the twin paradox. a quick note to the ‘reader’ this is intended as a supplement to my workshop...
TRANSCRIPT
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)
Qualitative: mapping the paradox
End
Calculations:computing the times
Intro to Spacetime Diagrams
The twin paradox &The doppler effect
Description of ‘Paradox’
From the Travelling Twin’s view
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.
Click on any box to go to that part, or just click anywhere else to continue to the next slide!
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 we carry our everyday ideas of time into our understanding. Even when we are trying not to!
Skip the Summary
Summary of Twin Paradox
Introduction to the Twin Paradox• In our study of special relativity we have
learned that moving clocks ‘run slow’.One tick of your clock One tick of moving clock
3.0m
Clock moves
> 3.0m
lightlight
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?
Hey sib! Better fix
your clock!
tearth
tship
v
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 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?
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.
x
ctc
1D space, 1D time
x
ctc
y
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’:
space
time Same point in
space at different times
Standing Still
1)
2)
3)
space
time
Running
Different points in space at different times
1)
2)
3)
space
time
1)
2)
3) The path in spacetime is called a “world-line”
space
time
1)
2)
3)
Notice that for a moving observer the world-line is slanted.
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 and B) ?
x
ct
c
stationary
ct'
x'
moving
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 space axis… it represents the “now” of the observer.
In terms of the moving observer’s space and time coordinates, what is the same for the two dots shown on this axis (marked D and E)?
A
B
DE
This is the moving observer’s time axis… it represents the location of the observer at different moments in time.
The size and direction of the coordinate axes change, depending on how the one frame moves relative to the other.
time
rest
space (now)
time
(her
e)
slow
c
space
time
(her
e)space (now)
fast
time (
here
)
space (now)
faster
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!)
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.
Ship(v)
x ship (
now for s
hip)
ct ship
Planet Earth
Planet Relativity
xplanet(now for planets)
ctpl
ane
t
Light
Starting out
Here the travelling twin leaves the earth in the ship, already travelling at 0.866c.
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”
This line shows the velocity of the rocket (its world-line)
This line shows the space axis of the ship (its now-line)
Which of these points in the history of the destination planet would someone on earth say is at “the same time” as the ship leaves earth?
Which is “the same time” as the ship leaves earth in the frame of the ship?
Ship(v)
Planet Earth
Planet Relativit
y
xplanet(now for planets)
ctpl
ane
t
ct ship
Light
x ship (now fo
r ship)
Half way
The travelling twin is now half way to the destination planet.
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?
Ship(v)
Planet Earth
Planet Relativit
y
xplanet(now for planets)
Light
ct ship
x ship (
now for s
hip)
Arriving at the Destination
• The ship has now reached its destination. The travelling twin must now slow down and stop.
Which point in the earth’s history corresponds to the time of the ship’s arrival in the earth’s frame?Which point corresponds to the arrival in the ship’s frame?
How do the times for the trip compare in the two frames?
ctea
rth
Ship(v)
Planet Earth
Planet Relativit
y
xplanet(now for planets)
ctea
rth
Light
ct ship
x ship (
now for s
hip)
Ship(v)
Planet Earth
Planet Relativit
y
xplanet(now for planets)
ctsh
ip
Light
x ship (now for ship)
Planet Earth
Planet Relativit
y
xplanet(now for planets)
Light
ct
ship
xship (now for ship)
Ship(v=0)
Arriving at the DestinationNow, 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)
Light
Ship(v=0)
Ship(v)
Planet Earth
Planet Relativity
xplanet(now for planets)
Light
ctship
xship (now for ship)
Ship(v)
Planet Earth
Planet Relativity
xplanet(now for planets)
Light
ctship
xship (now for ship)
Planet Earth
Planet Relativit
y
xplanet(now for planets)
Light
ct
ship
xship (now for ship)
Return
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!)
ctea
rth
Ship(v)
Planet Earth
Planet Relativity
xplanet(now for planets)
ctpl
anet
Light
ctship
xship (now for ship)
… and back again!
Ship(v)
Planet Earth
Planet Relativity
xplanet(now for planets)
Light
ctship
xship (now for ship)Ship
(v)
Planet Earth
Planet Relativit
y
xplanet(now for planets)
ctpl
anet
Light
ctship
xship (now for ship)
Finally the trip back, with the usual rotation factors.
(click to begin trip back!)
Planet Earth
Planet Relativity
xplanet(now for planets)
ctpl
anet
ctship
Trip OutNow with numbers!
v=0.866c=2
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
Planet Earth
Planet Relativity
xplanet(now for planets)
ctpl
anet
ctship
Trip OutNow with numbers!
v=0.866c=2
Time that has passed for stay-at-home twin = 12 years
How much time has passed for travelling twin:(slowed by a factor of )?
Planet Earth
Planet Relativity
xplanet(now for planets)
ctpl
anet
ctship
Trip OutNow with numbers!
v=0.866c=2
Time that has passed for stay-at-home twin = 12 years
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 )
Time that has passed for travelling twin = 6 years
How much time does the travelling twin say has passed for the stay-at-home twin during the 6 year trip?
Time on earth relative to SHIP’S NOW = 3 years
Planet Earth
Planet Relativity
xplanet(now for planets)
ctpl
anet
ctship
Trip Back is much the same!
Time that has passed for stay-at-home twin = 12 years
The return trip is a reverse of the trip out, with the same times all around.
Time that has passed for travelling twin = 6 years
Time on earth relative to SHIP’S NOW = 3 years
Planet Earth
Planet Relativity
xplanet(now for planets)
ctpl
anet
ctship
For the whole trip
What is the total time that has passed for the travelling twin?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)
Planet Earth
Planet Relativity
xplanet(now for planets)
ctpl
anet
ctship
Summary for the whole trip
Total Time that has passed for stay-at-home twin = 24 years
Total time that has passed for travelling twin = 6+6 = 12 years
The travelling twin sees the time on earth as 3+3= 6 years while travellingPlus 18 years swept over during the turn around.6 + 18 = 24 years on earth.
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-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!)
The real issue is what the twins are going to do about the asymmetry of number of Birthday Presents!!
The EndUnless 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 travelBecause 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.
ct
ct
A light signal from here
Is received 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.
Planet Earth
Planet Relativity
xplanet(now for planets)
ctpl
anet
ctship
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
Planet Earth
Planet Relativity
xplanet(now for planets)
ctpl
anet
ctship
What the stay-at-home twin sees:
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?
Coming back the ship is rapidly following its signals, so they will come in very rapidly.About how many are received per year?
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?
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)
From the point of view of the stay-at-home twin the ship sends 6 signals while moving away from the earth.
Earth
Earth Planet
x
ct
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?)
From the point of view of the travelling twin the signals from earth are spaced out as the earth moves away.
Ship
Earth Planet
x
ct
ctship
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)
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)
How many birthday greetings are received by the ship in total? (What is the total number of birthday messages the ship receives?)
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!
Show me all the gory details!
No thank’s… I’m satisfied.
Skip it!
This can also be seen in the travelling twin coordinates.
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?
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 past…The next signal to be received is the second birthday wish… how many years ago (relative to this new frame) was the signal sent?
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?
We’ve 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.
This time really…
The End
(And many happy returns!)
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!
