I finished watching the show Dark[archived] last night, and it's one of my favorite shows. It starts small and simple on the first season, expanding in seasons 2 and 3, with a very satisfying ending that ties everything up in a really nice way — and one that makes sense. I think I watched at most one episode per day, specially for the third season, and after each one I would discuss it with my girlfriend and read comments on the r/dark[archived] subreddit.

After finishing it last night I had some thoughts on how the physics of the show work, and wanted to write them down. If you haven't seen the show I recommended you to give a try, even though the first few episodes of season 1 can be somewhat slow.

Spoilers ahead!

The bootstrap paradox

One of the interesting things about the show is that in the knot worlds there's a bootstrap paradox[archived]. Also called a causal loop, this happens when something from the future affects the past in a way that causes its own origin. In Dark, an example would be Katharina Nielsen's name: Hannah traveled to the past and was using the name "Katharina Nielsen" instead of her own. She then met Katharina's mother years before Katharina was born. The mother liked the name "Katharina", and eventually gave it to her daughter. The same thing happened with the last name as well: Hannah had a daughter called Silja Nielsen, who had a daughter called Agnes Nielsen. Agnes' grandson Ulrich Nielsen married Katharina Albers, who changed her name to... Katharina Nielsen.

This is a recurring theme in the show: there's no beginning and no end. Adam's goal is to find the origin, to destroy it. In the show, it's initially implied and later confirmed that the characters have been stuck in this loop for an uncountable number of times (Claudia even says "infinite"). But how can that be? We know that the Adam and Eva's worlds were created from the origin world when Tannhaus tried to travel to the past. We know that some loops are different than others. So there must have been a start, when the two worlds were first created and things were set in motion by Tannhaus failedsuccessful experiment (he did achieve what he wanted!).

The infinite loops without beginning nor end reminded me of the Lorenz system[archived] (source[archived]):

The system was designed by meteorologist Edward Lorenz[archived] while studying atmospheric convection. Lorenz came up with a simplified model of convection that was based on 3 simple differential equations. But even though the model was relatively simple, Lorenz noticed that small differences in the initial conditions of the system would quickly diverge into different results — the model was chaotic, depending on its parameters and initial conditions. This became on of the basis for chaos theory, and shed light on why it's impossible to predict the weather after more than 10 days.

One peculiar feature of the Lorenz system is that it has two relatively stable points, around which the state will orbit. You can see it in the animated figure above, where the line is either orbiting one of the lobes (or "butterfly wings"), or the other. The line is attracted to one of the lobes, but at any time it might switch to the other. These attractors represent the semi-stable states in which the system tends to be, though transitions can happen at any time. More complicated and/or realistic systems might have more than two fixed point attractors.

What is the relationship between the Lorenz system and Dark? In Dark we have all the story lines repeating themselves in cycles, but I think we can assume that they're not exactly identical every time. Maybe in one of the cycles Katherine's mother never meets Hannah, and gives her daughter a different name. Maybe Hannah will then adopt that name in the next cycle, and the name will be passed again to Katherine, creating a different bootstrap paradox where the name is different. This would correspond to the line switching to a different fixed point in the Lorenz system. Maybe eventually the name switches back to Katharina.

We can think the same way for most of the paradoxes in the show. There's an initial condition, after the world is spun off from the origin world. As the timeline cycles and cycles it tends to stabilize in states where the past is influenced by the future in a way that it tends to recreate it. This is helped by the fact that Claudia, Adam and Eva are ensuring that the conditions in the past will reproduce the trajectories that hey have caused them to be. This is a very stable state, where the system itself makes sure that the conditions are just right for it to repeat.

In that case, the model of a limit cycle [archived] might be more appropriate:

Here we have one stable state, and the system naturally converges to it. Once it has reached the limit cycle (though it only does so asymptotically) it can never escape, unless there's an external force. Note how the lines all converge to the thick line, never diverging from it.

In summary, we can assume that initially, after emerging from the origin world, there were no bootstrap paradoxes occurring — at least not in the first cycles. They started occurring as people traveled back in time, and once they started occurring the whole system became more stable. After countless cycles, the timelines were basically just repeating themselves.

Why people can't die

The other thing that is interesting in the show is the fact that characters cannot die. For example, there's a scene where Jonas tries to shoot himself with a loaded gun, but it fails to shoot until Noah points the gun to a wall. Initially this made no sense to me, because it implies that there's a "force" ensuring that things happen again as they've happened before. This also contradicts my idea about the Lorenz system, which is free to exhibit any kind of behavior and even transition abruptly between stable states. How does this happen? How does the gun know that Jonas can't die, what is the thing preventing him from committing suicide?

The explanation I came up for this is loosely based on the "many-worlds" interpretation of quantum mechanics. When you roll a 6-sided die you can imagine the universe splitting into 6 different versions, each one with a different result. If you see the die rolling a 2, that defines which universe you're in. This happens constantly, with the universe spawning new multiverses every time there is a range of possible outcomes.

So what happened when Jonas pulled the trigger to his head? In most of the multiverses the bullet left the chamber and he's dead. In a small percentage of them the gun jammed and he's alive. In the former the cycles end, since without Jonas they can never go back to the stable point we see in the show. Those timelines are irrelevant to us. In the latter, Jonas lives and is able to repeat the cycle, thinking that it has to happen.

What we see in Dark is basically the version of the multiverse where guns jammed and things happened as they should. Because we're watching a version of the multiverse where a countless number of cycles happen, those things must've happened. It's similar to the antropic principle[archived]:

The anthropic principle is the philosophical premise that any data we collect about the universe is filtered by the fact that, for it to be observable at all, the universe must have been compatible with the emergence of conscious and sapient life that observes it.

This would explain why people cannot break the loop, until they eventually can (for example, when Adam refused to kill Eva).

I'm not 100% sure of how the multiverse theory plays with the creation of Adam's world and Eva's world from the origin world. Are those just different versions of the multiverse? The show might be inspired by the multiverse theory, especially when Eva (and later, Adam) exploit the fact that time stands still to create different versions of the timelines.

Closing thoughts

I think the show does a great job in tying up all the knots and building a framework for time travel that is interesting and consistent. Watching the final episode was really satisfying, and most of the questions I had about the show were answered. The thoughts I raise here are just a way I found of making sense of some of the things that I felt could be better explained in the show.