> Under the laws of special relativity an observer in a space ship travelling at 0.1c will see Earth speed up, while an observer on Earth will see the ship slow down.

No they won't. They'll both see each other slow down, since motion is truly relative. From the perspective of the Earth, the ship is moving away. From the perspective of the ship, the Earth is moving away. Neither is right or wrong. Both see symmetrical time dilation.

The unintuitiveness of this fact is why the twin paradox is so famous, despite not really being a paradox: https://en.wikipedia.org/wiki/Twin_paradox