I have the following doubt: let person A be in free fall being gravitationally attracted towards a massive body M. Suppose then that person B is completely isolated in space and in particular very far away from M so that it is not influenced by it. Now I think: A is in a free-float frame, so according to GR he is in an inertial reference frame. Further, B is completely isolated, so he is in an inertial reference frame too. However, A is attracted towards M, while B is not, so there is a relative acceleration between A and B. The question is: shouldn't all inertial reference frames move uniformly with respect to one another? Where is the flaw is the above reasoning?
The flaw is in using Euclidean geometry to make arguments about a curved manifold. The statement that inertial frames move uniformly with respect to one another works in special relativity, where the curvature of the metric vanishes at every point. The wiki article on inertial frames of reference covers this phenomenon under geodesic deviation. A crude analogy is that parallel lines (geodesics) are not equidistant on a sphere. We only work with local inertial frames on the curved spacetimes we encounter in general relativity.
If B has the information of the gravitational field of M, it is possible to know that the covariant derivative of the velocity of A is zero.
In GR the notion of inertia is modified that way. Instead of zero acceleration (derivative of velocity), ir is the covariant derivative of velocity that is zero. The body is said to follow a geodesic path in this case.
But in order to calculate the geodesics, it is necessary to know the metric, and in the case of spherical massive and slow rotating bodies it is the Schwartzschild metric.
It is the same idea of the route of a long airline trip east - west. The route Tokyo - Paris fly over Russia. It is easy to realize in a globe that it is the shortest path, but the paper maps will show a curve line. The map shows that the velocity vector is not constant, because it starts with a north component and finish with a south component. For a constant velocity, the compass needle should not move during the cruise part of the fly.
But using the metric of the surface of a sphere, it can be shown that the covariant derivative of that vector is zero, if the plane is following the shortest path. That results in part of a great circle in the globe, what is the geodesic for that metric.