The clearest explanation for the twin paradox

In the comments to my previous article and in the comments to video There have been many questions and incorrect comments about the twin paradox. As it turned out, my explanation was not as clear as I had hoped, so in this article I decided to explain the twin paradox as clearly, thoroughly and consistently as possible and answer some other questions.

For illustrations and animations, I wrote an interactive browser-based visualizer where you can move sliders, change modes and observe Lorentz transformations.

Let me briefly recall the essence of the paradox:

We take two twins, put them on a small light planet (light, so as not to take into account the influence of gravity), leave one motionless, and launch the second on a rocket to fly and come back. When they meet, it turns out that the flying twin has aged less than the stationary one.
The paradox is that it is not obvious why time flowed more slowly for the flyer. After all, it seems that the situation is symmetrical: in the reference frame of the flyer, it was a planet with a fixed twin that flew and returned, and it should have taken them less time to fly.

The twin paradox is very important because… This is the clearest way to see that the relativistic effect of time dilation is not just a mathematical artifact of the special theory of relativity or an illusion, but a very real physical phenomenon.

Let's ask a running cat to run to the right at 75% of the speed of light, then turn around and run back at the same speed.
Here is a visualization on the diagram (vertical axis of time, horizontal axis of space):

It shows that the running cat had less time running than the stationary cat, but it is not clear why.

To understand what is happening with each of the twins, you need to look at the situation from the perspective of each of them.

Let me remind you that in the special theory of relativity, when the speed of the observer changes, the points on the diagram shift not only along the space axis, but also along the time axis. All relativistic effects inevitably follow from this (time dilation, length contractions, relativity of simultaneity).

On the left is the classical Galileo transformation, on the right is the Lorentz transformation, which replaced it. The yellow lines illustrate the speed of light in both directions.

I talked more about why this happens in a previous article and video.
In short, the whole point is that, according to experiments, the same “beam” of light flies at a speed 299,792,458 m/s relatively any observer. No matter how fast or in what direction that observer is moving relative to the source of that light. In other words, no matter how fast you move, light still flies away from you at the speed of light.

This fact contradicts the usual Galileo transformation:

Let's say we are running on a cat at great speed (30% the speed of light). Then we run out from the surface of this cat on another cat with the same speed relative to the first cat. Then we do the same three more times.

In the usual Galilean world, it turns out that the distances between cats at each moment of time are the same, and we are moving away from the first one faster than at the speed of light (here the speed of light is shown in burgundy).

In order to reconcile the fact of the constancy of the speed of light with physics, it was necessary to change the Galileo transformations (where, when the speed of the observer changes, the points on the diagram shift horizontally) and turn them into Lorentz transformations (where the points also shift in time, asymptotically approaching the line of the speed of light). By the way, mathematically this is a rotation in 4-dimensional space-time with the Minkowski metric.

Pay attention to the blue and red segments. In the Galilean version, their lengths are preserved, which are calculated as(according to the Pythagorean theorem), and in the Lorentzian theorem, the interval is preserved, which is considered as (Minkowski metric).

You can do the cat trick as many times as you like, but the first cat will never reach the bard line. Its line will only asymptotically approach it and stretch. The light itself, at the same time, flies at the speed of light relative to any of the cats.

So, back to the twin paradox

Let's imagine that one twin is sitting on Earth, and we asked the second to fly to a neighboring galaxy and return. Let the galaxy be at such a distance that, according to the moving twin’s clock, the entire journey takes 8 seconds.

Let's start with the stationary twin on Earth:

Everything is simple here. We wait 8 seconds and nothing happens.

Now let's look at the situation from the perspective of the moving twin.

1) First, we quickly pick up speed (we accelerate very quickly, so we have not yet flown a significant distance).

It can be seen that the time in the galaxy has shifted into the future by more than 6 seconds. But we won’t notice this right away, because this time shift increases gradually along the axis from us to the galaxy.

Time graph:

Also, the distance to the galaxy has decreased (before acceleration, there were 7 cells between the Earth and the galaxy, but now it is ~3.5) and the passage of time in the galaxy has slowed down for us (the vertical distance between neighboring images of the galaxy has become larger).

At this moment, the flying twin does not have access to the information that time in the galaxy has shifted into the future and that the distance has decreased. After all, he sees only what directly reaches his eyes.

So that there is no doubt that everything is fair, all shifts of points on the diagram occur only according to the Lorentz transformation formulas (this can be checked using the visualizer sources).

So, the speed has been picked up. Now we fly at this speed until we reach the galaxy. This will take 4 seconds (for this I added a “Wait” slider to the visualizer):

While we were flying, we collected this accelerated and compressed light from the galaxy and observed its evolution in fast forward as we moved. As a result, at the destination point we see that according to our clock, 4 seconds have passed, and on the galaxy – 8.

Important point: We are flying towards the galaxy, so its time is slowed down for us. But with our eyes we see it, on the contrary, accelerated, because we collect the flattened light emitted from it. In other words, the distance to the galaxy has shrunk more than its time has slowed down.

Now all that remains is to turn around and fly back to Earth at the same speed:

And stop:

What do we end up seeing? While the mobile twin ran for 8 seconds, the immobile twin took 16.

When the moving twin flies without acceleration, the situation is symmetrical. Each of them believes that time is slowed down for the other. But it is precisely during the accelerations of the moving twin that the time of the motionless one is “eaten up,” which can be seen in the animations.

Result: The twin paradox is resolved by the fact that the acceleration of one of the cats introduces asymmetry into the system and leads to a difference in their ages. This effect really exists, it has been measured experimentally and clearly follows from the Lorentz transformations, like all other relativistic effects of the special theory of relativity.

Is general relativity needed to explain the twin paradox?

No, GTR is needed where gravity needs to be taken into account. In our case this is not necessary.

Visualizer: https://dudarion.github.io/Interactive-Minkowski-diagram/

Project on GitHub: https://github.com/Dudarion/Interactive-Minkowski-diagram

Thank you for your attention!