0:10

Last week we worked through the Twin Paradox and

if you've been thinking a little bit about it, you might be asking,

can we put it to some practical use here?

And so Alice has been thinking about that.

So here's what she would like to do.

She would actually like to take a trip to the center of the galaxy and back again.

So she's going to be in the spaceship.

Bob is going to be here on Earth observing her, here's my fanciful drawing of

the center of the galaxy, a lot of gas clouds around, we actually can't see it.

Invisible light we can study using infrared telescopes and the like,

and a lot of bright, blue young stars there orbiting around,

actually a super massive black hole with the equivalent mass about.

2 million solar masses, even more than 2 million solar masses.

So, very interesting region if we could actually get there.

And Alice thinks she can do it.

And so, let's think about this, using our analysis of the The twin paradox.

We know that in terms of Bob,

the observer, it's a fairly straightforward analysis of the situation.

Remember, Alice the traveler, it's a little more complicated there because we

had some relativity simultaneity leaving clocks slide going on, things like that.

But from Bob's perspective fairly straightforward.

So let's think about it from his perspective here.

He knows that it's actually about 30,000 light years to the galaxy,

this is a round figure.

It's not easy to actually figure out the exact distance,

the best estimate it's right now maybe around 27,000 light years or so.

So we'll just round it up to 30,000 light years there.

So we know the distance to the galaxy.

So if you think about it,

Bob is going to see even if Alice moves pretty close to the speed of light here.

If she could move the speed of light, it would take her 30,000 years to get there,

assuming we're just ignore the acceleration as needed for the moment,

just assume she's going to constant velocity of motion all the way there.

30,000 take 30,000 years traveling at the velocity of light from Bob's perspective

but remember if he were to observe her clock that she's carrying with her.

And that measures her time and her frame of reference.

The Lorentz factor comes in, the time dilation factor.

So we know that the we'll just do half the trip because we know

from Bob's perspective it's symmetrical, so coming back is the same thing.

So we know from Bob's Perspective that she travels,

not quite the speed of light but very close to it,

it's going to take about 30,000 years on his clocks.

3:56

So if this is 2, we have okay,

Alice, she's in a hurry, she wants to get there in 2 years.

Bob says you know what, even if you go at the speed of light,

it's going to take you 30,000 years.

But her clock, he's going to observe her clock running more slowly.

So if her clock is going to run 2 years on the trip,

from 0 all the way to 2 to get there, what does gamma have to be?

Well obviously very easily, 30,000, you'll bring the 2 over here,

30,000 divided by 2, this implies that gamma = 15,000.

So that is the Lorentz factor that Alice needs to get

to the center of the galaxy in two years, assuming constant velocity motion there.

Well, what velocity is that?

If you work that out using the Lorentz factor equation,

here's what you're going to get.

4:48

Assuming I got all the digits right here, it's something equal to this, or

very close to it.

It's a one, two, three, four, five, six, seven, eight, seven,

seven, seven, seven, seven, eight or something like that.

In other words, very, very close to the speed of light, and c here,

we have the c at the end.

So 0.99999 essentially times the speed of light to get a gamma factor of 15,000,

which will enable Alice to get there,

according to her clock that she was carrying with her.

And what she is aging according to, she'll get there in two years and then maybe

sling shot back in another two years back this way, so she's back in four years.

And she gets back and she's really excited to tell Bob, gosh that was fantastic.

All the great stuff, took some great photographs there.

Comes back and, If she asks okay, where's Bob?

It's like, well, Bob who?

If you think about it.

Alice has aged four years total.

People on Earth though, the time on Earth, 30,000 years there and 30,000 years back.

It's 60,000 years later on Earth when she gets back.

And in fact, so you can figure this somewhat as time travel into the future

would actually be possible here with the twin paradox analysis or

anything similar to do that.

So it's not only that point, it's Bob Junior, Bob III, Bob IV.

It's more like Bob the 600th, might still be around at that point.

So about 600 generations have passed, so be very interesting for Alice not only

the, to experience the center of the galaxy, what's going on there.

But to come back and see what Earth is like 60,000 years later,

if anyone is even still around at that point.

Another interesting question here, is a couple interesting questions.

One is, what would Alice actually see, if she could take this trip?

As she speeds along towards the center of the galaxy.

And of course, in films like Star Wars and Star Trek, you got the idea of,

you see the stars that are flowing by you, streaks of light.

In fact, the image we use for this course,

if you look on Coursera, it has sort of that effect, right?

sort of the stars streaming by on a blue background.

In actual fact, if you do the analysis it wouldn't be quite so dramatic,

unfortunately.

It would be more just like a fuzzy glow, that you would you see.

In fact, not only that, it'd be a very dangerous type of situation for it.

Because, here's what would happen, remember how we talked

earlier in the course about the Doppler effect, and train whistles?

We've got a train whistle coming towards us, or we're moving toward it,

you get the higher pitch, and then as it moves away, it's a lower pitch.

In other words, the frequency of the train whistle increases as you move towards it.

Same thing here with light, there's a Doppler effect and

even a relativistic Doppler effect.

But as Alice moves at that velocity and sees stars coming to her, the light

from those stars, would be shifted toward the blue end of the spectrum.

And even beyond that into the ultraviolet region, and beyond that into what's known

as extreme ultraviolet region, and beyond that to the X-ray region.

In other words, all those stars that she's approaching, the light from

those stars coming at her would be X-rays, and so she'd need some pretty good

shielding on her space ship to protect her from those X-rays during her trip.

Now of course we can't see X-rays so in other words, the idea here is the visible

light from those stars because of extreme Doppler shift the frequency shift there

would turn the visible light into X-Rays coming at her.

And some of you may know that throughout the universe,

there's what's called the cosmic microwave background radiation.

It's a sort of the remnant of the big bang, and

it's as the name implies, microwave radiation.

And what would happen is, and that sort of pervades all of space,

we see it coming from all different directions in space.

There's some slight differences in certain directions, but for

the most part, it's uniform throughout the universe as best we can tell.

Well that's in the microwave region, what would happen is that radiation,

which is like medical radiation, be shifted into the visible region.

And so this fuzzy glow would be all around Alice as she traveled

to the center of the galaxy and back at a very high velocity.

So she would actually see the cosmic background radiation,

cosmic microwave background radiation

as a fuzzy glow all around her in the visible region of the spectrum.

So that's what she would see not quite as dramatic as the stars flying by

9:58

So let's think about this.

We know that us humans, we can't stand more than certain types of acceleration.

And so we want a relatively gentle acceleration for the spaceship but

enough to get Alice up to speed fairly quickly.

So let's just say what if acceleration =

10 meters per second squared.

In other words what that is saying is ten meters per second per second.

So her velocity increases by ten meters per second every second.

That's literally what acceleration is,

ten meters per second squared is ten meters a second every second.

So it's increasing, again, ten meters a second every second.

That's roughly the acceleration to the gravity which is 9.8

meters per second squared and we know we can live in that type of environment.

So, our first space ship had a constant acceleration of ten meters per second

squared, she would feel pushing on her just

the normal force or Earth's gravity, and so presumably that would work.

So if we could have that constant acceleration, how long would it take

to get up to essentially the speed of light or very close to it.

Just do a simple calculation.

Well the formula is velocity is acceleration times time.

Just a times the elapsed time that you're going with it.

And so, we know the velocity here is essentially C,

0.999 times C, acceleration is 10.

So essentially the velocity is let's just say,

we'll get right up to the speed of light.

So we know that's 3 times 10 to the 8th meters per second.

In other words really 0.99999 times that, but we'll just round it to 3 times 10 to

the 8th meters per second equals acceleration here is 10.

That's what we're assuming times the time.

How long is it going to take us to get up to this

using acceleration of 10 meters per second squared.

Well, you do the math there, it's not too difficult.

Convert it into days, say, and

you find that t is approximately equal to 347 days.

I was like, wow, hey, that's not too bad.

If I can accelerate at a reasonable acceleration like that, for

347 days I'm up to, essentially, the speed of light.

And, so, Alice's trip then, looks maybe a little more promising.

Unfortunately, there's another catch here and

that catch is the energy involved to do something like this.

Because little later on this week we'll talk about the famous equation,

e equals mc squared is that also came out in the miracle year of

1905 in Einstein's September paper.

So, we'll get to that but just to let you know one of the results that come

out of that is The energy involved to accelerate as you get up to

speeds close to the speed of light, also increases by gamma.

13:00

And therefore, what happens is, remember as v gets very close to the speed

of light, gamma gets larger and larger and larger and tends towards infinity.

So it really, to get up to these speeds, close to the speed of light takes massive

amounts of energy for any relatively large mass like a spaceship.

We can actually get subatomic particles going that fast.

But even a subatomic particle takes a lot of energy to do that.

So, unfortunately not very practical there unless we could come up with some way

to generate a lot of energy and design a spaceship to do it and so on and so forth.

One more thing though, you might say,

well maybe we don't need to get all the way up to here.

What if we just said, what if we could just take the trip at,

9 tenth the speed of light, okay.

So, maybe that would reduce the energy requirement perhaps and

maybe it's more practical that way but remember what happens then to gamma with

vehicles 0.9c gamma is about 2.3 for that.

And with, gamma equals 2.3, well maybe the energy of it sort of

requirements sort of scale according to gamma, perhaps that isn't so bad.

But remember what that does to Alice's trip now.

Remember the time on her clock is 30,000 years

divided by gamma, as Bob sees it,

as he's observing her clock and as she's observing her clock tick as well.

And therefore if gamma is only 2.3, then the time on Alice's clock that

ticks off on her journey to the center of the galaxy is about 13,000 years.

And so clearly she's not going to make it even close,

nowhere near the center of the galaxy,

she's wanting to get the next star really before she gets old unfortunately.

So, travelling the galaxy, yes it is certainly possible in theory,

and we can imagine situations like that.

But the amount of energy involved, is just way beyond our capabilities at the moment.

And yet, we do see effects like this, certainly with,

as we've talked about before, in the twin paradox, in the sub atomic world.

Where we can actually get particles up to that speed, whether they're artificially

accelerated or naturally accelerated like the muon example we did.

So unfortunately we have to burst our bubble in terms of traveling the galaxy.

We don't have anything like warp drives that we know of at the moment.

But who knows perhaps someday in the far distant future we'll figure something

like that out.