs along the direction of motion.

So if you have a moving object going by you, as you see it go by and

you measure the length of it as you watch it go by,

you will get an answer the length from your perspective, from your frame of

reference looking at that moving object going by, will be 1 over gamma,

the length of that object if it's just sitting there at rest next to you, or

if somebody is on that object, spaceship, train car, or something like that.

They will measure one length for

it, the rest length sometimes actually called the proper length.

It's not the greatest term because it sounds like that's the correct length and

everything else is incorrect.

But the proper length or

perhaps rest length is the length of an object while it's at rest, okay.

So if it's a train car and somebody's riding along in it,

as far as they're concerned, that's their frame of reference.

They're at rest, they can get out the tape measure and measure it,

and get a length for that train car or a spaceship or whatever.

Somebody else watching it go by and

measuring the length as it goes by would get a different answer.

Different by this 1 over gamma factor, where again of course,

gamma is greater than or equal to one.

So the moving length is going to be shorter,

just a tiny bit shorter because gamma is very close to one until,

as we talked about, you get up near the speed of light.

But it will be shorter.

So we have this length contraction effect.

That perhaps raises the question, well does everything change?

Is everything suspect?

Are there any quantities that are not suspect?

So that's what we're considering in this video clip.

What is, is not suspect and

in fact as a side here, Einsteins theories become known as a special

theory of relativity, or the general theory of relativity later on,

just the theory of relativity and that wasn't his original name for it.

He actually preferred invariants theory, that a theory of invariance,

because if you actually look at the theory in more mathematically sophisticated ways,

more advanced ways, it'll really emphasize not the things that change,

they're important, but really the things that stay the same

from one frame of reference to another frame of reference.

So a theory of invariance.

We talk about what is invariant.

So really what we are considering here are a couple of things that are invariant

from one frame of reference to another moving frame of reference, and

actually in the next video clip we'll consider

something called the invariant interval which will be important for us later on.

So, anyway, let's get back to this.

So what is not suspect?

We know time seems to be suspect in terms of synchronization of clocks and

time dilation.

Length seems to be, not seems to be, is suspect as well.

What about other quantities?

And in particular we want to look at the width and height of,

we'll go back to our train car here.

So here's Bob going along at some velocity v along

these super fast you know train here.

And we'll say that the length of his train car is L,

as he measured it, so that would be the proper length.

The rest length of his car as he measure's it.

He gets out his tape measure there and does that.

Here's Alice on the station platform.

So this time Bob is moving and Alice is stationary from her perspective.

Of course, from Bob's perspective he's at rest.

Alice is moving that direction to the left, to his negative x direction.

So, let's look at the perspective that Alice saw.

Clearly from Alice's perspective, she would see this assuming that velocity

is high enough that you get some gamma there, significantly different from one.

So, she would measure the length Bobs moving train car as L moving times

one over gamma, whatever the rest length was, if he brought at the station or

if she could measure throughout the station platform.

Then he goes by, she gets a different answer for the length.

So now let's consider let's do a top view first here, okay?

And let's consider the width of the train car.

All along now so far really we've been emphasizing just one dimension in motion

along the x direction but we're still emphasizing that as a direction motion.

Let's just imagine we can expand it out just a little bit and consider width and

height here so here's the top view of our train car,

Bob's train car that goes along the tracks, some high speed v.

And the question is is there an effect there?

We have a length contraction effect.

Maybe as Alice looks at Bob's train car,

maybe there's a width contraction effect as well.

She gets up on top there she can see down on top of Bob's car.

Maybe there's width contraction effect as well.

So let's imagine there is this width contraction effect.

So what would happen if it was a substantial enough effect.

Bob's train car would contract, as far Alice is concerned.

Alice is observing the train car and obviously,

train wheels have rims such that this couldn't happen but

we'll assume that it's such that as the train car gets shorter,

less wide from Alice perspective, what would happy to the wheels?

The wheels would fall inside,

just imagine, the wheels are right here of course.

Right there and there, there and there on the tracks.

But as the train car narrows in width, the wheels are going to be here and

they're going to fall off the tracks, okay?

And if it narrows enough the whole train car would fall in between the tracks.

All right, obviously there are railroad ties and all that, but you get the idea.

That there'd be an accident involved because Alice would see,

if there's this width contraction effect Alice would see the train car

get narrower, it falls through the tracks that way, okay.

So, that's what Alice would see assuming this width contraction effect occurs.

Now, let's analyze it from Bob's perspective

because from Bob's perspective, again in his frame of reference, he's stationary.

He's at rest.

What does he see?

He sees Alice shooting back that way, but

he also sees the train tracks going by him, right?

In his frame of reference, he's at rest.

It's the train tracks that are moving in a negative x direction behind him.

He's watching train tracks just constantly go behind him at a high velocity, okay.

Well if the width contraction affect is true,

it has to apply for this situation as well, right?

Because, you know, as we mentioned, the length contraction effect,

the time dilation effects, they work no matter which perspective,

which frame of reference you're seeing here.

So in this case, if there is a width contraction effect, it's gotta be true for

Bob as well, because he assumes he is at rest.

Tracks are going by him and maybe you can see what's going to happen here.

If the tracks are what's moving with respect to him and

there's a width contraction effect, the tracks are going to get narrower.

So in this case, these

are our original where the wheels were, for him the wheels stay the same.

His car stays the same but it's the tracks that are getting narrower now.

Okay, so the traction is exaggerated for.

The tracks, get squeezed in.

And what happens?

The train car goes off the tracks, but certainly doesn't fall through the tracks.

It lands on top of the tracks, which are now underneath the car.

So you have, two very different conclusions.

Alice would see, the tracks get wider and the train car fall through.

Bob would see the tracks get narrower.

And the train car would come off the tracks, but would not fall through.

You've got to have one or the other.

You know, you could take a photograph.

They both have to agree on the photograph that it happens.

And therefore the conclusion is that there is no width contraction effect, okay?

That in that dimension.

Again, it's perpendicular really to the direction of travel.

So the length contraction effect is only in the direction of motion.

At least we just did width here and we'll do height here in a second as well.

Although another argument you could make here, physicists love to make symmetry

arguments, and you could say if there's no width contraction effect

you know we'll go do a height here in a minute, well you can just sort of rotate

around how there really is no difference between width, height, whatever.

It just depends on what direction you choose to be horizontal or

vertical or whatever.

So if there's no width contraction effect by a symmetry argument there should be no

height contraction effect either because you just as easily could define,

okay, here's the width or we could call that the height and this the width and

so on and so forth.

But let's look at the, let's do the actual analysis here for

the height contraction effect.

So that was the top view.

We show that if there is width contraction you get two different situations

in terms of what Alice and Bob observe.

They both have to observe the same thing if you take a photograph, and

therefore can't happen.

Now let's imagine going back up here to Bob along the tracks.

Say there's a tunnel coming up here.

So we'll just sort of try to draw a tunnel like that,

so going into the tunnel.

To Alice, the tunnel is stationary.

To Bob, of course, Bob is approaching the tunnel.

So to Bob, the tunnel's actually coming toward him.

And he's at rest as far as he's concerned.

In his frame of reference, the tunnel is coming toward him.

To Alice, the tunnel is at rest.

To Bob, he's moving toward the tunnel.

Let's analyze the situation and let's assume this tunnel is

just the right height here, from Alice's perspective.

Okay, that if they're at rest, let's just put the train car at rest there a minute,

we slowly move it up to the town, we say hey, it will just fit.

So the height of the train car is such that when everything's at rest,

it just fits inside the tunnel.

Now, what's Bob going to see, however?

Remember we just did the analysis with Bob?

The train car, he's at rest, the tracks are actually moving by him.

Bob sees the tunnel moving toward him at a very high velocity.

If there is a height contraction effect, okay?

So, height contraction means height is going to get shorter.

Bob going to be very worried here, scared out of his wits probably.

He sees this tunnel coming toward him at a very high velocity and

if he sees the height contracted he will see that he's not

going to be able to get through that tunnel.

The tunnel is going to smash into him.

Meanwhile, what does Alice see?

So Bob definitely sees a big problem coming ahead here, right?

Alice sees Bob, what are you talking about?

Again, see, and I'm using see in the sense of observe.

Alice observes that there's a height contraction effect

but she sees the train car contracted because to her, that was moving.

If there's a high contraction effect on a moving object,

it's the train car is getting shorter and then she says hey, Bob, no problem at all.

You have plenty of clearance to get through the tunnel.

Again clearly one or other thing has to happen when he gets to the tunnel or

the tunnel gets to him.

Either he smashes into it, tunnel smashes into him or he makes it through.

And therefore since again you have this contradiction,

you could at that instant time you could take a photograph,

you decide hey does he make it through the tunnel, is it too short?

Is it plenty of room even larger room?

More height than we start off with, again, both Bob and

Alice would have to agree on that photographic evidence.

So, this conclusion is that there is no height contraction effect or

a height lengthening effect, either.

It doesn't go either way.

So, that height is going to be unchanged.

It is not suspect.

It is invariant.

So when you have motion in one direction, there's a length

contraction effect in that direction but no effects in other dimensions.

No effect on width, no effect on height, those stay the same.

So again, that's what is not suspect or what is invariant.

And in the next video clip, we'll consider, again, as I mentioned,

something called the invariant interval, that is another quantity that's invariant.

Not quite as simple as height and width, but we'll see what that is all about.

What is not suspect

Understanding Einstein: The Special Theory of Relativity

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