The Twin paradox is probably the most famous
of all of the seeming paradoxes of special
relativity.
I made a video on this that had some math
in it and it seems that some viewers wanted
more explanation on the phenomenon.
I do read the comments from time to time and
decided to do another video with more explanation
and less math.
We’ll see which one the viewers like better.
In its simplest form, a pair of twins performs
an experiment.
Their names are Dr. Don and Dr. Ron.
Dr. Ron heads off in a rocket ship at high
speeds towards a distant star and returns
home.
Dr. Don remains stationary on Earth.
Upon Ron’s return, they find that the traveling
twin, that is to say Dr. Ron, is younger-
perhaps much younger- than Dr. Don.
This is a well-established consequence of
relativity– moving people age more slowly
than stationary ones.
This example is called a paradox because it
involves Dr. Don and Dr. Ron and they are,
well- a pair of docs.
The real reason that it’s called a paradox
is because it seems to break one of the assumptions
that goes into relativity.
This assumption is that all people can equally
well claim that they are unmoving and that
people around them are the moving ones.
Under this assumption, Dr. Ron, who is in
the rocket, could say that he wasn’t moving
and the Earth and the star did.
Since Dr. Ron wasn’t moving, it would be
reasonable to claim that the he was the older
and Dr. Don was the younger.
In such an experiment, there can be only one
outcome.
Only one person is older.
And yet we have two participants, who have
different points of view.
It seems that both can claim that they are
unmoving, which means that both can claim
that they are older than their twin.
And, obviously, they both can’t be right.
Hence the name paradox.
It turns out that this really isn’t a paradox.
There is an explanation why both people- Dr.
Don and Dr. Ron- agree that Dr. Ron– the
guy on the spaceship– is younger.
Now you’ll find many books and videos that
will tell you that the reason that this isn’t
a paradox is that there is something that
distinguishes between the two observers and
the reason is that one of them experiences
acceleration and the other one doesn’t.
Because of the acceleration, Ron and Don have
different experiences and that’s what people
claim is the answer.
The problem is that this answer is not fundamentally
correct.
I mean, it does have some merit and we’ll
get to that point shortly.
But at the deepest level, acceleration isn’t
the explanation.
To show this point, we can come up with a
thought experiment in the spirit of Einstein.
Let’s imagine three observers in constant
motion and in which none accelerate.
The observers have the unimaginative names
A, B, and C.
Observer A sits stationary on the Earth.
Now, there are some of you that remind us
that the Earth is moving, but one of the premises
of relativity is that everybody can consider
themselves to be stationary, and in this case,
the Earth is moving around observer A. So
in this case, it's important to remember it
really is legitimate to say that observer
A sits stationary on the Earth.
Observer B is traveling towards a distant
star located some distance away.
The distance doesn’t matter but let’s
call it L. Observer B is traveling at constant
speed as seen by Observer A. Observer C is
a distance 2 L away, traveling towards Earth,
this time with a velocity minus v.
Observer A watches Observer B head away from
earth and Observer C head towards it.
B and C’s paths cross at the distant star
and C heads back to the Earth.
During this entire exercise, all three observers
(A, B and C) all move at constant velocity,
meaning- and this is important- none experience
any acceleration.
So Observer A uses the symbol T stationary
to represent the time it takes for B to go
from Earth to the star.
And because C travels at the same velocity
as B does, just in the opposite direction,
Observer A says that C takes T stationary
to go from the star to Earth.
Thus Observer A says that the total time she
measures from when B passes her on the outward
leg and when C passes her on the return leg
to be simply two T stationary.
What time does the traveling observers experience?
Well, here I’m not going to do any math.
I’m just going to tell you the answer.
If you want to know the mathematical details,
I recommend you watch my other Twin Paradox
video.
It’s all worked out there.
So when Observer B passes Observer A, they
start a stopwatch.
When they get to the distant star, they stop
the stopwatch and hold up a huge sign that
displays the time they experienced on the
outward journey.
The sign is big enough that Observer C can
see it as they pass by on their way back to
Earth.
Observer C writes down that number.
We’ll call that T moving.
Also, as Observer C passes the star, they
start their own stopwatch to record the time
it takes them to travel back to Earth.
When they pass Earth, they stop the stopwatch
and see how much time they experienced on
the return trip.
It turns out that they also experience T moving,
which is the same amount of time as Observer
B experienced on the outward leg.
Observer C holds up another sign that Observer
A can see, which displays the time experienced
both by B on the outgoing leg and C on the
return trip.
Observer A writes them down.
So the travel time for B to head outward and
C to return is simply 2 times T moving.
How do the times experienced by the earthbound
A and moving B and C compare?
Well you can use the equations of relativity
to calculate this and I did it in the other
video.
But here I’ll just tell you the answer.
T moving is equal to T stationary divided
by gamma.
Gamma is a term that shows up everywhere in
relativity and it's related to the relative
velocity between two observers.
It's always greater or equal to one.
Thus we can then simply say that the time
experienced by the B and C as they travelled-
call this moving trip time- is equal to the
stationary trip time divided by gamma.
And since gamma is greater than or equal to
one, that means that the moving trip time
is shorter than the stationary trip time.
So that’s the bottom line.
The moving people really do experience a shorter
amount of time than the stationary one.
So why is that?
It’s not acceleration.
After all, in my example, there is no acceleration.
So that just can’t be it.
The difference is that Observer A is in a
single and unchanging reference frame, while
to get the moving frame, you need to add up
the time experienced in two frames- B’s
outgoing and C’s incoming.
Now, judging from the comments, this statement
wasn’t very clear.
So let’s talk about it.
Some people correctly pointed out that, for
example, Observer C saw A and B as being in
two different frames and that’s true.
The same is true for Observer B.
But it’s very important to remember that
we are comparing times at the location of
Observer A. That’s not the thing that fixes
the paradox, because, well, remember in the
classic Twin paradox with Ron and Don, the
times were also compared where Don was.
But perhaps it reminds us that when we ask
about durations what matters is not only the
time experienced, but the location where the
time is experienced.
In fact, I made yet another video in which
I showed that a moving clock runs slower or
faster depending on the location at which
the measurement is being made.
That video is a little technical, but it makes
so many important points that if you want
to really understand it, I recommend watching
it again.
Another important point is that all observers
agree A is in one reference frame and B and
C are in two.
I mean, you could look at this from the point
of view of Observer C. Observer C sees A moving
constantly at one velocity.
It's also true with Observer B. But when you
ask what time Observer A saw, that time was
experienced in just one frame.
Even Observer C sees that adding up time experienced
in the B and C frame is two distinct frames.
Imagine a fourth observer D who sees all three
observers A, B and C moving off at some other
common velocity.
Observer D still sees A as having a single
and unchanging velocity and B and C as having
two different ones.
Yet there is no acceleration in any of these
examples.
It’s not the acceleration.
In the comments of the earlier video, some
people insisted that frame jumping, which
is the term they often used, requires acceleration.
And that’s true to change the motion of
one observer.
In the classic twin paradox example, Ron certainly
did experience acceleration to change frames.
But that’s why the thought experiment explained
here is so important.
It’s because it removes that confusing factor.
Time dilation occurs because of constant motion,
not because of acceleration.
Finally, there were a few people who claimed
that the time dilation was built into the
example because A, B, and C started out stationary
and B and C were accelerated prior to the
start of the example.
But that’s not true either.
I mean, it could be that way.
But you could redo the problem with B staying
stationary and A and C having been accelerated
in the past, or C staying stationary and the
other two having been accelerated.
Or you could have all three of them having
been accelerated.
Since I didn’t give you the acceleration
history in my thought experiment, you can’t
know it and it therefore doesn’t matter.
What matters is one and only one thing.
And that’s the fact that we are comparing
the time experienced by an individual in one
frame to the time experienced in two frames.
Acceleration, while obviously necessary for
an individual to move between frames, is an
incidental factor here.
It’s only time and distance and velocity
that matter here.
It’s just the coolest thing.
Okay, so this video covers some of the same
material as in the previous video, but with
a very different explanatory approach.
I’m very curious to hear what you think
about the two different approaches so be sure
to share your thoughts in the comments.
We’d hope that you’ll like and share the
video so other people know how to use relativity
the right way.
And, of course, remember- physics is everything.
