0:03

So, we've set up now the twin paradox.

You can see I've got some things covered up here because I don't want to get to far

ahead of ourselves, and distract ourselves from some things will be doing here.

But this is a twin paradox part two, and remember how we set it up,

the paradoxical part here is that both Bob and Alice is, here's Bob.

Alice takes her trip to the star, three light-years from Bob, and then back again.

Both of them see the other's clocks running slow.

Time dilation as we've done a number of times before.

But, when they get back, Alice is younger than Bob, and so Bob can understand.

He says all right as much as possible

I can understand the time dilation effect yes that's going on.

And therefore Alice is traveling with respect to Bob,

and so he sees her clock running slow and also running slow on the way back.

1:02

We talked about, remember, the acceleration issue here.

In fact, we can analyze the twin paradox without having to do anything

with acceleration, the general theory of relativity, and so on and so forth.

We can understand that just from the concepts of the special theory

of relativity.

So, Bob sees Alice go out and back again.

And understands that with time dilation her clocks are running slower.

She should be younger, but Alice sees the same thing for Bob.

So the question is why is Alice younger?

That's the paradox.

How can she understand that she's younger when it seems like maybe Bob

should be younger.

And so that's what we want to do in this part two of the twin paradox.

And we're going to use our space-time diagram,

to see if we can understand it graphically.

You can see Ive added some green dash lines here,

we will get to those in due time.

But essentially what we want to do here is break

this basic diagram up into two parts.

We're going to look at outbound trip then we're going to look at the inbound trip.

So let's take a look at the outbound trip here, just doesn't

take off my much of the diagram here.

Okay so, what we've done like we've done a number of times before is we have

put both of their space time diagrams on the same plot both Alice and Bob.

So Bob is in black Alice is in red.

So we can see Bob's got the 90 degree angles.

Alice for a velocity of 0.6 c we've drawn the appropriate angle of each of these.

This is x sub a axis, this is the t sub a axis.

In fact, remember this really is the world line for Alice.

She's moving along the t sub a axis and we can see that as she moves along,

when she gets to four, four years on her clocks,

she's at the start, three light years away.

And on Bob's clocks, that reads five Light-years.

In the next part, we're going to do this quantitatively.

We'll actually go to put the numbers in.

So, they do check out, here.

But, in terms of three light-years, velocity is .6C.

Clearly, Bob sees Alice go three light-years in,

it takes five years to go three light-years, at .6C.

And, Alice From the time dilation effect the gamma equals 1.25 factor

takes her four years as Bob is observing her clocks.

3:36

Now the key thing here to remember we've done this before is the lines of

simultaneity.

So the dashed red lines are the lines of simultaneity for

Alice, the black lines, the horizontal lines this lines is simultaneity for Bob.

And as we've noticed before, this can show us

that both of them see the other's clocks running more slowly than theirs.

So for example, if we just pick out when Alice's clock is at four here

on her time axis, the line of simultaneity for and she had her lattice or clocks.

Remember the lines of

represent the lattice of clocks in each frame of reference.

And so she has her lattice of clocks here all synchronized.

And so when this clock here is at 4 all these clocks for her read 4.

That's the definition of a line of of course.

And line of same time, and therefore she sees,

she takes a photograph right here, with Bob's clock in the photograph as well.

She sees this clock is actually 3.1, 3.2, something right in there,

so here clock is at 4, at that instance, in turn the line of simultaneity,

Bob's clock is at 3.2 the same.

So she see's Bob's clocks running slower than hers.

In other words hers as she travels along here

her clocks have run through four years of time.

And she's observed Bob's clocks run through only about 3.2 years of time.

Meanwhile, Bob observes the same thing.

His clocks are going one year, two years, three years, four years, five years.

The lines are so the black dash lines represent his lattice of clocks,

at any given instant of time.

And, so, we can see, when he gets up to five years,

Alice's clock on that line of simultaneity is only at four years.

And, of course, it's because they're using different lines of simultaneity.

Simultaneity is relative, as we've said many times before.

So we can understand that from our spacetime diagram here.

5:36

So, just to reiterate this actually the outbound trip

is similar to the things we've done before but just to make sure we understand here.

So, here comes Alice, she's traveling along, in reference of course,

she's traveling along the X-axis here.

And she gets to three light years out, that's where the star is, and

at that point, she is going to start her return trip.

So really, this line right here,

which is her world line travelling through Bob's frame of reference,

is the same as this line right here, travelling out to the star.

And we've just put the lines of simultaneity in, and also the scale

markings for Alice's clock and actually her a axis as well.

Okay, so that's the outbound trip we see that when Alice reaches the star

Bob's clock reads five, Alice's clock reads four.

And so then, the next thing we have to understand is the inbound trip back again.

So let's take a look at that now.

Actually, before we do that, want to mention one reminder.

We've mentioned this before but

haven't used it as much as we've done this diagram.

This is combined diagram.

And that is this is a diagram for Alice,

in this case, moving to the right with positive velocity.

If we have a situation where the person or object whatever is moving to the left.

Remember it's a different combined diagram we still have I'll

just do it right here to sketch, and then we'll see exact diagram here.

So we still have something like this where we have in this case T sub V,

so this is Bob being stationary, his frame of reference.

Then, if we've got the other frame of reference moving to the left,

we get something like this for the other axis.

Let's see if I have any red left, here.

7:28

And, like that roughly speaking.

Okay, where this is d a and

this is x a not d there.

Okay, we haven't done as much with that just because normally

we have things moving to the right.

It's just easier to work with that one direction.

Then we can reverse things if we have to to go left.

But if we were going to do a combined diagram of something moving to the left

as Bob is observing it here from his frame of reference.

Then just think about this the t sub a axis here is a line of.

I'm sorry.

8:07

Not the line's it's a world line for that object moving.

Clearly, it's moving to the left on the x axis.

As time goes on its x value is increasing negatively there, and

then the lines of if we put those in.

Again Earl is parallel to the x axis and so just roughly speaking,

the lines of simultaneity are going to look something like that.

8:36

This is going to be a key point, actually, because what happens is

when Alice turns around at the star, her frame of reference is changing.

Instead of going positive velocity, now she's going in the opposite direction.

And, when that changes, her lines of simultaneity change as well.

And, so that's what we want to analyze now,

using a diagram that's going to look something like this.

So, let's take a look at that, for the inbound trip.

So, now we're going to look at this part of the drip going back

from the star remember everything is really happening on the X axis.

Here she goes out to the star turns around and comes back again.

And this is the world line out and the world line back,

and ends up at the planet again if that's where above is.

Okay so [SOUND] that goes there Here is

what the inbound trip looks like.

And again, it's useful for you to reproduce this.

If you have graph paper, great, but

the key thing is just get a general sketch of what's going on here.

So, here's what we've done, we've taken Alice out do three light years away.

So now, we're starting at three light years.

Okay so this right here is this segment of her trip back.

And note just like we had up here a moment again we'll look at this one more time.

Note the similarity here between this is what we have.

A combined space time diagram when something is moving to the left.

Notice what we have here okay.

So the x a axis going down like that t a axis t sub a axis

tilted to the left and the lines of simultaneity this is really why we have

the x sub a axis in this diagram because we want those lines of simultaneity for

her new frame of reference.

10:24

One frame the reference going out,

velocity going that way now as turn around coming back this way

as a new frame of reference she's changed her frame of reference.

So let's see what we've got here, again we put the scale markings for her.

And once again if we want you can look and see that each of

them thinks the others clocks are running slow.

10:50

And what happens though as she moves back towards Bob position at 0 here,

you can see moving that she ends

up at that point when she gets back her clocks have ticked off eight total.

In other words, she started, remember, to get to here, she went to four.

And, then, going back, it's just another four.

So, that's very symmetrical, that way.

11:14

The question, though, again is, where does that extra time for Bob go?

Because, again, Alice should be thinking,

Alice sees Bob clocks running more slowly than her clocks.

And therefore, she should think shouldn't Bob be younger when I get back.

Now, here's the key thing and we've tried to bring it out with these dots here,

the green dash lines.

Let's take a look at that.

When she's at the planet, from the outbound trip she gets there.

And at that instant when she arrives,

presuming she hasn't slowed down quite yet.

So she's still going with velocity v towards the planet.

Her line of is with that frame of reference pointing back down here.

Bob's clocks read 3.2 actually.

Looks like 3.1 it's really 3.2, and

we do it in the next part of the Twin Paradox video.

So it points there, okay.

12:26

She's at the planet right there.

So really this point and this point are the same.

If we combine the diagrams and

it's really this point right here, which is that's a planet.

And so in that instant of turning around all of a sudden

she's in a new frame of reference with different lines of.

Her first frame of reference the lines of go down like this.

And so when she's there this green dashed line represents our line of simultaneity.

As soon as she turns around and we're assuming not quite instantaneously, but

fast enough such that we can ignore the acceleration and

deceleration compared to the long time of just going at constant velocity.

As soon as she turns around essentially what that acceleration,

deceleration does is it puts her into a new frame of reference,

she decelerates and the accelerates again very quickly up to her velocity of 0.6c.

So now she's heading back here, but

she's in a new frame of reference, the lines of simultaneity now point this way.

So look what happens in that a few instance of turning around

13:29

by before she turns around if she compared took photographs of her clocks and

any Bob's clock back here at the home planet this clock would read 3.2.

As soon as she goes to the process of turning around and then looks

at her lattices clocks, and sees where Bob's clocks are back at the home planet.

Look where it is, it's up at 6.8 actually, okay?

So in that turning around procedure, her,

as far as she's concerned she's observing Bob's clocks throughout that time,

his clocks speed up from 3.2 here, to 6.8.

And that's where the lost time as it were comes in,

it's during that turning around process.

Because then and as we'll see quantitatively in the next

video clip that then it just ticks off normally.

She sees this clock's running at 3.2 so essentially on the Outbound trip.

She see's his clocks reading a total of 3.2 years as we'll see in the next clip.

And then on the inbound trip, she sees from 6.8 to 10 is another 3.2 years.

Well, her clocks read 4 years on each leg.

And then again what happens is, so

if you think about 3.2 years she observes Bob's clock's running slow.

Another 3.2, that's 6.4, so when she gets back she says, hey, Bob,

I thought you'd be 6.4, you would have aged 6.4 years while I aged 8 years.

But, again, that turning around,

that switching of frames of reference is accounted for.

Or, really, the clocks, then, in Bob's frame of reference, or

really Bob's clocks, from Alice's perspective,

from Alice's frame of reference, jump from 3.2 to 6.8.

So, when she does get back, Bob has aged ten years and she has aged eight years.

So what we want to do in the next video clip then is do this quantitatively.

With some of the procedures and machinery we developed.

Time dilation and so on and so forth, the Lorenz transformation.

And see if we can get the numbers to work out.

Again conclude that there's this jump when Alice's changes frame of reference,

there's this big jump in the timekeeping as she is observing Bob's clocks and

that's the key to understanding the twin paradox.