Physics 12SL�Option A�Special Relativity 7

Twin Paradox

The Twin Paradox as described by Mr. Freeman

• 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!

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)

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?
• 12 years according �to the traveling twin!

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 traveling twin returns they will be significantly younger than their identical Earth twin!

���

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.

The issue

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

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



To paraphrase

• 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!

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?

?



Who’s younger?

• Since the travelling twin sees the Earth as moving, they will see the stay-at-home twin’s clock running slow, not theirs.
• So shouldn’t 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! ☺

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?

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 twin’s point of view!)
• But the stay-at-home twin doesn’t 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 that’s what we’re exploring here!

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.

Cut to the chase! Mapping the twin paradox

To make the numbers simple we will regard the travelling twin as travelling at 0.866c during the trip (γ=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.

Starting out

Here are the positions of the earth, the ship, and the destination planet at the start of the trip (as seen by the earth)

The earth is not moving (in its frame) so it stays in the same place at all times (this is its world-line aka its time axis)

Beginning the Trip

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.

Half way

Arriving at the Destination

How do the times for the trip compare in the two frames?

ct_earth

Arriving at the Destination

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)

When the ship comes to rest (relative to the earth and planet) it is in the same frame as the planet and the earth…

so it has the same “now” line as the earth does (a horizontal line in the Earth coordinates).

“Now” has changed its meaning!

Light

Ship�(v=0)

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!)

ct_earth

… and back again!

Finally the trip back, with the usual rotation factors.

(click to begin trip back!)

ct_ship

Trip Out

Now with numbers!

v=0.866c�γ=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



ct_ship

Trip Out

Now with numbers!

v=0.866c�γ=2

Time that passed for stay-at-home twin�(as seen from earth’s now):

Time that has passed for stay-at-home twin = 12 years

ct_ship

Trip Out

Now with numbers!

v=0.866c�γ=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 γ)?



ct_ship

Trip Out

Now with numbers!

v=0.866c�γ=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 γ)?

Time that has passed for travelling twin = 6 years

ct_ship

Trip Out

Now with numbers!

v=0.866c�γ=2

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 γ)

How much time does the travelling twin say has passed for the stay-at-home twin during the 6 year trip?



ct_ship

Trip Out

Now with numbers!

v=0.866c�γ=2

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 γ)

Notice that the earth and the ship disagree about how much time has passed on the earth during the trip. This is because the ship’s “now” and the earth’s “now” are very different.����

The earth and the ship do not agree as to the time on earth that is at “the same time” as the ship’s arrival at its destination!

ct_ship

Trip Back is much the same!

The return trip is a reverse of the trip out, with the same times all around.

ct_ship

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)

ct_ship

Summary for the whole trip

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.

If you got all the way here, you’re amazing!

Assignment:

The Freeman Project

Listen to:

Chris Hadfield - Space Oddity

(2013)

Originally by David Bowie