Principle of equivalence

In special relativity, we defined that an inertial frame had the property that in the absense of external forces, all objects maintain their relative velocities.

In general relativity, with gravity at play, this definition does not apply. Instead we note Einstein’s principle of equivalence: that in the freely falling frame, objects feel the same acceleration due to gravity, and maintain their relative velocities in absense of non-gravitational forces.

Locally we cannot distinguish between gravity and uniform acceleration. Globally in general relativity there are no uniform gravitational fields sourced by finite sized objects. But locally this approximation is useful.

In special relativity, i.e. a spacetime that is well described by a global Lorentz frame, objects following parallel trajectories will remain parallel forever. Even under uniform acceleration they remain parallel. In general relativity, where gravitational acceleration is never globally uniform, this is not the case. This is what we mean when we say spacetime is curved — its geometry is non-Euclidean.

The Minkowski spacetime, although also non-Euclidean, describes flat spacetime. We use other spacetimes to describe general relativity — as well as other, more complicated metrics.