The Twin Paradox
A quick note to the ‘reader’
• This is intended as a supplement to my workshop on special relativity
at EinsteinPlus 2012
• I’ve tried to make it ‘stand alone’, but in the process it became rather
didactic and lecturey, as pointed out by the excellent Roberta Tevlin,
who was kind enough to look it over (all mistakes remain my own).
• I have gone back and tried to ask more and tell less… but since I want
someone to be able to go through this on their own I couldn’t resist
keeping some answers in… So on some pages there are questions,
and a little symbol will bob up at the bottom of the page:
• Clicking the symbol should take you to a ‘hidden’ page that has the
answers or other comments. Clicking elsewhere (or using the arrow
keys, etc.) should navigate normally! I hope you find this helpful!

Outline:
Title &Intro
(you are here)
Description of
‘Paradox’
Click
on
any
box
to
Because there are a number of choices you can make as
go
to
that
part,
or
you go through this presentation, I thought it might be
just
clicktoanywhere
helpful
give you an outline of the different parts right
Intro to Spacetime
else
to
continue
to
away.
Diagrams
the next slide!
Qualitative:
mapping the paradox
Calculations:
computing the times
From the Travelling
Twin’s view
The twin paradox &
The doppler effect
End
The Twin Paradox
• One of the hardest things to get used to about
relativity is the way that time can be different for
different observers.
• This includes not just how quickly time passes, but also
what different observers call “now”
• Let’s take a little time to look at
what is behind the ‘twin
paradox’… which isn’t really a
paradox at all, just an example
of how weSummary
carry our
of everyday
Twin into
Paradox
ideas of time
our
understanding. Even when we
are trying not to!
Skip the
Summary
Introduction to the Twin Paradox
• In our study of special relativity we have
learned that moving clocks ‘run slow’.
One tick of your clock
3.0m
light
One tick of moving clock
light
> 3.0m
Clock moves
Light must travel further in moving clock. But light has the same speed relative
to all observers, so one tick of the moving clock takes longer than one tick of the
stationary one (as measured in the stationary frame)
A long trip
• If we have two identical twins, one on earth and one in a
spaceship which is moving at a speed close to light speed
(relative to the earth), what will the stay-at-home twin say
about the travelling twin’s clock?
• Suppose that the travelling twin’s clock is running at half
the rate of the stay-at-home twin. If the trip takes, say, 24
years on the stay-at-home twin’s clock, how long will it take
on the travelling twin’s?
tship
v
tearth

Hey sib!
Better fix
your clock!
Different times
• We can see that a long trip will take a very different
amount of time according to the two twins.
• When the twin returns they will be significantly
younger than their identical twin!
• This isn’t actually the oddest combination… how
about a daughter who is much older than her
mother?
• What do possible situations like this say about our
ideas of age and time?
The issue
• So far this is weird… but it isn’t a paradox. There is nothing
contradictory about this except language.
• But here’s the rub: Motion is relative.
What would a round trip, as the spaceship goes to another
planet and returns, look like to the stay-at-home twin?
Imagine or sketch the motion of the ship as seen by the twin
staying on earth.

From another viewpoint
• What would this same round trip look like from the
point of view of the twin in the spaceship?
• Remember that they don’t see themselves as
moving, it is the earth that goes away and comes
back!
• Imagine or sketch the motion of the ship as seen by
the twin who is travelling on the ship

Who’s younger?
• What does the twin on the ship (the “travelling
twin”) say about the Earth’s motion?
• Whose clock does the travelling twin see as
running slow?
• Which twin should be younger according to the
travelling twin?

The In-Between
• We know that during the trip out and during the trip back
both the travelling twin and the stay-at-home twin see
the other twin as moving near light speed.
• What will they say about one another’s clocks?
• What will the travelling twin experience at the turnaround point (what would it feel like on the ship)?
• What will the stay-at-home twin experience at the turnaround point (what would it feel like on the earth?)

• What is different?
What you need to know:
• For this explanation to make sense you need
to understand a few things about spacetime
diagrams.
• There is another powerpoint about this, which
you can look at. I’ll give a quick summary here
or you can skip that and go straight to the
explanation.
Intro to
spacetime
diagrams
Cut to the chase!
Mapping the
twin paradox
Space, time, and spacetime
• One of the key ideas which emerges from special
relativity is the fact that space and time are not
separate things, but components of one thing,
spacetime.
• Thus we can measure time in metres, or distance in
seconds.
• And different observers can have their time and
space axes pointed in different directions (which is
responsible for all the ‘strange’ effects of special
relativity)
Spacetime diagrams
• Spacetime diagrams show this 4D spacetime with 2
(or sometimes 3) dimensions by showing only one
direction in space (sometimes 2), and using the other
direction for time.
ct
ct
c
c
y
x
1D space, 1D time
x
2D space, 1D time
Spacetime diagrams are like traditional position-time
diagrams BUT time goes vertically by convention.
So as time passes things are ‘copied up’:
Same point in
space at
different times
2)
3)
time
time
3)
Different points
in space at
different times
2)
1)
1)
space
space
Standing Still
Running
time
2)
3)
time
The path in
spacetime is
called a
“world-line”
3)
2)
1)
1)
space
space
Notice that
for a moving
observer
the worldline is
slanted.
The time and space axes for a moving observer tilt in toward
the light speed line (45 if time is converted to the same units
as space by multiplying by c)
This is the moving observer’s
time
it represents
the
This isaxis…
the moving
observer’s
location
of it
therepresents
observer the
at
space axis…
different
in time.
“now” of moments
the observer.
ct'
ct
c
B
A
D
E
x'
x
In terms of the moving
observer’s space and time
coordinates, what is the same
for the two dots shown on this
axis (marked A
D and B)
E)??
stationary
moving

The size and direction of the coordinate axes change, depending on
how the one frame moves relative to the other.
c
rest
How do the time axis
(“here”) and the space
axis (“now”) change as the
relative speed increases?
time
faster
fast
slow
What is the limit as speed
gets bigger and bigger?
(click to increase speed!)
space
Changes in rate are due to the changing direction of the time axis… but the
changes in what “now” means are also important to understand the resolution
of the twin ‘paradox’.

Details
• The next few slides show the trip, relative to
the stay-at-home frame.
• We will use this frame because it remains
constant throughout the trip. Later you will
have the chance to see the trip from the
travelling twin’s view too (the result is the
same)
• The key idea to keep in mind is that the point
where the ship turns around, although brief, is
very important.
To make the numbers simple we will regard the travelling twin
as travelling at 0.866c during the trip (=2) to a planet 10.4 ly
away (this distance was chosen so that the trip time to
destination = 12 years in earth frame).
Ship
Earth
Destination
Planet
The time and space axes of the stay-at-home frame are in black.
The axes of the travelling frame are in blue.
et
ctplan
Starting out
This line shows
the velocity of
the rocket (its
world-line)
Light
Notice that right away
the ship and the earth
would describe very
different
times
asin“the
Which
of these
points
the
history
the destination
sameoftime
on the planet
would
someoneplanet
on earthassay is
destination
at “the same time” as the ship
the time the ship left”
This lineleaves
showsearth?
the space Here
axis the travelling
Which
“the same time” as the
of the ship
(its isleaves
twin
the
earth
ship
leaves
earth
in
the
now-line) in the ship, alreadyframe
of the ship?
travelling at 0.866c.
xplanet(now for planets)
Planet
Earth
Ship
(v)
Planet
Relativity

et
ctplan
Half way
Light
How much time has
passed on earth at this
point, from the point of
view of the travelling
twin?
The travelling twin is
How half
doesway
thistocompare
now
the
to the time that
has
destination
planet.
passed on earth, from
the point of view of the
stay-at-home twin?
xplanet(now for planets)
Ship
(v)
Planet
Earth
Planet
Relativity

ctearth
Arriving at the Destination
Light
Which point in the
• earth’s
The ship
has now
history
reached its to the
corresponds
destination.
The
time
of the ship’s
How
do in
the
times
travelling
twin
arrival
the
forearth’s
the now
trip
must
slow
frame?
down and
stop.two
compare
in the
Which
point
corresponds to the
frames?
arrival in the ship’s
frame?
xplanet(now for planets)
Planet
Earth
Ship
(v)
Planet
Relativity

Arriving at the Destination
ctearth
Now, as the ship slows down to turn around, watch what happens to the earth time
that corresponds to the ship’s NOW. (click to begin)
ctship
Light
Ship
(v=0)
Planet
Earth
Ship
(v)
Ship
(v)
Planet
Relativity
xplanet(now for planets)
ctearth
Return
Light
Light
ctship
Light
Ship
(v)
Ship
(v)
Planet
Earth
Ship
(v=0)
Planet
Relativity
Now the travelling
twin must begin
the trip back.
After you click,
notice how the
travelling twin’s
“now” continues to
sweep across the
world-line of the
stay-at-home twin.
(click to begin trip back!)
xplanet(now for planets)
t
t
ctplane ctplane
… and back again!
Light
Light Light
Finally the trip
back, with the
usual rotation
factors.
(click to begin trip back!)
Ship
(v)
Ship
(v)
Ship
(v)
xplanet(now for planets)
Planet
Earth
Planet
Relativity
ctplanet
v=0.866c
=2
Trip Out
Now with numbers!
How much time does the trip
to the planet take according to
the stay-at-home twin
(as seen from earth’s now)?
Distance =
10.4 ly
xplanet(now for planets)
Planet
Earth
Planet
Relativity

ctplanet
v=0.866c
=2
Trip Out
Now with numbers!
How much time has passed
for travelling twin:
(slowed by a factor of )?
Time that
has passed
for stay-athome twin
= 12 years
xplanet(now for planets)
Planet
Earth
Planet
Relativity

ctplanet
v=0.866c
=2
Time that
has passed
for stay-athome twin
= 12 years
Trip Out
Now with numbers!
To the travelling twin it is
the stay-at-home twin who
is moving at 0.866c, and so
the stay-at-home twin’s
clock that is slow:
(by a factor of )
How much time does the
travelling twin say has
passed for the stay-at-home
Time that has twin during the 6 year trip?
Time on earth
relative to
SHIP’S NOW
= 3 years
passed for
travelling
twin = 6 years
xplanet(now for planets)
Planet
Earth
Planet
Relativity

ctplanet
Time on
earth relative
to SHIP’S
NOW = 3
years
Trip Back is much the same!
Time that has
passed for
travelling
twin = 6 years
Time that
has passed
for stay-athome twin
= 12 years
The return trip is a
reverse of the trip
out, with the same
times all around.
xplanet(now for planets)
Planet
Earth
Planet
Relativity
ctplanet
What is the
total time that
has passed for
stay-at-home
twin?
For the whole trip
What is the total
time that has
passed for the
travelling twin?
The travelling twin sees the
time on earth as partly having
passed during the trip, and
partly “swept over” during
the turn around.
How much earth-time does
each of these correspond to?
xplanet(now for planets)
Planet
Earth
Planet
Relativity
(summary)
ctplanet
Summary for the whole trip
Total time that has
passed for
travelling twin =
6+6 = 12 years
Total Time that
has passed for
stay-at-home
twin = 24 years
The travelling twin sees the
time on earth as
3+3= 6 years
while travelling
Plus 18 years swept over
during the turn around.
6 + 18 = 24 years on earth.
xplanet(now for planets)
Planet
Earth
Planet
Relativity
So that’s the resolution of the ‘paradox’
• Everyone agrees about how much total time has
passed for each twin.
• The apparent symmetry between the two trips is
broken by the act of changing frames, during
which the travelling twin’s ‘now’ “sweeps
through” the missing time.
Extra: See the trip
from the travelling
twin’s coordinates
too!
The change of frames of the travelling twin is
not relative, and the views are not symmetric!
• The act of turning around makes the view of the stay-athome twin different from the travelling twin, no matter
whose point of view you follow.
• When the travelling twin changes frames, the meaning of
“now” changes for the traveller, and their coordinates
are very different, including their own view of their past
motion.
• Thus changing frames (accelerating) is not relative. But
we knew that…
(you can FEEL an acceleration, even in a closed room!)
The real issue is what the
twins are going to do about
the asymmetry of number of
Birthday Presents!!
The End
Unless you want a quick aside on what the twins actually SEE each other’s
clocks doing on the trip (not the same as the times they calculate).
click this button for the extra notes, anywhere else to end!
Signals take time to travel
Is received here
ct
Because signals (or
images or whatever) can
travel no faster than the
speed of light, the times
when signals from earth
reach the spaceship (or
signals from the
spaceship reach earth)
are not necessarily
spaced out just according
to the rate time seems to
flow.
A light signal from
here
To understand what we ‘see’ we
have to track the signals
• This travel time means that we actually see events when
their signals catch up to us (or we intercept them).
• For example, we saw that during the ship’s turn around
the ship’s ‘now’ sweeps through 18 years of the earth’s
time.
• But that doesn’t mean that the twin on the ship “sees”
18 years pass on earth – it means that 18 years of earth
history that they called ‘future’ they now call ‘past’. But
news from that past still has not reached the ship.
‘What you gets is what you sees’
• Let’s track signals to see what you would actually
receive in the way of signals from earth if you were
the travelling twin.
• We’ll assume that the ship sets out on the twin’s
birthday, and each twin sends the other a birthday
greeting each year.
ctplanet
What the travelling twin sees:
On the trip out the signals
have to catch up to the
ship. Estimate, from the
graph, how much time
passes on the ship before
the first birthday greeting
is received?
From the graph, about
how many signals a year
does the ship encounter as
it returns?
Ticks mark
birthdays
xplanet(now for planets)
Planet
Earth
Planet
Relativity

ctplanet
What the stay-at-home twin sees:
Coming back the ship is
rapidly following its
signals, so they will come
in very rapidly.
About how many are
The
stay-at-home
received
per year?twin
also gets only infrequent
birthday greetings during
the outward part of the
trip. How many years
apart are the birthday
messages?
xplanet(now for planets)
Planet
Earth
Planet
Relativity

If you do the math on this expansion/compression of time
(and frequency) you get exactly the relativistic Doppler
effect… which perhaps is not a surprise if we think about it!
Calculate the ratio between the frequency of signals sent and
received at the relative speed of the two ships.
When does the travelling twin get slowed down signals?
When do they get signals that are sped up?
What about the stay-at-home twin?

From the point of view of the stay-at-home twin
the ship sends 6 signals while moving away from
the earth.
ct
Earth
Going that
the other
wayofthe
ship sendsis63.73,
signals,
Given
the ratio
frequencies
how
but now
they
are
received
1/3.73
years
many
years
will
pass
on earth
before
all apart.
those
How many
years will pass on earth before all
signals
are received?
(Include
least 1 decimal
place in your results)
those atsignals
are received?
(Include at least 1 decimal place in your results)
How much time passes on earth during
this whole process? (What is the total
time taken to get all the birthday
messages?)
x
Earth
Planet

From the point of view of the travelling twin the
signals from earth are spaced out as the earth
moves away.
As thethat
earth
the ship again
signals
Given
theapproaches
ratio of frequencies
is 3.73,
how
are received
morefrom
often.
Howare
many
many
birthdaymuch
greetings
earth
signals are
bythe
theearth
ship during
received
by received
the ship as
movesthis
away?
(Include
at least
decimal place in your results)
part of
the1 voyage?
ct
Ship
(Include at least 1 decimal place in your results)
How many birthday greetings are
received by the ship in total? (What is the
total number of birthday messages the
ship receives?)
x
Earth
Planet

This can also be seen in the travelling twin coordinates.
We looked at what messages the travelling twin receives, but we still
used the earth coordinates while doing so!
You might want to look at this using the actual (changing) ship
coordinates.
Warning: it’s a little messy, because we have to switch frames half
way through. But you can see what the messages are really like from
the travelling twin’s perspective. Your call!
No thank’s…
I’m satisfied.
Skip it!
Show me all
the gory
details!
The 1st part of the trip in Ship coordinates. Notice that during the
first 6 years for the ship the earth moves away from the ship, but not
all signals sent are received by the ship. How many signals will the
ship receive, given that the ratio of frequencies is 3.73?

Given
ratio of
signals
is 3.73,
many signals will be received in the
NOW that
herethe
is where
the
shift of
framehow
happens!
next
ship
years? frame. In this frame what WAS the present on earth is
The 6
ship
changes
What
is the
total number
of signals
twin? wish…
now the
past…The
next signal
to bereceived
receivedby
is the
the travelling
second birthday
What
is the years
total number
of signals
sent
byframe)
the travelling
how many
ago (relative
to this
new
was thetwin?
signal sent?

When
the twin
on earth will
the 24th
We’vethey
seenfinally
that ifmeet
we count
the messages
we be
getcelebrating
just what our
birthday
since the
the spacetime
travelling twin
left, while
the travelling
twin twin
will be
analysis using
diagrams
requires:
The travelling
th!
celebrating
their 12messages
sends 12 birthday
and gets 24, and the stay-at-home twin
sends 24 ang gets 12. They are aged just the amount we calculated.
The twins can still celebrate together, but they are no
longer the same age!
This time really…
The End
(And many happy returns!)