Relativity and intertial reference frames in hyperbolic space I was watching this video (https://www.youtube.com/watch?v=zQo_S3yNa2w) about Hyperbolica, a game that is set in a non-euclidean space. On 7:47 the author mentions that in curved space, moving objects experience a "tidal force", with the magnitude of this force being dependent on the velocity of the object. In hyperbolic space, this force stretches the objects, while in spherical space it compresses them.
I thought of the following thought experiment. We can make some device that will measure this tidal force on all axes, and then adjust it's velocity (let's say, with rocket engines), until the magnitude of this tidal force on all axes is 0, after which it will turn off and continue to move inertially though space. What will happen if we release many such devices in different points? Will they all have a relative velocity of 0 at the end of their operation? If yes, does this mean that in curved space there exists an absolute frame of reference all observers can agree on (namely, such that the tidal force is 0)? If not, where does my reasoning break down?
Since we are on Earth, and live in a gravitational well, we also live in a curved space. What will happen if we actually build such a device and let it run on earth? Will it go into some orbit? Will it just land on Earth and turn off?
Sorry if the question is dumb or already has been answered.
 A: 
If yes, does this mean that in curved space there exists an absolute frame of reference all observers can agree on (namely, such that the tidal force is 0)?

Yes. A hyperbolic or spherical space does have a unique “rest” frame.

Since we are on Earth, and live in a gravitational well, we also live in a curved space. What will happen if we actually build such a device and let it run on earth?

The same reasoning doesn’t actually apply. The reason is time. In “Hyperbolica” there was curved space, but not time. In a gravitational well you have curved spacetime, not just curved space.
Why does that make a difference?
The source of the tidal force was the fact that parallel lines diverge. To remove that effect we had to stop moving. That made it so that instead of tracing lines we had just points. There is no parallel or diverging for points.
But since time is also a curved part of curved spacetime and since we cannot stop moving in time we cannot turn into points in spacetime. We remain “worldlines” as we go through time, even if we stop moving in space. And since we are lines in spacetime we cannot avoid the tidal effects.
