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Okay, the twin paradox, now part four.

want to do a brief review of where we've been with this.

And, also as you can see,

we will do something with the Lorentz transformation here.

If this is going to distract you, I suggest pause the video right now, write

that down so you can focus on other things we're going to be talking about here.

And we'll get to this is a few minutes.

But remember, here's the situation, in a smaller diagram this time.

We had Alice in the rocket going at 0.6 c, gamma factor of 1.25.

Bob in the lab frame, and

this time we're going to specifically talk about the rocket and lab frame.

And as you can see over the equations here, we've got R and L standing for

rocket and lab.

Not right and left, but rocket and lab, just to try to keep things straight here.

Because sometimes we've have Alice moving and

sometimes we have Bob moving in previous analysis and examples.

And so sometimes, if you just keep using A and B, Alice and Bob,

then we forget who's moving and who isn't, it gets even more confusing perhaps.

So, Alice is moving this time in the rocket frame,

we measure things in light years and years,

and went out to a star three light years away and then traveled back again.

So, here's what Alice's world line looks like and

Bob's frame of reference, out to three light years, and then back.

So back in the negative direction, leftward direction and

ends up, according to Bob, 10 years later, he observes her,

takes 5 years to get to the star, 5 years times .6 time

the speed of light is 3 light years and then, another five years back.

So this analysis, remember is fairly straight forward, he expects,

he'll see 10 years tick off on his clock.

And because of time dilation, the factor that 1.25, 1 over gamma factor,

he sees Alice's clocks tick off four years in each case.

So that was Bob's analysis via time dilation.

We did Alice's analysis, where she sees Bob, since she's going away and

the star coming toward her and then back again.

And for that, it's a little more complicated.

We had to use, not only time dilation, but length contraction,

because she sees the distance to the star not as 3 lightyears as Bob does, but

a contractive distance.

So we had to take that into account.

And once we did that, we got the correct number of years ticking off on her clock,

4 years.

And then, though, the real puzzle was

we also saw that Alice of course saw Bob's clocks running more slowly and

the number we got for that was 3.2 years, again the gamma factor.

4 years on Alice's clock, she see's Bob's clock ticking more slowly

compared to her lattice of clock, she compared it to by factor 1.25,

which will lead 3.2 years, so why does an Alice see Bob only age 6.4 years?

Remember, the key was that when we analyze the turn-around here, essentially,

Alice's changing frames of reference.

And so, her lines of simultaneity were parallel to this,

the green dash line that slopes down like this.

And when she changes frames of reference,

then her lines of simultaneity are paralleled this way.

And at that turn-around, there's a jump that takes place between

frames of reference and therefore, Bob's clock speed up

compared to Alice's clock at that point and tick faster as it where.

You can actually delve into the details of that turn around and

analyze it and show in more details if you want, but that actually occurred.

But for our purposes, we just said that hey,

change in frame of reference, therefore, we can see on our diagrams that we did

that we get this jump in time from 3.2 years up to 6.8 years.

And then, during the second leg of the trip,

again Bob's clocks, as far as Alice observes, tick off 3.2 years.

Add it all together and you get 10 years.

So both of them come up with the same answers,

all of by different reusing with them.

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Another thing, just to mention here, and as because some of you

are probably still thinking, well, It just seems, aren't they symmetrical?

In other words, Bob watches Alice go away and come back in.

Alice and her spaceship, from her frame reference, watches Bob go that way,

the star came this way, and then back again.

Just think about it, it seems very symmetrical.

But we made the point that only Alice has acceleration involved,

and Bob actually does not feel any acceleration.

We mentioned of course, if you're in a car that's accelerating, you can feel being

pushed back in the seat, or decelerating you're thrown forward a little bit, or

even moving side to side around around the curve.

Another, maybe, more physics way of doing that is to do a simple experiment.

Drop a ball, and if you're moving at constant velocity motion, and there's no

wind or anything like that, if you drop a ball, it will drop right down at your

feet, because the ball participates, again we're talking about slower speeds here.

We obviously did the analysis with relativistic speeds,

but the ball participates in the motion of the object as you're moving.

If you're accelerating,

however, what happens is the ball falls behind you for large enough acceleration.

For smaller acceleration, it actually does,

but you just can't notice it as much.

And so, essentially, what happens is, as you drop it,

the ball has the velocity at that point, so it would be going forward with you, but

you sort of accelerate out from underneath it and therefor, it falls behind you.

Or if you decelerate, it falls ahead of you, because it still has that initial

velocity, you've slowed down now, and so it moves ahead of you.

So Bob could do that dropping ball experiment all he wants,

and during this whole time, it would just fall at his feet.

He would not undergo acceleration, whereas Alice would

when she was at the turn around point, decelerating and accelerating.

So if that helps a little bit to see what's going on,

that even though it seems symmetrical, it actually is not.

And of course, the acceleration or

deceleration acceleration at the turnaround point is the key factor,

because that's what changes the frames of reference for Alice.

And then you get the jump going on there,

the jump in time as Alice observes Bob's clock.

So in one sense, we've talked about how the special theory of relativity is all

about inertial frames of reference.

It only applies to that, not In the case of acceleration and deceleration, but

you really don't need, say, the general theory of relativity here which deals with

acceleration and deceleration because you just assume the acceleration and

deceleration occurs over a short enough time

that you can essentially analyze it using the special theory.

And it's really the change in reference frame,

which is covered by the special theory, that gives us the results there.

So, yes, acceleration deceleration is involved, but

we can analyze the results and get the correct results using the special theory.

So, all that being said, what about the Lorentz transformation?

What we want to show is that you can get the same numbers using Lorentz

transformation, perhaps a little faster as well.

The downside is, first of all, you've got to use the correct formulas of course,

the correct versions of the formulas, so you've got to be careful about that.

But the downside is, you don't get as much feel for

what's going on as you do when you break it down via time dilation,

length contraction, and leading clocks lag, the relativity of simultaneity.

I do want to show that you can do this.

Again, our basic equation that we suggested you memorize,

if you're in the mood for that and want to use this a lot, is this one here

where we have the rocket moving away in a positive x direction l 4 lab And

one way to memorize whether or remember whether it's a plus or minus here.

Is just think about, okay.

The rock is moving away from me If I'm in the lab.

That mean's any rocket value for X would be a bigger value for me.

because it's moved away and so, If they measure something ten meters ahead of them

that's going to be maybe thirty meters, for me it's a twenty meters In that time.

So, if you can remember that basic situation, rocket moving away,

it's going to be a plus sign here in the basic forms of the equation.

Then from that you can derive pretty easily just in your head

that the other forms of the equation here.

So let's start over here with A.

Calculating Alice's results using Bob's Values and

so Bob's values are the lab frame values and

to Alice here Bob is moving in the negative x direction to the left.

And that's why we got minus signs in this versions.

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equals 3 light years right, right there and the time is 5, 5 years.

T l Equals 5 years.

And so that's for the outbound trip.

So the question is, okay, those are Bob's measurements right there.

What are the measurements x and t for Alice in the rocket at that point.

And if you plug the numbers in here,

gamma's 1.25, plug these values in for xL and tL.

V is 0.6 c.

We're using c = 1 because we're doing light years per year for c.

Plug that all in and you'll find you get 0 for here, and you get 4 for that.

Think a minute, does that make sense?

Well yes, because this is Alice's world line right?

She in her frame of reference hasn't moved.

She is at 0 in her frame of reference.

And her clocks tick off four years.

And that's what we got before.

So I guess that does make sense there.

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Between two spacetime points

essentially is c squared delta t.

And here again we'll use lab and rocket frame again.

So l and r so i'll just do L squared minus

excuse me delta X just to remind ourselves

it's a difference in coordinates

delta XL squared equals C squared delta

TR squared minus Delta X R squared.

And again what's nice if we're using everything in light years per year,

light seconds per second then C just becomes 1 so we don't have to worry about

the, the C we're just looking at the difference in the time coordinates and

so for the outbound trip, outbound For

the outbound trip, we have delta xL is three.

Okay, so let's write that down, delta xL.

Now as it goes to three, minus zero, so

difference is three light years in this case.

Delta tL Again, we're assuming we've done Bob's analysis.

Delta tL is five years for the outbound case.

The other nice thing with the invariant interval if we're doing delta xR for

Alice As we've already mention a couple times, that's just zero, right?

Coz she stays, from her frame of reference she's not moving,

she's just in her spaceship.

And so we can see we have everything we need here except, so we got delta xL,

and delta t sub L, we have delta xR over here which is zero.

So that disappears.

So we're just left with delta R.

And when you do that plug in the numbers here.

We get, delta t sub R equals four.

That's easy to verify that if you want.

So, that's the outbound trip for the inbound trip very similar.

Again, so again you can see here, it's an even

easier calculation than the Lorentz transformation if you know the values.

And depending on what the situation is,

sometimes it's very easy to see what those are.

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And it seems logical once you've starting from Einstein's principles and

sort of building on those.

And yes, we can see how with the light clock we get time dilation,

length contraction, relativity simultaneity, but because it's so

far out of our every day experience it's sometimes just hard to know.

So I mentioned for example the mewon experiment.

They've also done experiments with mewons in particle accelerators

where they essentially have some mewons just at rest and

then other mewons circling around them.

And from those they can show that the particles at rest

will decay faster than those in motion.

So again it shows the clocks slow down the ones in motion.

So that's well understood and

demonstrated but I've also mention these airline experiments they've done.

The first one, just if your interested a little bit.

And this was done in 1971 I believe.

Where they took they had two accurate atomic clocks on the ground.

And they took some portable one's up in commercial airliners actually and

flew around and some flew east bound, some flew west bound.

And they had to take in count a number of things going on including the rotation of

the earth.

And the fact that clocks actually run more faster at higher altitudes and lower

altitudes in a gravitational field like the earth effect next week we'll talk just

a little bit about that, to see how time dilation occurs in a gravitational field.

So they had to take those things into account as well as the velocity of

the airplanes.

And the eastbound and westbound was a little different because earth was moving

and therefore the clocks on the ground where not completely stationary, but

they were moving on the Earth, and so on and so forth.

So what did they find?

I won't give you all the numbers here, but I'll give you an idea.

For the westbound clocks that went on the planes, they had a prediction

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the result was about 30, almost 40 nanoseconds.

I think 39.8 nanoseconds.

And the result was about, I think,

39 plus or minus 2 nanoseconds.

You don't have to remember these numbers of course.

But the point is they're getting results that are very close to experimental or

the theoretical predictions.

And also just the fact that these are very very small time amounts here.

Billionths of a second are what nanoseconds are.

They've also done, in the intervening years since 1996,

not that long ago actually, but there's an experiment done in 2010

at the National Institute of Standards and Technology.

Yeah, NIST which is in Boulder, Colorado in the United States.

Other countries have these things, have these institutes as well that

are in charge of standards and making very precise measurements and things like that.

So in the United States it's the National Institute now the standards and

technology, used to be the National Bureau of Standards but

they changed the name a while back.

Anyway, 2010, they did an experiment with

a very very accurate form of clock that they had developed.

And instead of taking it on a jet airline or anything, they could just do it within

the lab at various speeds even as low as four meters per second.

Okay, so it's about 12 feet per second, if you want to think in those terms.

Up to maybe about 90 miles an hour.

So they did a variety of speeds, but slow speeds.

We're not talking about jetliner speeds.

And the experimental results

matched up nearly perfectly within the very small error bars they had.

To the predictions of the special theory of relativity.

So, the theory of relativity is well understood, theoretically.

It's a well supported, experimentally.

It doesn't mean that it's possible we could discover something that