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I recently watched the film Timecrimes, which got me thinking about the theories of time travel. Now I'm not a physics expert, but I have thought about these ideas for a while now and thought it was time to write them done.

I'll try not to spoil the film if you haven't already seen it, but this discussion might give away too much. The general idea in the film was the main character Hector was chased into a time machine and was made to travel back a few hours in time. The film did a good job of explaining this to the viewer with the aid of a diagram and naming of the Hectors. I have reproduced their diagram and will use it for my discussion.

Hector 1 was being chased, and when he reached point A in time he entered the time machine and goes back. He is now at point B in time and is called Hector 2. Now the interesting thing in the film is that Hector 2 turns out to be the person chasing Hector 1 into the time machine! This is somewhat of paradox which I hopefully will explain.

Hector 2 is convinced he needs to chase Hector 1 into the machine, and try to recreate events exactly as before, otherwise his existence will end. Now what would happen if Hector 2 decides not to chase Hector 1. Will Hector 2 disappear? I don't think so, because Hector 2 has to chase Hector 1 as it has already happened. I think this contradicts the multiple universes theory where every possible action that can happen, does happen. Here, the only thing that can happen for Hector 2 is to repeat exactly the events that Hector 1 experienced in his time frame. Without Hector 2 even knowing, he will be repeating exactly what he originally experienced.

If Hector 2 was able to decide not to chase Hector 1, then I think the order of events would have never happened, and Hector 1 would have never entered the time machine. So as soon as Hector 2 is created, he has no choice (or free will) to change the events that Hector 1 (his former self) experienced.

The thing I find interesting here is how the chain of events happened in the first place, and this is where I have a theory. In mathematics many equations can be solved iteratively, and there are typically two outcomes. The equation converges to a solution, or the iterative process does not converge and quickly goes off to infinity. A fractal is a perfect example of this, as the colours of a fractal represent how many iterations it takes to solve the equation, or if it does not solve it is coloured black. This can be seen with the simple Mandelbrot set fractal with equation Z_n = Z_n-1² + c.

Taken from Wikipedia under the Creative Commons Attribution-Share Alike Licence.

Now, I'm of the opinion that for Hector 2 to have gone back in time and influenced his own previous time line (ie Hector 1's), there must have been an iteration of events, which lead to a constant non-fluctuating time-line that Hector 2 chased Hector 1 into the machine. For example, without Hector 2 influencing Hector 1, there must have been a series of events that caused Hector 1 to enter the time machine. After he enters the time machine, events keep changing until we get to a constant state where Hector 2 chases Hector 1 into the machine, and each time round the iteration the exact same set of results occur.

Now I realise the last paragraph completely contradicts my previous statement that Hector 2 was unable to change what was going to happen next in his time-line. This is obviously the paradox. However, I think once the order of events is in a steady state, the solution is solved and this is what happens. Until then, the time-lines can keep changing, until either a stead state is achieved, or it goes to infinity, in which case all the events never happened, and that branch of the multiple universe theory is not travelable.

The steady state may not be a single time-line where someone influences their previous time-line. It could be a iteration between two (or more) steady states. For example, lets use the Mandelbrot equation, and say that Z₀ = 0 and we start with c = 1. The sequence of solutions is, 0, 1, 2, 5, 26, and keeps going to infinity. In time travel terms this could have never happened. However, let c = -0.5, we get a long sequence which eventually converges at ~-0.366. This is the kind of solution we observed in the film with Hector 1 and 2. As there are some starting events, which cause a sequence of time-lines to eventually converge where the previous iteration is the same as the next.

Finally lets imagine the situation where there are multiple solutions with a period. If we let c = -1 , then we get the sequence 0, -1, 0, -1, 0, etc. We can see this iteration has two solutions, 0, and -1, which flip between each other. Now, imagine this kind of solution happening with time travel. There are three players, Bob 1, 2 and 3. Bob 2 carries out a sequence of events that causes Bob 1 to turn into Bob 3. Now Bob 3 carries out a sequence of events that causes Bob 1 to turn into Bob 2. Therefore, depending on which iteration of events we are in, Bob 2, or Bob 3 is created from Bob 1.

One additional component in the film is that actually a third Hector is formed from Hector 2. Hector 3 played a minor part influencing Hector 2. Now this would be an equation in the form: Z_n = Z_n-1 + Z_n-2.

Hopefully my ideas of iteratively solving time travel makes sense, and I'd appreciate any comments. As far as I know these are my own ideas, and I haven't previously read this anywhere. Also, as with most of my knowledge it has been collected from watching far too much Sci-fi :)