I am following Carroll, he states that the geodesic equation is the the generalization to curved spacetime $\vec f = m \vec a$, for $\vec f = \vec 0$. This leads me to wonder what is the correct to generalize this further to:
- Paths that are not geodesics
- Cases where $\vec f \neq 0$
Suppose for example that an observer is sitting at on the surface of a neutron star in Schwarzschild geometry. I imagine that (1) this observer does not follow a geodesic since he is not falling toward smaller values of $r$ like test-particles would and (2) there is something present which the observer would interpret as a force, $\vec f = \vec f_g \neq 0$. What would be the acceleration such an observer feels. In particular, this means that I do not want to arrive at newtons law of gravitation.
What generalization of the geodesic equation is appropriate in situations like this, and how would it be applied? In any possible answer, I would prefer if point (1) and (2) would be addressed separately if possible.