Spacetime curvature is relative? I have the following conceptual doubt.
These are my assumptions:
1) The geometry of spacetime is the same for all observers, regardless their motion
2) All motion is relative (both uniform and not uniform)
Now follow this reasoning:


*

*If we neglect the effects of gravity (away from masses and energy), the spacetime is flat in good approximation 

*If in this flat spacetime one particle is submit to a force, it will accelerate (no more geodesic-path)

*From the particle perspective, there is a local gravitational field (equivalence principle: acceleration <--> gravity)

*The particle will deduce that the spacetime is locally curved.

*But for a comoving free falling particle, the spacetime clearly appears flat!
So we have two particles, in the same spacetime local region, who disagree about the effective geometry of spacetime. They can't both be right, because this would violate assumption 1). And it can't be that one particle is "indeed moving", while the other is "indeed at rest", because this would violate assumption 2).
So... where is the wayout?
 A: Assumption 2 is false, not all motion is relative. For example, inside a closed box, with no access to anything external, it is possible to use accelerometers to establish the difference between free fall, proper acceleration, and rotation. The reading of an accelerometer is an invariant, and therefore those motions are invariants and thus not relative. 
Also, step 4 is incorrect. A local gravitational field in this sense does not imply spacetime curvature. Spacetime curvature is related to tidal gravity, not gravitational acceleration (which is related to the Christoffel symbols). Since there is no tidal gravity in this scenario the spacetime would remain flat even for the particle in the “gravitational” field. 
So what can we say about curvature? It is a rank-4 tensor, so like any tensor it is a geometric object which is the same in all frames. However, all of its components are relative to the given reference frame. There are also several invariants of the curvature tensor, including the Ricci scalar. 
