The Twins Paradox
Excerpt from a book I am currently writing on Einstein and Relativity:
It is the year 3000. Steve and Arianna are twins. Arianna has just completed a trip to a
faraway planet and back to Earth at relativistic speed. Steve has remained on Earth. As
a result of the great speed Arianna experienced on the rocket relative to Steve on Earth;
she returns significantly younger than her twin brother. This is an example of the socalled Twins Paradox. The excerpt below attempts to explain why it was the traveling
twin, Arianna, who aged less than the stay-at-home twin, Steve.
“I still don’t get it,” Steve says to Arianna. They tell us that this time dilation works both
ways. You know, if you and I are in relative motion, then I see your time slowing down,
and you see my time slowing down. Right?”
“Ya, I think that’s it,” responds Arianna.
“Then how come it was you on that rocket that aged more slowly. I mean from your
point-of-view, your rocket was standing still, and I was in motion. So why wasn’t it me
who aged less?”
“Well,” says Arianna, “We can’t both be younger. I mean if there is a change in time,
only one of us can be younger than the other.”
“Ya, I know. But I still don’t get it. You’d think after all the time I’ve spent on rockets,
I’d understand the theory behind this stuff.”
“I have an idea,” Arianna responds “I’m scheduled to take a rocket trip to Space Station
Einstein in a couple of weeks. It’ll be a short day trip. You know the space beacons that
that flash every hour on the hour? How ‘bout we get two of them. I’ll take one on the
rocket, and you keep one here on Earth. That way we can keep in touch with our relative
times through the whole trip.
“Let me see if I understand this,” says Steve. “Let’s say here on Earth I get signals from
you in the rocket every hour on the hour. Then I would know that your time on the rocket
is running at the same rate as mine on Earth. Right?”
“Ya.”
“OK, so if I receive your hourly signals say only once every two hours, then I know that
your time is running slower than mine.
“That’s it.”
“Great idea, Sis! Let’s do it.”
The Einstein Space Station is 3 light-hours (about 2 billion miles) from Earth. Assuming
Arianna’s rocket travels at a speed of 0.6 c or 60% the speed of light, excluding time on
the station, the round-trip will take her:1
Time = distance/speed = 2 x 3 light-hours / 0.6 = 10 hours Earth time
Now according to special relativity, for a speed of 0.6c, the Lorentz transform factor is:
LTF = sqrt (1 – (0.6)2 = 0.8
So to Arianna on the rocket, the round-trip should take 10 years x 0.8 = 8 hours rocket
time.
OK, but why is it that Arianna’s time runs slow, and not Steve’s.
In their now long-forgotten flight school classes, Steve and Arianna were taught that the
relativity of their respective times is actually determined by two effects. The first is time
dilation. And it works just as Steve said. He sees Arianna’s time running slower, and she
sees his time running slower.
The second effect is a discovery made nearly forty years before the birth of Albert
Einstein, and is actually considered part of Newtonian physics. It is an effect caused by
the finite speed of light and relative motion. It is called the Doppler Effect; named after
its discoverer, Austrian astronomer and mathematician Christian Doppler.
Perhaps you have experienced the Doppler Effect without realizing it. If you have ever
been to an auto race, you may have noticed that when a race car speeds towards you, you
hear a high-pitched roar. Then as the car travels away from you, the sound drops in
frequency to a low pitched rumble. This change in sound frequency is due to the vehicle’s
motion relative to you (and the finite speed of sound).
This same effect occurs for light. Imagine the race driver car has a monochromatic (one
color) light source on the top of his car. Just like for sound, you would see the light at a
higher frequency (shifted towards the blue) when the car is coming towards you, and a
lower frequency (shifted towards the red) when it is moving away from you. However,
this effect on light is very slight at race car speeds, because light travels so fast.
The Doppler Effect also effects light in another way. First let’s consider when Arianna’s
rocket is on the launch pad, at rest relative to the Earth. (Fig. 8-12A). She turns on the
space beacon to check it out. It sends out light flashes in one hour intervals as expected.
From the Earth frame of reference, the expanding light signals from the beacon form
concentric circles over time.
But what about when the rocket is in uniform motion (with respect to the Earth)?
How Often Does Steve on Earth Receive Signals from Arianna?
Rocket Outbound. Let’s look at when the rocket is traveling outbound (Figure 8-12B).
Every time the rocket sends out its hourly signal, the rocket has moved further away from
the Earth. As a result, its light flashes are spread further apart as seen from Earth. So
Steve on Earth receives Arianna’s (red-shifted) rocket signals less often when the rocket
is traveling away from him, or outbound.
Rocket Returning. When the rocket is moving towards the Earth, the light flashes
“bunch up” towards the Earth. (Figure 8-12C). Thus Steve receives Arianna’s (blueshifted) rocket signals more often when the rocket is returning to Earth.
How Often Does Arianna on the Rocket Receive Signals from Steve?
Outbound. As the separation between Arianna and the Earth grows, Arianna receives
(red-shifted) light signals from Steve on the Earth less frequently.
Rocket Returning. When Arianna has turned around and is heading towards the Earth,
the distance between her and Steve is getting smaller. So she receives (blue-shifted) light
flashes from the Earth more frequently.
Thus Arianna sees the same effect as Steve!
In Summary

Outbound Trip – flashes received less often:
-
Rocket and Earth moving away from each other.
Both Steve and Arianna receive each other’s signals less
frequently.
Time
A) Rocket at Rest
Rocket
B) Rocket Outbound
Earth
C) Rocket Returning
Earth
Not to scale
Figure 8-12. Doppler Effect - Hourly Light Flashes from Rocket. A) Rocket at rest
sends out hourly light flashes in concentric circles. B) Rocket Outbound: Earth receives
flashes from rocket at lower rate (less often). C) Rocket Returning: Earth receives flashes
from rocket at higher rate (more often). (Rocket sees same effect for signals from Earth.)

Return Trip – flashes received more often:
-
Rocket and Earth moving towards each other.
Both Steve and Arianna receive each other’s signals more
frequently.
Time Dilation and the Doppler Effect
It is the combination of time dilation and the so-called Doppler Effect which determine
how often and how many flashes Steve receives from Arianna. And also how often and
how many flashes Arianna receives from Steve.
Let’s construct a spacetime diagram to get a visual picture of this; one that shows both
Steve’s and Arianna’s worldlines through spacetime. (See Figure 8-13)
First we show the hourly light flash from Steve on Earth to Arianna on the ship. They are
the grey dashed arrows in Figure 8-13A. Because we have chosen units of light-hours for
space and hours for time, the speed of light is again a line at a 45 degree angle. So the
hourly light signals from Steve’s worldline are 45 degree lines to Arianna’s worldline.
Then we do the same for Arianna’s light flashes to Steve in the figure 8-13B.
The spacetime diagram tells the story in a single picture. Let’s look at what is says about
how many light flashes are received by Steve and by Arianna.
A) How many flashes does Arianna receive from Steve on Earth?
It takes 4 hours rocket time to travel outbound, and 4 hours rocket time to return to Earth.
According to the spacetime diagram (Figure 8-13A), during this trip Arianna receives 2
flashes outbound and 8 flashes on her return trip. (Last flash is on landing.) In other
words, she receives flashes less often outbound and more often on her return, just as
predicted by the combination of time dilation and the Doppler Effect.
A) Arianna receives signals from Earth
Time (hours)
Time (hours)
10 hours
8
B) Steve receives signals from rocket
10 hours
8 hours
Arianna’s
6 worldline
6
8
8 hours
6
6
4
Steve’s 4
worldline
e
2
4
4
2
Speed
of light
2
2
Speed
of light
Space
Space
0
(light(lighthours)
hours)
Grey arrows are hourly light flashes
Figure 8-13. Spacetime Diagram - Hourly Light Flashes. A) Steve sends hourly Earth
flashes to Arianna’s rocket. B) Arianna sends hourly rocket flashes to Steve on Earth.
0
B) How many flashes does Steve receive from the rocket?
The round-trip rocket trip takes 10 hours Earth time. According to Figure 8-13B., Steve
receives 4 flashes over the first 8 hours. He then receives 4 more flashes, but over only
the last 2 hours. (Last flash on landing.) So he receives flashes at a slow rate for 8 hours,
and then at a high rate for 2 hours.
Why does Steve receive low rate outgoing flashes for so long, and high rate flashes
incoming over such a short time? Ah, here is the key to the whole puzzle. It explains why
Arianna’s time runs slower than Steve’s.
A kink in (Arianna’s) path . . . makes all the difference.2
- E.F. Taylor and J. A. Wheeler.
Let’s look at Arianna again. She receives low rate flashes during the outgoing portion of
her trip. And when she turns her rocket around at Space Station Einstein, she begins
receiving flashes from Steve at a high rate immediately. This is because she is present at
“event B”, the turn-around point in spacetime.
But for Steve, it’s a different story. To him, Arianna’s outgoing flight takes 5 hours Earth
time. OK, so far. But he is not present at “event B”. He is not at the turn-around point. He
is on Earth, 3 light-hours away from the turn-around point. Thus it takes 3 hours for light
from the turn-around point to reach Steve on Earth. There is a delay in Arianna’s turnaround, as seen by Steve.
During this 3 hour delay, Steve continues to receive low rate flashes. So in total, he
receives low rate flashes for 5 plus 3 equals 8 hours.
Once Steve “sees” the turn-around at Space Station Einstein, then and only then does he
begin to receive flashes at the high rate. The total trip is 10 hours. But Steve sees the high
rate flashes for 10 – 8 or only 2 hours.
This delay is due to the finite speed of light. It means that Steve on Earth receives more
flashes at the slow outbound rate and fewer flashes at the faster return rate. More flashes
at the low rate; less flashes at the high rate. So fewer flashes overall. This asymmetry
means that Steve receives fewer flashes overall than Arianna does.
Remember, we said that fewer flashes received means slower time for the one who sent
the flashes. So since Steve receives fewer flashes than Arianna, he concludes that time on
her rocket is running slower than his time on Earth. Based on this, Arianna ages more
slowly than Steve.
What about Arianna? Since Arianna received more flashes from Steve, she concludes that
his time on Earth is running faster than her time on the rocket. Based on this, Steve ages
more quickly than Arianna. This is summarized in Table 8-4.
V = 0.6, LTF = 0.8, distance to Space Station = 3 light-hours (in Earth RF)
Time Duration
Flash Period
Number of Flashes Received
By Arianna:
Outbound
4 hours
every 2 hours
2 flashes
Return
4 hours
every 0.5 hour
8 flashes
TOTAL
8 hours
10 flashes from Earth
By Steve:
Outbound
5 + 3 = 8 hours every 2 hours
4 flashes
Return
10 – 8 = 2 hours every 0.5 hour
4 flashes
TOTAL
10 hours
8 flashes from Rocket
Table 8-4. Hourly Flashes as Received by Arianna and Steve. Arianna receives a total
of 10 flashes from Earth during her short round-trip. She concludes that 10 hours have
passed on Earth. Steve receives a total of 8 flashes during the rocket’s short round-trip.
He concludes that 8 hours have passed on Arianna’s rocket.
So it is the turn-around (and the finite speed-of-light) that gums things up, that destroys
the symmetry, that results in Arianna being younger than Steve. Arianna is present at
“event B”, the turn-around; Steve is not. And in the turn-around, the rocket has to change
directions. Per the physics definition, this constitutes an acceleration! (Remember
acceleration is any change in uniform motion, whether change in speed or change in
direction.) Arianna feels the push and pull on her body during the turn-around. Steve on
Earth feels no such thing. Unlike Arianna, Steve is in uniform motion during the entire
trip; she is not.
So it is Arianna who undergoes acceleration. She is the twin whose time runs slower. In
fact, Arianna is actually in two different uniformly moving (inertial) reference frames on
her rocket trip; one outgoing to Terra and the other returning to the Earth. (They cannot
be the same frame of reference, because the rocket had to accelerate to get from one to
the other.)3
The mathematical details for Arianna’s short trip are presented below:4
Relative velocity is 0.6c, LTF = 0.8, distance from Earth to Space Station is 3 light-hours (Earth
frame)
So in Steve’s Earth frame:
Round-trip rocket trip time = distance/speed = 2 x 3 light-hours / 0.6 = 10 hours
In Arianna’s rocket frame:
Round-trip rocket trip is time (Earth frame) x LTF = 10 hours x 0.8 = 8 hours
LIGHT FLASH RATE:
Light flashes are transmitted by Steve on Earth and by Arianna in rocket once an hour (in their
respective frames), starting one hour after launch. How often they receive each others flashes is
given below.
Outbound trip
Both Steve on Earth and Arianna in the Rocket are equally affected by the Doppler Effect and
time dilation. Thus they each see the others light flashes at the same rate:
• Doppler Effect – Rocket speed is 0.6 light-hours per hour. This means that both Steve
on Earth and Arianna on rocket are constantly moving away from each other. Every hour, their
separation increases by 0.6 light-hours. So based on the Doppler Effect alone, they receive a light
flash every: 1 + 0.6 = every 1.6 hours.
• Time Dilation – Arianna sees Steve’s time running slow, and Steve sees Arianna’s time
running slow. This further affects how often each receives the others light signals. At a relative
speed of 0.6c, the LTF is 0.8. We divide the flash period by the LTF. Thus they each receive light
flashes from the other every: 1.6 hours / LTF = 1.6 / 0.8 = every 2 hours.
Return trip
Again both Steve on Earth and Arianna in the rocket are equally affected by the Doppler Effect
and time dilation. Thus they each see the others light flashes at the same rate.
• Doppler Effect – Now Steve and Arianna are constantly moving closer to each other.
Every hour, their separation decreases by 0.6 light-hours. Thus based on the Doppler Effect only,
they each receive light flashes from the other every: 1 – 0.6 = 0.4 hours.
• Time Dilation- Arianna still sees Steve’s time running slower, and vice-versa. This
further affects how often each receives the other’s flashes. They each receive light flashes from
the other at a rate of: 0.4 / LTF = 0.4 / 0.8 = every 0.5 hour.
HOW MANY FLASHES? – RECEIVED BY ARIANNA:
Recall that rocket trip is 8 hours round-trip in Arianna’s frame. Thus it is 4 hours each way.
Outbound Trip
Arianna receives flashes from Earth every 2 hours over a time period of 4 hours. Thus she
receives: a flash every 2 hours for 4 hours. So you divide the time duration by the period or 4 /2 =
2 flashes outbound.
Return trip
Arianna receives flashes from Earth every 0.5 hours over a time period of 4 hours. Thus she
receives: a flash every half- hour for 4 hours or 4/0.5 = 8 flashes returning.
Summary
Arianna receives 2 flashes from Earth on her outbound trip and 8 flashes on her return trip, for a
total of 10 flashes. Thus she concludes that 10 hours have elapsed on Earth.
HOW MANY FLASHES? – RECEIVED BY STEVE:
Recall that in Steve’s Earth frame, it takes Arianna 10 hours to round-trip, thus 5 hours one-way.
Outbound Trip
After 5 hours Earth time, Arianna’s ship turns around at the Space Station. At the moment of
turn-around a light flash is sent to the Earth. However, the distance form the Earth to the Space
Station is 3 hours. This means that Steve on Earth does not receive this flash until 3 hours after
the turn-around. Now this is key: The time duration for outbound flashes is thus 5 hours plus 3
hours = 8 hours to Steve. So Steve receives: a flash every 2 hours for 8 hours or 8/2 = 4 flashes
outbound.
Return Trip
To Steve, the total round-trip rocket flight takes 10 hours. Since he receives outbound flashes for
8 hours, he will receive inbound flashes for 10 – 8 = 2 hours. So Steve receives on Arianna’s
return flight: a flash every half an hour for 2 hours or 2/0.5 = 4 flashes returning
Summary
Steve receives 4 flashes on Earth from the rocket on its outbound trip and 4 flashes on its return
trip, for a total of 8 flashes. Thus he concludes that only 8 hours have elapsed on the rocket.
There is no way that two people can travel in relative uniform motion with respect to
each other, and then meet to compare times without one of them having undergone
acceleration! It is this person; the one who has experiences acceleration whose clock runs
slower, who experiences slower aging.
There are in fact a number of ways to analyze the Twins Paradox; using either special or
general relativity. However, they all conclude it is indeed the “travelling” twin; the one
who experiences acceleration, who ages more slowly.
Still skeptical? Good for you. A good scientist doesn’t totally accept a proposition based
on analysis alone. In physics such issues are not settled just by argument but by
experiment.”5 So let’s look at one such experiment.
We don’t have the technology (yet) to accelerate macroscopic objects such as rockets and
people to relativistic speeds; but we can do it with microscopic objects. Let us turn again
to our particle friend, the muon. In 1975, an international team of physicists led by Italian
particle physicist Emilio Picasso sent muons on a series of “merry-go-round” rides
around and around in circles. The muons were steered by magnets to follow a great
circular path some 46 feet in diameter inside the then brand-new Muon Storage Ring at
CERN. The test was conducted to check out a fundamental point in electron force theory,
but also gives us a way to test the acceleration issue.
A circular path is one where the direction is constantly changing; and since change in
direction is a form of acceleration, the orbiting muons are in fact constantly undergoing
acceleration.
The muons achieved speeds of some 99.94% the speed of light relative to the laboratory.
Based on lifetimes of muons at rest, the traveling muons would on average live only long
enough to make 14 to 15 trips around the Ring.
Now if Einstein is correct, we would expect that the accelerating muons undergo a
slowing of time, and thus have a longer lifetime (as observed in the laboratory frame of
reference). Repeated testing showed that the traveling muons lasted long enough to make
on average about 400 orbits around the Ring. Muon lifetimes were extended nearly thirtyfold. Careful measurement showed that this agreed with Einstein’s prediction to 1 part in
500.6
So this experiment tells us that it is the accelerated muons (or by analogy the accelerated
twin) who experience the slowing of time. Numerous laboratory tests since that time, as
well as atomic clocks on airplanes, rockets, and satellites have all confirmed the same
principle; the reference frame which experiences acceleration is the one which shows the
slowing of time. Einstein is proven right again!
IME
7/4/09
Copyright © Ira Mark Egdall, 2009
Endnotes
This subsection is based on “More Relativity: The Train and The Twins”, lecture by
Michael Fowler, University of Virginia Physics11/28/07, website:
http://galileoandeinstein.physics.virginia.edu/lectures/sreltwins.html and Einstein Light, School
of Physics, University of New South Wales, Sydney, Australia, website:
:www.phys.unsw.edu.au/einsteinlight/jw/module4_twin_paradox.htm.
2
E.F. Taylor, J. A. Wheeler, Spacetime Physics, p. 125.
1
3
Another excellent explanation of why the traveling twin is the one who experiences
time dilation (and not the stay-at-home twin) is given by E.F. Taylor and J. A. Wheeler,
Spacetime Physics, pp. 124 -129. Taylor and Wheeler tell us to think of two separate rockets; one
going form Earth to Terra, and the other returning from Terra to the Earth. These represent two
uniformly moving frames of reference; outgoing and incoming. We should also imagine Arianna
“jumping” from the first rocket to the second at Terra.
From the outgoing rocket frame, it is stationary and Earth is moving at 0.9c. Thus to the
rocket, time on the Earth is running slower; by 9.82 years X LTF = 9.82 x 0.436 = 4.28 years.
The same is also true from the second rocket frame. But when the rocket returns to the Earth, it
sees the Earth clock reading 45.1 years. How can this be? Taylor and Wheeler tell us that,
according to special relativity, there is a “jump” in time when the rocket turns around
(accelerates) at Terra. “This jump appears on no single clock, but is the result of the traveler
(Arianna) changing frames at (Terra)”. Steve and Arianna “have a “consistent and nonparadoxical interpretation of the sequence of events” in their respective reference frames. But
they “infer misbehavior in frames other than their own,”
4
The astute reader will notice that the spacetime diagram is presented in Steve’s (Earth)
frame of reference. As noted, there are two other frames of interest here; Arianna’s outgoing
frame and her incoming frame. The total number of flashes sent and received by Steve and
Arianna is the same in all three reference frames. (An example of Twins Paradox spacetime
diagrams in all three reference frames is given in Einstein Light, School of Physics, University of
New South Wales, Sydney, Australia, website:
:www.phys.unsw.edu.au/einsteinlight/jw/module4_twin_paradox.htm.)
5
Nigel Calder, Einstein’s Universe, The Layperson’s Guide, p. 156
6
Nigel Calder, Einstein’s Universe, The Layperson’s Guide, p. 158.
General relativity gives us one way to see why the circulating muons are the ones which
experience a slowing of time. As we shall learn in Part II of this book, general relativity tells us
that acceleration and gravity are equivalent. Thus the constantly accelerating muons can be seen
as in the equivalent of a gravitational field. Also according to general relativity, a gravitational
field slows down time. Thus the traveling muons experience the equivalent of a gravitation field
(in addition to Earth’s); so time for the muons slows down relative to stationary muons which see
no additional gravitational field because they do not undergo acceleration.