Some context for the above question is warranted.
While reading Hartle's Gravity, the following statements made recurring appearances: "Gravity is not a force, it is the geometry of 4D spacetime. ... In GR, the mass of the sun curves the surrounding spacetime, and the Earth moves on a straight line path in that curved spacetime."
The phrase 'a straight line path' seems to imply that any particle in the presence of only massive objects and nothing else (i.e. a free particle, since gravity is not a force; it is just geometry) will follow a trajectory that minimizes the distance. However, in mechanics, the trajectory of a particle is obtained by extremizing the action.
So, my questions are as follows:
Are the Lagrangian and action still defined the same way?
If we write out the usual Lagrangian (with time as a parameter) without a potential term (free particle) and incorporate the geometry in the kinetic energy term with constraints relevant to the geometry, will extremizing the action be the same as minimizing the distance on the suitably chosen metric? This is true for cartesian and spherical geometries; is it true for all others as well?
In Newtonian gravity, the potential term gives us accurate trajectories for most purposes. So, how does including the potential term equate to simply changing the metric?
Is it always possible to rewrite a Lagrangian by replacing the potential term with a suitable geometry?
