The Twin Paradox: Telescopes

Saturday, November 03, 2018

The Twin Paradox: Telescopes

The Twin Paradox Part 3 Telescopes

Check out the first 2 versions of the twin paradox if you haven't seen them already!

In this final version of the twin paradox, we will consider what the two twins actually see. Sam will stay on Earth while Rob flies away in a rocket and then returns. We know that Sam's frame is the only valid one since it is inertial and so we know he is right in saying Rob is the one who experiences time dilation.

During the first half of the journey both observers are inertial and so both their claims are valid. How does the problem suddenly sort itself out? How does Rob accelerating make Sam age quicker so that he is older than Rob when he gets back?

Using a telescope is simply looking at the light coming from each person. To make it easier to see what is happening, each twin will send out a flash of light after each year passes on their clock.

Rob's perspective

During the first part of Rob's journey, light from Sam takes a long time to reach him. This is because Sam is moving away from him and so each flash has a longer distance to travel than the previous one. He is experiencing the relativistic Doppler effect because the frequency of light emitted is greater than the frequency of light observed. Each flash marks a year passing for Sam and so Rob is concludes that years are taking a long time to pass for Sam. This matches up with what Rob knows about time dilation.

Now consider what happens on Rob's return journey. Sam appears to be moving towards Rob and so each flash has a shorter distance to travel than the previous one. This means the flashes arrive with a higher frequency than the one they were emitted with. This means during the return journey, Sam appears to be aging quickly because flashes marking each year arrive frequently.

Sam's Perspective

During the first part of the journey, each of Rob's flashes take a longer time to reach him. This is because Rob is moving away from him and so each flash has a longer distance to travel. In other words, he is experiencing the relativistic Doppler effect. Sam therefore concludes that each year is taking a longer time to pass for Rob. This observation matches up with what he knows with time dilation. So far so good.

Let's think carefully about the part when Rob turns around.

Rob's View

When Rob turns around, flashes from Sam are already on their way towards him. This means the next flash will reach him shortly. This means shortly after turning around, he experiences the relativistic Doppler effect and flashes start to reach him more frequently.

Sam's View

Rob sends out a flash and then immediately turns his rocket around. From Sam's perspective it is as if Rob is chasing the first flash. This means when that flash reaches Sam, it won't be long before Rob arrives as he was seemingly chasing the flash.

The time period in which Sam experiences the relativistic Doppler effect is between the first flash and last flash of Rob's return journey. This time period is short however because Rob arrives soon after the first flash. Rob turns and emits that first return flash. He continues to emit these return flashes after each year passes but is chasing them in his rocket. When the first finally arrives, all arrive quickly because subsequent ones had a shorter distance to travel. Rob then arrives shortly after as he was chasing the flashes. For Sam the return flashes come very quickly but there are not many of them since Rob arrives shortly after the first.

We can now see where the asymmetry comes from. From Rob's point of view, the effect of accelerating is seen immediately and Sam appears to start aging quickly.

From Sam's point of view however, the effect Rob's acceleration takes a long time to kick in. When it finally does kick in, the flashes come very quickly but Rob arrives shortly after and so he only sees Rob age quickly for a short period of time. This means overall, Rob is still younger than him as the time dilation during the first part of the journey is greater than the seeming speeding up of time during the second.

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