The Twin paradox, from both sides now (with apologies to Joni)

 I will give an exposition on the twin paradox, using an example I read in a book about 4 decades ago, so I’m relating this from memory.

Imagine that one of the twins goes to visit an extra-terrestrial world 20 light years away in a spaceship that can travel at 4/5 the speed of light. The figures are chosen because they are easy to work with and we assume that acceleration and stopping are instantaneous. We also assume that the twin starts the return journey as soon as they arrive at their destination.

 

From the perspective of the twin on Earth, the trip one-way takes 25 years because the duration is T = s/v, where s = 20 (light years) and v = 4/5c. 

So 5/4 x 20 = 25.

 

From the perspective of the twin on the spaceship, their time is determined by the Lorentz transformation (γ).

 

γ = 1/√(1 – v²/c²)

 

Note v²/c² = (4/5)² = 16/25

So √(1 – v²/c²) = √(9/25) = 3/5

 

Now true time for the space ship (τ) is given by τ = T/γ

So for the spaceship twin, the duration of the trip is 3/5 x 25 = 15

So the Earth twin has aged 25 years and the spaceship twin has aged 15 years.

 

But there is a relativistic Doppler effect, which can be worked out by considering what each twin sees when the spaceship arrives at its destination. 

 

Note that light, or any other signal, takes 20 years to come back from the destination. So the Earth twin will see the space ship arrive 25 + 20 = 45 years after it departed. But they will see that their twin is only 15 years older than when they left. So, from the Earth twin’s perspective, the Doppler effect is a factor of 3. (3 x 15 = 45). So the Doppler effect slowed time down by 3. Note: 45 years has passed but they see their twin has only aged 1/3 of that time.

 

What about the spaceship twin’s perspective? They took 15 years to get there, but the Doppler effect is a factor of 3 for them as well. They’ve been receiving signals from Earth ever since they left so they will see their twin only 5 years older because 15/3 = 5, which is consistent with what their twin saw. In other words, their Earth twin has aged 5 years in 15, or 1/3 of their travel time. 

 

If spaceship twin was to wait another 20 years for the signal to arrive then it would show Earth twin had aged 5 + 20 = 25 years at the time of their arrival. But, of course, they don’t wait, they immediately return home. Note that the twins would actually agree on each other’s age if they allowed for the time it takes light to arrive to their respective locations.

 

So what happens on the return trip? The Lorentz transformation is the same for the spaceship twin on the return trip, so they only age another 15 years, but according to the Earth twin the trip would take another 25 years, so they would have aged 50 years compared to the 30 years of their twin.

 

But what about the Doppler effect? Well, it’s still a factor of 3, only now it works in reverse, speeding time up. For the spaceship twin, their 15 years of observing their Earth twin is factored by 3 and 15 x 3 = 45. And 5 + 45 = 50, which is how much older their twin is when they arrive home.

 

For the Earth twin, their spaceship twin’s round trip is 50 years, so the return trip appears to only take 5 years. And allowing for the same Doppler effect, 3 x 5 = 15.

When the Earth twin adds 15 to the 15 years they saw after 45 years, they deduce the age of their spaceship twin is 30 years more (against their own 50). So both twins are in agreement.

 

Now, the elephant in the room is why do we only apply the Lorentz transformation to the spaceship twin? The usual answer to this question is that the spaceship twin had to accelerate and turn around to come back, so it’s obvious they did the travelling.

 

But I have another answer. The spaceship twin leaves the surface of Earth and even leaves the solar system. It’s obvious that the spaceship didn’t remain stationary while the solar system travelled through the cosmos at 4/5 the speed of light. There is an asymmetry to the scenario which is ultimately governed by the gravitational field created by everything, and dominated by the solar system in this particular case. In other words, the Lorentz transformation only applies to the spaceship twin, even when they only travel one way.