The%20Twin%20Paradox

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The Twin Paradox
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A quick note to the reader

• This is intended as a supplement to my workshop
  on special relativity at EinsteinPlus 2012
• Ive 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 couldnt 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!

?
3
Hey! That one was just an example!! ) But your
keenness does earn you a reward! Should you find
that you have questions that are not answered or
answers to questions I should have asked but
didnt or anything else you can drop me a line
at PhilipF_at_sphericalcows.net We now return
you to your regularly scheduled power point!
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Outline
Title Intro(you are here)
Description of Paradox
Click on any box to go to that part, or just
click anywhere else to continue to the next slide!
Because there are a number of choices you can
make as you go through this presentation, I
thought it might be helpful to give you an
outline of the different parts right away.
Intro to Spacetime Diagrams
Qualitative mapping the paradox
Calculationscomputing the times
From the Travelling Twins view
The twin paradox The doppler effect
End
5
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

• Lets take a little time to look at what is
  behind the twin paradox which isnt really a
  paradox at all, just an example of how we carry
  our everyday ideas of time into our
  understanding. Even when we are trying not to!

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Introduction to the Twin Paradox

• In our study of special relativity we have
  learned that moving clocks run slow.

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)
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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 twins clock?
• Suppose that the travelling twins 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 twins clock, how long will it take
  on the travelling twins?

Hey sib! Better fix your clock!
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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 isnt 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?

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The issue

• So far this is weird but it isnt a paradox.
  There is nothing contradictory about this except
  language.
• But heres 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.
?
10
The issue

• The ship travels to the destination planet, then
  turns and comes back this is what we usually
  view as THE motion.

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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 dont 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

?
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From another viewpoint

• From the point of view of the twin on the ship
  the ship stays in one place (right where the twin
  is!) while the earth leaves and then reverses
  and comes back!

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Whos younger?

• What does the twin on the ship (the travelling
  twin) say about the Earths motion?
• Whose clock does the travelling twin see as
  running slow?
• Which twin should be younger according to the
  travelling twin?

?
?
14
Whos younger?

• Since the travelling twin sees the Earth as
  moving, they will see the stay-at-home twins
  clock running slow, not theirs.
• So shouldnt it be the stay-at-home twin who is
  younger?

?
There is no quick answer here! This question is
what the whole power point is about! ?
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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 anothers clocks?

• What will the travelling twin experience at the
  turn-around point (what would it feel like on the
  ship)?
• What will the stay-at-home twin experience at the
  turn-around point (what would it feel like on the
  earth?)
• What is different?

?
16
The In-Between

• We know that during the trip out both the
  travelling twin and the stay-at-home twin see the
  other twins clocks running slow.
• And that the same thing is true during the trip
  back.
• So we might expect that if something weird is
  happening it must be during the moments BETWEEN
  the trip out and the trip back.

That is indeed the case!
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TURNING AROUND

• If the ship turns around very fast then the
  travelling twin will feel some very strong forces
  as the ship reaches its destination!(or the
  destination reaches the ship, from the travelling
  twins point of view!)
• But the stay-at-home twin doesnt feel anything
  at all, even when the travelling twin sees the
  earth reverse and come back!
• There is something very different about the
  frame(s) of the two twins.

And thats what were exploring here!
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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. Ill 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
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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)

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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.

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Spacetime diagrams are like traditional
position-time diagrams BUT time goes vertically
by convention.
So as time passes things are copied up
22
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 observers time axis it
represents the location of the observer at
different moments in time.
This is the moving observers space axis it
represents the now of the observer.
In terms of the moving observers space and time
coordinates, what is the same for the two dots
shown on this axis (marked A and B) ?
In terms of the moving observers space and time
coordinates, what is the same for the two dots
shown on this axis (marked D and E)?
?
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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)
The moving observers time axis represents the
location of the observer at different moments in
time (all the same place for that observer). The
dots on this axis are all at the same place
relative to this observer. So A and B are the
same place (at different times) for this observer
The moving observers space axis represents the
now of the observer (all the same time for that
observer). The dots on this axis are all at the
same time relative to this observer. So DE
happen at the same instant (for the moving
observer)
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The size and direction of the coordinate axes
change, depending on how the one frame moves
relative to the other.
c
How do the time axis (here) and the space axis
(now) change as the relative speed
increases?What is the limit as speed gets
bigger and bigger? (click to increase speed!)
time
space
?
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The faster the one frame moves relative to the
other, the more the axes for the moving frame
converge toward the light speed line.
c
Notice how both the time axis (here) and the
space axis (now) shift and stretch as the
relative speed increases.
time
space
The diagonal line (light speed) is the middle
line for all frames. It is the ultimate limit as
speed increases (since the axes will meet).
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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
  twins 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.

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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).
The time and space axes of the stay-at-home frame
are in black. The axes of the travelling frame
are in blue.
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Starting out
Notice that right away the ship and the earth
would describe very different times as the same
time on the destination planet as the time the
ship left
?
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Starting out
The travelling twin in the ship and the
stay-at-home twin on earth see different events
in the destination planets history as the same
time as the ship sets out.
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Half way
How much time has passed on earth at this point,
from the point of view of the travelling twin?
How does this compare to the time that has passed
on earth, from the point of view of the
stay-at-home twin?
?
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Half way
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Arriving at the Destination
ctearth
Which point in the earths history corresponds to
the time of the ships arrival in the earths
frame? Which point corresponds to the arrival in
the ships frame?

• The ship has now reached its destination. The
  travelling twin must now slow down and stop.

How do the times for the trip compare in the two
frames?
?
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Arriving at the Destination
How do the times for the trip compare in the two
frames?
ctearth
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Arriving at the Destination
Now, as the ship slows down to turn around, watch
what happens to the earth time that corresponds
to the ships NOW. (click to begin)
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Return
ctearth
Now the travelling twin must begin the trip
back.
Light
After you click, notice how the travelling twins
now continues to sweep across the world-line of
the stay-at-home twin. (click to begin trip back!)
Ship(v0)
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and back again!
Finally the trip back, with the usual rotation
factors.
(click to begin trip back!)
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Trip Out Now with numbers!
v0.866c?2
ctship
How much time does the trip to the planet take
according to the stay-at-home twin(as seen from
earths now)?
Distance 10.4 ly
?
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Trip Out Now with numbers!
v0.866c?2
ctship
Time that passed for stay-at-home twin(as seen
from earths now)
Time that has passed for stay-at-home twin 12
years
 
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Trip Out Now with numbers!
v0.866c?2
ctship
How much time has passed for travelling
twin(slowed by a factor of ?)?
Time that has passed for stay-at-home twin 12
years
?
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Trip Out Now with numbers!
v0.866c?2
ctship
How much time has passed for travelling
twin(slowed by a factor of ?)?
Time that has passed for stay-at-home twin 12
years
 
Time that has passed for travelling twin 6 years
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Trip Out Now with numbers!
v0.866c?2
To the travelling twin it is the stay-at-home
twin who is moving at 0.866c, and so the
stay-at-home twins clock that is slow(by a
factor of ?)
ctship
How much time does the travelling twin say has
passed for the stay-at-home twin during the 6
year trip?
?
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Trip Out Now with numbers!
v0.866c?2
ctship
Notice that the earth and the ship disagree about
how much time has passed on the earth during the
trip. This is because the ships now and the
earths now are very different.
To the travelling twin it is the stay-at-home
twin who is moving at 0.866c, and so their clock
is slow(by a factor of ?)
 
The earth and the ship do not agree as to the
time on earth that is at the same time as the
ships arrival at its destination!
43
Trip Back is much the same!
ctship
The return trip is a reverse of the trip out,
with the same times all around.
44
For the whole trip
What is the total time that has passed for the
travelling twin?
ctship
What is the total time that has passed for
stay-at-home 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?
(summary)
45
Summary for the whole trip
ctship
46
So thats 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 twins now sweeps
  through the missing time.

Extra See the trip from the travelling twins
coordinates too!
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The trip in the coordinates of the stay at home
twin. Watch how the positions and coordinates
behave. What changes? What stays the same?
Click to Start
Next
48
The turn around. What changes? What stays the
same?
Click to Start
Next
49
The trip back in the coordinates of the
stay-at-home twin.
Do you wonder why we bothered dividing this up?
Wait for the next view!
Click to Start
Next
50
The 1st part of the trip in Ship coordinates.
Here the Earth leaves the ship and the planet
comes to it. Notice how (except for the values
of the coordinates) this is very similar to the
previous.
Click to Start
Next
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The turn around. Watch closely. How is this
different from the view of the stay-at-home twin?
What changes and what is the same? Why is this so
different?
Click to Start
Next
52
The earth returns to the ship in the view of the
travelling twin. Notice how the first part of the
trip looks completely different. Was that true
for the stay-at-home twin?
Click to Start
Next
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In this frame the earth has been moving toward
the ships present position the whole time, at
0.866c.
Relative to this frame the earth is moving toward
the ships position, but the in the first part of
the trip the ship was moving toward this position
faster. The ship travelled 41.6 ly in 42 years,
so relative to this frame it was going at 0.990c.
This is exactly what relativistic velocity
addition gives.
It will arrive after 24 years on earth 48 years
in this frame (since the moving earths clocks
run slow by a factor of ? 2
 
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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-at-home 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!)

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The End
Unless you want a quick aside on what the twins
actually SEE each others clocks doing on the
trip (not the same as the times they
calculate).click this button for the extra
notes, anywhere else to end!
56
Extra What would the twins really see?

• We are sometimes rather loose with the way we
  talk about things in discussing the different
  frames in relativity.
• We will say things like the stay-at-home twin
  sees the travelling twins clock running slow.
• But what we mean is that relative to the
  stay-at-home twin the travelling twins clock is
  running slow and if the stay-at-home twin works
  out how quickly the travelling clock is running,
  or waits to see what is reported when the ship
  arrives, they will find the time is slowed
• You cant totally blame us for the shortcut but
  its misleading!

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Signals take time to travel
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.
58
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 ships turn
  around the ships now sweeps through 18 years
  of the earths time.
• But that doesnt 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.

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What you gets is what you sees

• Lets track signals to see what you would
  actually receive in the way of signals from earth
  if you were the travelling twin.
• Well assume that the ship sets out on the twins
  birthday, and each twin sends the other a
  birthday greeting each year.

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What the travelling twin sees
ctship
Ticks mark birthdays
?
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What the travelling twin sees
Notice that on the trip out the signals have to
catch up to the ship, so only 1 is received along
the way (and one hasnt quite gotten there when
the ship arrives). The messages are more than 3
years apart (about 3.7 years if you measure
carefully)
ctship
The returning ship encounters just over 22
birthday greetings in 6 years, or about 3.7 a
year!
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What the stay-at-home twin sees
ctship
Coming back the ship is rapidly following its
signals, so they will come in very rapidly. About
how many are received per year?
The stay-at-home twin also gets only infrequent
birthday greetings during the outward part of the
trip. How many years apart are the birthday
messages?
?
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What the stay-at-home twin sees
ctship
Coming back the ship is rapidly following its
signals, so they will come in very rapidly. All 6
are received by the stay-at-home twin during less
than 2 years as the travelling twin returns more
than 3 a year!
The stay-at-home twin gets just 6 birthday
greetings in the first 22 years of waiting more
than 3 years apart (3.7 years apart in fact).
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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?
?
65
 
The stay-at-home twin gets slowed down messages
as the ship travels away and then faster messages
while the ship travels back toward the Earth
The travelling twin gets slowed down messages as
the Earth travels away (6 years) and then faster
messages while the Earth travels back toward the
ship (another 6 years)
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From the point of view of the stay-at-home twin
the ship sends 6 signals while moving away from
the earth.
Going the other way the ship sends 6 signals, but
now they are received 1/3.73 years apart. How
many years will pass on earth before all those
signals are received? (Include at least 1
decimal place in your results)
ctship
Given that the ratio of frequencies is 3.73, how
many years will pass on earth before all those
signals 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?)
?
67
From the point of view of the stay-at-home twin
the ship sends 6 signals while moving away from
the earth.
Given that the ratio of frequencies is 3.73, a
total of 6 ? 3.73 22.4 years pass on earth
while waiting for those messages to arrive.
ctship
Going the other way the ship sends 6 signals, but
now they are received 1/3.73 years apart. So
6/3.73 1.6 years for those signals to arrive.
So all the birthday messages take a total of
22.4y 1.6 y 24.0 years to arrive. So 24 years
pass for the stay-at-home twin, who gets 12
birthday greetings from the travelling twin!
68
From the point of view of the travelling twin the
signals from earth are spaced out as the earth
moves away.
As the earth approaches the ship again signals
are received much more often. How many signals
are received by the ship during this part of the
voyage? (Include at least 1 decimal place in
your results)
Given that the ratio of frequencies is 3.73, how
many birthday greetings from earth are received
by the ship as the earth moves away? (Include at
least 1 decimal place in your results)
ct
ctship
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
?
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From the point of view of the travelling twin the
signals from earth are spaced out as the earth
moves away.
Given that the ratio of frequencies is 3.73,
there is time to receive only 6/3.73 1.6
signals
ct
ctship
As the earth approaches the ship again signals
are received much more often at a rate of 3.73 a
year during the next 6 years 3.73/y ?6y 22.4
signals
So during the 12 years that pass for the ship a
total of 24 birthday messages are received which
agrees with our calculations!
x
Earth
Planet
70
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 its 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 twins
perspective. Your call!
Show me all the gory details!
No thanks Im satisfied. Skip it!
71
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?
?
72
The birthday wishes from the stay-at-home twin
are received by the travelling twin every 3.73
years, so a total of 6.0y/3.73y 1.61 signals
are received.
73
Given that the ratio of signals is 3.73, how many
signals will be received in the next 6 ship
years? What is the total number of signals
received by the travelling twin? What is the
total number of signals sent by the travelling
twin?
NOW here is where the shift of frame happens!
The ship changes frame. In this frame what WAS
the present on earth is now the pastThe next
signal to be received is the second birthday
wish how many years ago (relative to this new
frame) was the signal sent?
?
74
The second birthday wish was sent 38 years ago
(relative to the new frame) but is just now
reaching the ships position. The remaining
signals will arrive by the time the earth reaches
the ship.
So, in the next 6 years a total of 6 ? 3.73
22.4 signals are received (most of these sent
long before but only just now reaching the ship).
Thus we see that the travelling twin gets a total
of 1.6 22.4 24 birthday greetings from the
stay-at-home twin. The travelling twin sent 66
12 birthday greetings.
75
When they finally meet the twin on earth will be
celebrating the 24th birthday since the
travelling twin left, while the travelling twin
will be celebrating their 12th! The twins can
still celebrate together, but they are no longer
the same age!
Weve seen that if we count the messages we get
just what our analysis using the spacetime
diagrams requires The travelling twin sends 12
birthday messages and gets 24, and the
stay-at-home twin sends 24 ang gets 12. They are
aged just the amount we calculated.
(And many happy returns!)
76
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