MARKUS KLUTE: Welcome back to 8.20, Special Relativity.
In this section, we talk about the famous twin paradox.
It's probably the most famous paradox in special relativity.
I want to get to the bottom of this,
and understand really where there
is a conflicting or contradictory statement
in this story.
Let me just first say that this is personal to me.
I do have a twin brother, and you
can see three pictures of myself and my twin brother here.
We were very little on our first day of school in Germany.
You get a little box of candy when
you go to schools to make it more attractive to actually
learn and study.
And then a picture, which is probably already
about 10 years old.
What you can take away from here is clearly
moving clocks are slow.
It turns out that my twin brother
lives in the very same village in Germany
where I grew up, where we both grew up,
while I traveled the world constantly and constantly
on the road between France and Geneva
and Switzerland and the United States.
And again, I think there is no dispute here.
It can be seen from this picture that your professor looks
much younger.
I even have a more recent picture.
This is two years ago.
The German Kris Kringle Market, where I asked my brother
to take this picture for this class, for 8.20.
And again, I think the answer to the question is clear.
Professor Klute has aged less.
All right.
On a more serious note, we're going
to quantitatively understand and analyze the situation.
And we use Bob and Alice again.
In this situation, here Bob stays local.
Alice has a spacecraft, and she moves with the velocity of 0.6,
six times the speed of light, a gamma factor of 1.25.
The travel takes her to a distant star,
which is in this example It's three light years away
from Bob, measured by Bob.
The journey takes her, on Bob's clock, five years,
and the return takes another five years.
She doesn't spend much time.
She wants to go home as quickly as possible.
If you analyze this, from Alice's perspective,
we see that for Alice, the journey takes four years,
and the distance traveled for her in a spacecraft is 0.
From Bob's perspective, the journey, as seen by Alice,
is only 3.2 years long.
And so we find that there's already a conflict.
If you add the times together, both ways, the inbound
and the outbound ways, 6.4 years is not equal to 10 years.
So there's already a contradictory statement
in this story.
But the key to the understanding of this problem
is that Alice, in order to return,
has to change reference points.
And there, we do have to resynchronize the clocks,
if you want, or add a specific extra factor.
And we'll go back to this when we
look at space and [INAUDIBLE].
So the time, as seen by Bob, is 3.2 years
for the outbound journey, and then 3.6 years
in order to resynchronize the clocks on the return,
on the turning around.
And then 3.2 years on the return, which makes 10 years.
And so that observation of Bob, of Alice,
is in agreement with Bob's own clock.
All right.
So we saved the day here.
Let's look at space-time diagrams.
The outbound journey is shown here.
You see I've got it--
in addition to Bob's reference frame,
I've plotted Alice's reference, and it makes it easier
to understand what's going on.
So we see in Alice's reference frame,
the journey takes four years.
If you then go back to the position in which Bob is,
3.2 years have passed.
So this is iffy.
At the time when Alice arrives we go back
to the position of Bob, 3.2 years have passed.
We then turn around and ask the very same question.
At that time-- it's still four years--
we go back the other direction now to Bob,
we are already much further ahead,
3.2 years plus 3.6 years.
And then the journey continues, and we add another 3.2 years
to the journey.
When Alice and Bob reunite, Alice
aged by eight years, 2 times 4, and Bob aged by 10 years.
So the question now is, there may be a paradox here.
Is it possible that we missed somehow that by--
and try to understand why this is a probably not symmetric.
Why can I not just use the other reference frame,
and just declare that Alice stayed stationary
in her spacecraft while Bob moved away with Earth
and then came back?
Why are those two things not consistent?
The answer is that it's not Bob who
has to change reference frame, but Alice.
It's Alice who has to do this.
There is where the asymmetry is.
You can argue if you want that, in order for Alice to do that,
she actually has to accelerate.
But we don't have any sort of discussion
of how the acceleration actually went about.
It's really the change in reference frame
which is crucial in this discussion,
and causes the asymmetry between Bob and Alice.