Question

How do we derive force/acceleration vectors from Einstein's field equations?

Answer 1

Not a relativist, but I'd guess that (1) GR doesn't speak to anything but gravity so you can only ask about gravitational forces directly and (2) you just transform from the local inertial frame where the motion is trivial into whatever non-inertial frame you standing in when you ask about the gravitational forces and a pseudo-force pops out just like the centrifical pseudo-force in rotational motion

Answer 2

geometry where the scalar product of nonzero vectors is only allowed to be ... of the Einstein field equations, the foundation of modern gravitational ..... describing an infinitesimal distance on a manifold.