I recently asked a question here about if the direction we travel matters in relativity theory: Does it matter in which direction I travel in relativity theory?
After I got answers and making more research about the topic, I found out that I was talking about speeds relative to the earth all the time. So, when I for example said that I travel away from earth at 50% of the speed of light, I automatically assumed that this is also the speed I can use to calculate the time dilation.
But since the earth already travels at some always varying speed and thus is not standing still in space, I think I also need to put this into the calculation.
I mean, if I would travel in the exact opposite direction the earth currently "flies" and at exactly the same speed, this would mean that I'm standing still and that the time for me would actually go as fast as possible, while the time on earth is still slowed down, as it always is due to its speed.
So: What I wanted to show you is that it really seems to depend from where I actually measure the speed I travel and what I consider the "root".
Moving away from earth at some speed can mean moving faster in space OR moving slower in space and thus also mean slower or faster time flow than the one on the earth.
Am I right with this? Or am I totally wrong and it's valid to measure the speed I travel by how much the distance from me to the earth gets bigger?
Actually, I think I have understood the answer of @John Rennie, and it really seems to explain a lot of the issues I ran into before.
His answer was, at least to my understanding, the following: If two objects in space stay together, they share the sime timeflow. If the distance between the objects changes over time, we cannot say which one will see the others clock going slower and which one will see the others clock going faster. To say that, we need to go back to the change of velocity that led to the increasing/decreasing distance. Here, we need to determine which of the objects changed it's velocity by the method John mentioned. And this will lead us to the answer.
This also explains the twin paradox perfectly. But after doing some thought experiments I came across a scenario that I cannot explain:
Imagine we humans have indeed built a death star as a moving base in space. This death star has also a lot of smaller fighter ships on board. Now the death star starts to fly away from earth at, say, 0.1c. According to the people on earth, the time on the death start now goes slightly slower than the time on the earth. After a while, a fighter on the death star begins to fly twice as fast (0.2c relative to the earth, 0.1c to the death star) in the same direction to explore the forthcoming space. According to the earthlings, the time on the fighter goes even slower than the time on the death star. The guys on the death star confirm that and notice that the time on the fighter seems to go slower compared to their time. After a while, the fighter found some enemy and decides to return to the death star to alert them.
Here, the fighter has 2 options:
1) since the death star is also on its way, he could just stop and wait until the deathstar comes across.
2) Stopping, and moving into the other direction to inform the death star faster and before he runs into the enemies.
The fighter goes for option 2) and flies back at 0.1c. For the earthlings, the fighter is now getting closer at 0.1c, while the death star is still moving away at 0.1c. For the death star, the fighter is getting closer at 0.2c.
And that is the point where this starts getting weird: For the earthlings, the fighters speed decreased from 0.2c to 0.1c, which means that the fighters clock is now going faster, while still being slower than that on the earth. Actually, since the death star and the fighter are moving at the same speed relative to the earth, they share the sime timeflow. For the deathstar, the fighters speed increased from 0.1c to 0.2c, which means to them that the fighters clock is now going even slower than that on the death star.
Can it really be?