### How special relativity predicts time dilation, length contraction and simultaneity loss

[Two ships in the night. Bob sees Alice travelling to the right at half the speed of light. From Bob’s frame of reference, he is at rest and Alice’s clock is running slow. Alice’s frame of reference has Alice at rest while Bob travels to the left. According to Alice, Bob’s clock is running slow. When they meet again, who will have aged more?]

The twin paradox is one way Einstein’s special relativity messes with time. Alice and Bob are twins. Alice takes off in a rocket to Alpha Centauri and returns several years later. Less time passes for Alice than for Bob. We call this time dilation. Alice is now younger than her twin.

But wait! Doesn’t relativity say that Alice will see Bob’s clock run slow as well? Then won’t he be the younger twin when she gets back?

Paradox! The universe implodes. Marty McFly is never born. Mary Hatch becomes a spinster librarian.

Special relativity hangs on two commandments:

1. Thou shalt not differentiate between inertial frames of reference.
2. Thou shalt keep the speed of light constant in all reference frames.

An inertial frame of reference is one for which the net force is zero. A spaceship moving in a straight line at a constant speed has an inertial frame of reference. It’s indistinguishable from a spaceship at rest. We don’t speak of velocity in absolute terms. We can only speak of relative velocity.

The quick answer to the paradox is that Alice does not always have an inertial frame of reference. She breaks from this frame three times:

1. She accelerates when she leaves Earth.
2. She turns around to return.
3. She decelerates when she arrives back at Earth.

We’ll chart Alice’s journey from each reference frame. We’ll begin with what happens. This will help us see that Bob and Alice both agree. Upon her return to Earth, Alice is the younger twin. After this, we’ll derive some formulae to determine why.

During our exploration, we’ll examine three effects of special relativity.

1. Time dilation — a clock in motion runs slow
2. Length contraction — a body in motion is contracted along the axis of motion. In the frame of the moving body, distances traveled are contracted.
3. Loss of simultaneity — a clock in the rear of a moving body will be ahead of a clock in the front (if they are synchronized when the body is at rest).

### The What

Bob’s Frame

Alice takes off, traveling at ⅗ light speed. Three clocks are synchronized:

• Bob’s clock: 0
• Alice’s clock: 0
• Alpha Centauri clock: 0

Alice’s 10-metre ship has contracted to 8 metres (length contraction). Her clock runs slow: 4 years for every 5 of Bob’s years (time dilation).

Alpha Centauri is 5 light years away from Earth. When Alice arrives, 8⅓ years have passed. Because Alice’s clock is slow, only 6⅔ years have passed on her ship. (Fear not. We’ll justify these numbers shortly.)

Once Alice arrives back to Earth, another 8⅓ years have passed. She has been away for 16⅔ years. She has aged only 13⅓ years.

Alice’s Frame

From Alice’s frame, she and her ship are at rest. Bob and the Earth are receding at ⅗ light speed. Alpha Centauri approaches her at ⅗ light speed. Bob and the star are both contracted along the axis of motion. The distance to the star is also contracted to 4 light years.

Besides length contraction and time dilation, Alice notices a third effect. The clock on Alpha Centauri is ahead of the clock on Earth by 3 years. However, it runs at the same rate as Bob’s clock: 4 years to 5 of Alice’s years.

After 6⅔ years, the Earth is now 4 light years behind Alice. The star’s clock has advanced by 5⅓ years. It was ahead 3 years to begin with, so it now reads 8⅓ years.

Once Alice is back on Earth, another 6⅔ years have passed. Her total time away: 13⅓ years.

Bob’s clock has been running slow. It advances 5⅓ years during Alice’s 6⅔ years. Once the Earth begins its approach, the time gradient pushes Bob’s clock ahead of Alpha Centauri’s clock by 3 years. Now Bob’s clock reads 11⅓ years. After 5⅓ years (Bob time), he has aged 16⅔ years to Alice’s 13⅓ years.

Common Frame

Alice and Bob each differ on the progression of events. However, some events are frame-independent.

• Bob’s and Alice’s clocks are synchronized at 0 when Alice departs.
• Alice’s clock reads 6⅔ years when she is at Alpha Centauri.
• The clock on Alpha Centauri reads 8⅓ years when Alice is there.
• Alice’s clock reads 13⅓ years when she arrives back at Earth.
• Bob’s clock reads 16⅔ years when Alice arrives back at Earth.

These frame-independent events all share one feature. Each event is a snapshot taken when Alice is at either the star or the Earth. Alice could have crashed her ship into Alpha Centauri, breaking both clocks. Her clock would be stuck at 6⅔. The star clock would be stuck at 8⅓. Any observer should agree.

### The Why

The Gamma Factor

When Alice is training for the Periluminal Marathon, she and Bob measure the effect that motion has on time.

Bob and Alice each have identical running clocks. A light pulse travels from bottom to top. A detector at the top registers one tick.

As Alice runs, Bob notices that the light from the bottom of her clock must take a longer route to the top. However, the speed of light is the same for each clock. Bob’s clock counts one tick before Alice’s clock. From Bob’s point of view, Alice’s clock is running slow.

From Alice’s point of view, it’s Bob whose clock is slow.

So far, we have described this effect — time dilation — qualitatively. We’ll now quantitatively compare the two frames.

The diagram below shows distances traveled in terms of velocity × time. The subscripts identify the reference frame. The speed of light, c, is frame independent. So is the relative velocity, v.

Pythagoras helps us perform the algebra. Our result is a conversion factor between the two reference frames.

The conversion factor which results is ubiquitous in relativity calculations. Thus it gets its own name: the Gamma Factor. Its symbol is the lower-case Greek letter: γ. In the following expression, velocity is in terms of light speed. Half the speed of light is v = ½.

In Bob’s frame, Alice’s clock is slow when she leaves Earth. We can find out just how slow.

Loss of simultaneity

Bob and Alice can both see the Alpha Centauri clock with their phenomenal telescopes. Each sees the same: the clock reads -5 years. They each adjust for the time needed for light to travel from the clock.

Bob

Bob knows that Alpha Centauri is 5 light years away. The light from the clock to 5 years to arrive. Since it left 5 years ago, the clock must now read 0.

Alice

Alice has more work to do. She knows that Alpha Centauri is 4 light years away. But it’s been moving toward her at ⅗ light speed. How long ago did it leave the clock?

Alpha Centauri has traveled ⅗ the distance the light has traveled. The light has traveled an addition 4 light years to reach Alice’s telescope. That amounts to the remaining ⅖ of the light’s journey. The light has traveled for 10 years.

Alice also knows that the clock is slow. She applies the Gamma Factor of 1⅖ and determines that the clock has advanced 8 years since it read -5 years. She concludes that the clock now reads 3 years.

Here is Alice’s calculation, generalized:

The derivation of this formula relies on time dilation and length contraction. We can also make this calculation from first principles, as in the following video.

Following the formula in that video, we take the Andromeda clock reading according to Bob, and add a factor of Lv = 5×⅗=3. Here, L is the distance in Bob’s frame, and v is Alice’s velocity in the clock's direction. We express velocity as a fraction of light speed. Otherwise, the expression is divided by c².

Length Contraction

It may seem we snuck this in with little justification. If we derive loss of simultaneity on its own, we can then use it to explain length contraction.

Bob’s Frame

Alice flies to Alpha Centauri, a distance L at a velocity v. The time elapsed on Bob’s clock is L/v [5/(⅗)=8⅓]. Alice’s clock is slow, so it measures only L/γv [5/(⅗ × 1⅖) = 3⅔].

Alice’s Frame

The reading on Alice’s clock is frame independent. Alpha Centauri has approached her at velocity v. The trip took a time of L/(γv) by her clock.The distance traveled is time × velocity, L/γ (5 /1⅖ = 4).

Time dilation and length contraction imply one another.

### Conclusion

We have thus described and explained the twin paradox. Duing Alice’s journey, she and Bob each see the other’s clock running slow. However, the situation is not symmetrical. Alice has to change velocity three times. At the end of her journey, she is younger than her twin.