# 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)

• 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?

• all motion is not relative. There is distinction between inertial frames and non inertial even in general relativity. – Umaxo Sep 10 at 10:50