In this video lecture we want to spend a little time exploring time dilation and

length contraction, which we introduced last week.

And so here's the idea.

We're going to do a little star tour here.

Image a trip to a star five light years away.

So if we could go there at the speed of light It would take presumably,

it would take five light year, not five light years, five years to,

get there in a certain frame of reference as we will see here.

So for our trip we're going to assume that our velocity here's earth, here's a star,

compressed together there, but the idea is that's five light years away.

Here's maybe it's Bob's space ship.

And traveling at velocity v.

And we're going to assume v is 0.943 times the speed of light.

The reason we chose that it gives us a gamma factor of three.

So a nice even gamma factor there, Lorenz factor.

So here's what we're going to do is we want to analyze

the situation first from the perspective of an Earth observer watching Bob,

so maybe Alice is on Earth watching Bob travel to the star.

And then analyze it from the rocket observers perspective,

frame of reference, that would be Bob in this case.

So, let's just think about this from somebody watching on Earth, watches Bob go

by at velocity v, presuming he's had a lot of time to get up to speed here so

he's going at a nice constant velocity as he goes by.

We also set up here, so again, imagine for

the Earth frame of reference it's really the Earth star

frame of reference because they are stationary with respect to each other.

So.

Once again we have to imagine a lattice of clocks going all along in the Earth star

direction here and all those would be synchronized in the Earth frame.

And so the idea here is that when Bob goes by earth, we'll set it up such that

the Earth clock, the clock on earth is time equals zero and Bob's clock

that's going along with him is also time t equals zero at that instant as he goes by.

And all the Earth clocks are synchronized in the Earth star frame of reference.

And Bob too has this imaginary lattice of clocks extending in

both directions to infinity and all those clocks are synchronized in

his frame of reference which of course his frame of reference he's not moving.

He's stationary.

And it's really the Earth and star system that is moving.

We get to that in a minute.

But let's look at the Earth observer first and just measure how long does it

take at 0.943c, gamma factor three for Bob to get from the Earth to the star.

So we have travel time here.

It's simply going to be five light years.

Five ly for light years.

That's the distance from the earth to the star divided by the speed which is 0.943c.

0.943 as the speed of light.

And as we talked about in previous video clip here.

The nice thing about using distances like light years or even light months,

light seconds.

Although particularly we use light years, especially for astronomical distances.

Is that the speed of light is simply one light year per year, almost by definition.

The definition of a light year is the distance light covers,

travels, in one year, technically in a vacuum.

Which of course, outer space isn't quite a perfect vacuum, but pretty close to it.

And so we get one light year per year for c so essentially

we're just doing five divided by .943 as long as we're using units of years.

If we're using units of light months here then the answer would be we'd have one

light month per month for c and the answer would be in months and so on and so forth.

So as long as we keep tracking we're using light years here.

the answer of five divided by .0943 is going to be the number of years travel

time to get there and if you do that it comes out to be 5.30 years.

And, it's always good to check does that make sense, well five light years.

And it takes, you're not going quite at the speed of light,

just a little less than the speed of light.

So you would assume it takes a little longer than five years to get there.

And of course that's the answer we get, 5.3.

So that checks out in terms of how we think about that.

Now another thing we want to do here from the Earth observer's perspective.

So that's the first thing.

The second thing we want to do is, as Bob's goes by here, as our rocket ship

goes by, takes a photo, right at that instant on the earth star latus of clocks.

Takes a photo both of the earth clock there and Bob's clock.

They both read t equal zero.

And then when Bob the rocket gets to the star 5.30 years later, take another photo.

And so in that photo the star clock will read 5.30 years

because this is from the Earth observer's perspective.

And Bob's going along a 0.943c and when

he gets there, 5.30 years have elapsed on the Earth star system of clocks.

So the question then is, really what does that photo show?

We know the photo shows 5.3 years for the star clock,

what does it show on Bob's clock as far as the earth observer is concerned here?

Well this is where we think okay

Bob's clock is a moving clock with respect to the Earth star system.

Time dilation factor is what enters in here.

So in terms of what the Earth observer sees, say Alice sees as Bob goes by.

Is Bob's clock runs slow.

Compared to the Earth's star system of clocks.

It runs slow, it runs slow by a factor of gamma.

Remember our basic equation is,

we'll say, we've written it different ways in terms of, you now,

Alice versus Bob, moving versus rest frame, here I'm able to do,

rocket in lab frame, where the lab frame is earth start, system.

Or actually let's just call it the earth frame.

So, delta t for the rocket here is one over gamma,

the elapsed time on the rocket is going to be

one over gamma times the lapsed time in the Earth star frame.

Okay, and gamma is three, we've set that up.

So if delta T on Earth is 5.3, then the Earth observer sees,

let's not use see we try to avoid that although we use it sometimes.

Observes, observes in this case, and in this case observes means photographs,

so might as well say observes/photographs rocket clock

at star at the star and,

the time on that clock is going to be the elapsed time on the Earth clock.

5.3 years, divided by three.

Divided by gamma.

And so that's going to be equals 5.3 years over gamma which of course is three year.

And to do 5.3 divided by three you get 1.77 years.

Of course I worked out the numbers beforehand so

I can't do that in my head just automatically but

just remember the numbers we're using here.

So, that's, in other words,

when Bob gets here in his rocket ship And the photograph is taken.

The star clock, which belongs to the Earth Star lattice system, will read 5.3 years.

The rocket clock will read 1.77 years, and then you're going now,

hold on a minute, wait a minute here, now how can that be?

Because Bob is going at 0.943c, about 0.9 times the speed of light.

It's five light years to get there.

How can it only take 1.77 years for him to travel that distance?

Clearly he would have to be going faster than the speed of light or it seem so.

And here's where we have to remember the different frames of reference in

different perspectives.

So now let's take a look at what the rocket observer says,

what Bob in this case not says but would see, would observe as it goes along here.

So, from the rocket perspective, and by the way,

you may have noticed this already.

This is very similar to our muon problem.

In fact, really an exact analog, just another version of the muon problem,

the muon problem is a version of this.

Although we're going to take this a little bit farther than we

did with the muon problem.

So, having that hint then, think a minute, what does Bob, the rocket observer,

see as he's travelling along in his frame of reference?

And actually, that's a misnomer,

he's not travelling along in his frame of reference.

He's stationary in his frame of reference.

And it's the Earth Star system that's moving with respect to him.

The star is rushing towards him and therefore,

here's where another aspect comes in, that's not time dilation.

It's length contraction for him.

He observes, let me just back up, clearly when he gets here at the star,

and the photo's taken 1.77 years on his clock, he has to agree with that.

He agrees, yes, absolutely, I can see the photograph there.

Clearly it says 1.77 years.

Clearly I can see that.

And the Earth observer, in their frame of reference, the clock for

the star is 5.3 years.

And to understand that discrepancy completely, will take a little while here,

but for the moment, there's no disagreement about the photograph here.

It's not like the number is wrong.

This is what's on Bob's clock.

This is what's on the Earth Star system clock at that point.

So now we're going to switch from the Earth frame of reference

to Bob's frame of reference.

Say okay, what is he observing here?

And so again, the idea is he's stationary, it's the star that's rushing towards him.

And so this distance here, essentially between the Earth and

Star, is contracted for him.

Because that distance is moving towards him.

Imagine if you have a rod there or something to connect the two, right?

It moves that way.

That length of that rod, or length of the distance here between the Earth and

Star, that's going to be contracted to Bob because that's a moving

frame of reference at a negative v going in that direction to him.

And so what is that distance?

So the rocket observer observes that the Earth Star distance,

So our distance, is contracted.

It's less than it is in its proper frame of reference,

in its regular frame of reference.

In other words, the proper length is shorter when it's moving.

And so the proper length here in the Earth Star frame of reference, between

the Earth and star, is not 5.3, it's 5 light years where we put here, up here.

5 light years, okay?

To Bob however, the rocket observer, it's 5 light years divided by gamma.

Because again, in the moving frame here, and

let's not get confused here with rocket and Earth, so I'm going to use

moving and rest frame, because you can switch back and forth between the frames.

And in this case, for Bob, he's in the rest frame.

It's the Earth Star system that is moving.

So the length in a moving frame is 1 over

gamma times the length in the rest frame, when it's at rest.

And so, when it's at rest, it's 5 light years.

But Bob sees it moving at a velocity v 0.943c.

And therefore, if he measures that distance, he's going to get 1 over

gamma times 5 light years, and gamma we've chosen to be 3.

And so he sees, we'll call it D,

the distance between the Earth and Star, so

we'll say D E-S, the distance between the Earth and Star.

That's not a very good D there.

sort of like a lopsided zero or something.

There we go, a little better D.

Distance between Earth and Star, according to,

this is in the rocket frame now, the distance between the Earth and

star in the Earth Star frame is just 5 light years.

In the rocket frame, it is 5 light

years divided by gamma, which is 3.

And when you do 5 divided by 3, you get 1.67.

Equals 1.67 light years, okay?

So to Bob, the rocket observer, he sees that distance as 1.67 light years, okay?

And now let's see, okay, so he's saying 1.67 light years.

He's saying, I see the star rushing toward me at 0.943c.

And by the way, a question that I think's come up in discussion forums and

a very excellent question actually,

that we never quite addressed directly is, what if velocity changes?

It seems length changes, time changes.

Does velocity change between frames as well?

Do we have to worry about that?

We'll actually be talking about that a little later this week.

So we will see that the short answer is no.

The velocity is a relative velocity between frames of reference and

it's going to be the same for Bob in this case and the same for Alice,

saying the Earth Star system, the relative velocity between them.

We will see cases though where we'll do an example where Bob,

say he has an escape pod and he shoots off the escape pod from

his spaceship at a certain velocity, then what velocity does Alice see that as?

So we'll do examples like that.

But back to our situation here just to indicate yes,

v is going to be the same for either one of them.

Alice sees Bob traveling along at v in that direction.

Bob sees the star moving toward him at 0.943c as well.

So from Bob's perspective then,

from his frame of reference, he sees the distance that the star has to travel to

get to him as 1.67 light years because it's contracted by a factor of gamma.

And how long does the star travel toward him until it reaches him?

Well, it's the time on his clock, 1.77 years.

We know that the star and Bob,

the rocket, are in the same place when Bob's clock reads 1.77 years.

And so, what does that mean here?

Does this connect with that?

So let's think about this a minute.

Distance the star has to travel to get to Bob, according to Bob,

is 1.67 light years.

It's traveling at 0.943c to him, so let's do that calculation.

Let's just say that time for

star to reach Bob, we'll say.

So Bob is in the rocket.

Time for the star to reach Bob.

Is simply the distance divided by the speed equals 1.67 light years.

1.67 light years / 0.943c,

that's how fast the star's moving toward Bob in his rocket.

And again, we're doing c is 1 light year per year.

So 1.67 / 0.943, the answer's going to be in light years.

Do you know what you get if you do it?

You get 1.77 years, in other words,

it's all consistent, okay?

So that from Bob's perspective he sees the star rushing toward him at 0.943c.

He sees the distance the star has to travel as 1.67 light years.

And he sees his clock tick for 1.77 years when the star reaches him.

And the photo is taken at that point and his clock says 1.77 years.

Meanwhile of course, from Alice's perspective on Earth,

she sees Bob driving towards the star at 0.493c.

In her frame of reference, that distance is 5 light years,

therefore it takes Bob 5.3 years.

So when he gets to the star, on the Earth-star clocks,

Alice's lattice of clocks there, it'll read 5.3 years.

And Alice observes Bob's clock, though, running slow by the time dilation factor

and therefore observes Bob's clock to be 1.77 years, okay?

So in that sense Alice says, it makes sense to me.

This certainly makes sense to Alice and she says Bob, your clock is running slow.

I don't know what the problem is over there but I get it as 1.77 years.

Meanwhile, Bob says well,

I dont know about this 5 light years distance you've measured.

because to me it's clearly 1.67 light years when I measure that distance.

But you are right it takes me, because it's a shorter distance in my perspective.

Obviously, your measuring system is messed up, it looks like.

But for 1.67 light years at that speed the star is coming toward me,

0.943c, it takes me 1.77 years.

And therefore, at least, they both agree that the 1.77 years is correct,

although for different reasons there.

Because Alice will say, hey your clock is running too slowly.

And Bob is saying no, it's not my clock that's running too slowly.

It's your distance measurement that is messed up.

But they both agree on the 1.777 years.

It's a little more difficult though.

And this is what we're going to be heading toward this week to a certain extent,

to explain how does Bob explain this 5.3 years factor here.

So that's something we're going to have to do.

But there's another thing here too.

We say okay, I sort of see how it hangs together.

There seem to be some loose ends we're going to have to worry about.

It's one of those things you have to just ponder, as well, and

let it sink in for a while.

And sometimes it just just takes a while.

I've been in classes before where it didn't really

sink in until after the class was done, unfortunately.

But sometimes that's the way it is.

Let's think about something here, okay?

Let's go back to Bob's frame of reference.

And he's observing Alice's lattice of clocks,

the Earth-star lattice of clocks.

You can imagine a whole bunch of them there, all synchronized for Alice.

But as Bob is observing that lattice of clocks go by,

let's think about this a minute.

Doesn't time dilation work in that direction as well?

And the answer is yes, works both ways.

And so if Bob is seeing the elapsed time for

the star to get to him is 1.77 years,

how much elapsed time does he see pass on the Earth-star clocks?

Well, time dilation applies.

The exact same relationship applies here,

except let's make it a little more general here, one we had before.

So it's delta t moving, okay?

In other words, I'm observing a clock in a moving frame of reference, okay?

And this is rest so if I'm at rest and I'm observing a clock,

my clock and my frame reference, and then I observe a moving clock,

that moving clock is going to run slow by the factor of gamma.

So Bob in his frame of reference has his clock there in the cockpit with him.

He's observing Alice's clocks rush by him in the Earth-star system,

as the star comes towards him.

Therefore, he will see those clocks running slowly by a factor of gamma.

And so when he sees, on his clocks ticking away nicely, 1.77 years,

how much elapsed time will you see on Alice's clocks?

And the answer is, we wouldn't have room over there.

So we'll squeeze it in right here, say, okay?

So this is Bob's, Observation.

Sorry, squeeze that in there.

So that's observation of we'll say, Alice's.

Okay, just remember Alice here, so here's Alice.

Get her up here, here's Bob, B for Bob.

Bob's observation of Alice's, Elapsed time.

Okay, according to her clocks,

it's going to be 1.77 years because he's got his clock right there.

It's ticking away nicely.

When the star reaches him, yes, they take the photograph it's at 1.77 years.

But the clock's at 5.3 years, but we're not going to worry about that for

the moment.

She thinks it's correct, he has other ideas about that.

But what is the elapsed time from Earth to the star, or

really the star getting to Bob here that Bob sees on Alice's clocks?

And it's 1.77 because that's the time on his clock.

So it's 1.77 years divided by the gamma factor.

It's always 1/gamma here in this case, when we're going in that direction.

So that's divided by gamma which is 3.

And the answer you get if you do that is, I think,

0.59 years.

And now you really go right?

How can this make sense?

Now we can sort of see yeah, okay I get time dilation.

Alice will observe Bob's clocks running more slowly and so

instead of 5.3 years gets 1.77 years.

And sort of get Bob over here, the rocket observer.

The distance is contracted, right, and therefore shorter distance.

And therefore his clock will run for 1.77 years, okay, that.

But then Bob looking at Alice's clocks running slowly,

just as Alice sees Bob's clocks running slowly.

Bob is seeing Alice's clocks running slowly by the factor of gamma.

You get his elapsed time divided by gamma is 0.59 years.

How in the world, you've got 5.3 years,

isn't that the elapsed time on Alice's clocks?

That's the elapsed time she's getting.

How is Bob getting a value of 0.59 years for the elapsed time on Alice's clocks?

And to understand this quantitatively,

we're going to have to do some work this week.

We're going to develop something called the Lorentz transformation because that's

going to help us understand this, among many other things.

It's useful for much more than just that.

So it's going to help us understand this but I'll give you a hint on it.

And that is when you think about the special theory of relativity,

you can't just focus on time dilation.

As we've seen here, time dilation and length contraction go together in

the analysis, often from different frames of reference.

In one case you're using time dilation.

From the other perspective it's length contraction that is the issue.

But there's a third thing here too, that we cannot forget.

And that is the relativity of simultaneity.

That clocks synchronized in one system frame of reference

are not synchronized in the other frame of reference, moving at a velocity v,

an inertial frame of reference.

And that is the key to understanding this.

And earlier on in the course,

qualitatively we talked about how leading clocks lag.

That if you have two clocks, they are a series of clocks moving past you at,

obviously, for this to really be observed it has to be a pretty high velocity.

But in principle, it's for any velocity.

As it's moving past you, if you're observing those clocks go by,

the clock in front is going to lag the clock behind.

The clock behind will be ahead of, time-wise, the clock in front.

And that is the key to understanding this discrepancy here.

And we want to be able to understand it quantitatively.

We want to be able to show that add the numbers together in some way and yes,

you actually do get the 5.3 years back in.

So to do that, we have some work to do this week.

But this is again, exploring time dilation and length contraction.

Originally, I thought I would do this in a couple a weeks,

when we talk about the twin paradox.

But I decided it was good to introduce it this week

to try to mull over a little bit how time dilation and length contraction works and

also the relativity of simultaneity.

So that's where we're heading over the next few video lectures,

to do the Lorentz contraction.

And then do leading clocks lag in a quantitative sense,

as well as looking at some aspects of velocity, and so on and so forth.

Exploring time dilation and length contraction (part 1)

Understanding Einstein: The Special Theory of Relativity

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