General relativity: A field theory or a mechanical theory? First let me introduce my terminology:
A "Mechanical theory": A theory which describes time-evolution of a particle or a system of particles regardless of the fields affecting the particle/system. e.g. Classical mechanics, Quantum mechanics, etc.
A "Field theory": A theory which describes time-evolution of a particle or a system of particles with taking the effect of fields on the particle/system into account and also a theory which describes time-evolution of the fields themselves. e.g. Classical gravitation, Classical electromagnetism, Quantum field theory, etc.
But I am always confused that "General relativity" falls into which category. I know "Special relativity" just applies a modification to "Classical kinematics" to build "Relativistic mechanics". But "General relativity" is talking about "Relativistic gravitational fields" while also talking about "Non-inertial frames of reference".
Now this is my question:
Is "General relativity" a field theory or it's a mechanical theory? If it's a field theory is there any other way to study non-inertial frames in the context of "Relativistic mechanics" without bringing any special field into play?
 A: *

*General relativity is clearly a field theory because the Einstein field equation
$$ G_{\mu\nu} = k T_{\mu\nu}$$
for a constant $k$, a function $G$ of the metric $g_{\mu\nu}$ and the stress-energy tensor $T$ is an equation of motion for the field $g_{\mu\nu}$ - at each point in time, the metric of spacetime (which is position-dependent hence a "field") is determined by the energy-matter distribution. However, it also includes "non-field" equations of motion if you consider single particles of matter, which move (in the absence of other forces) along geodesics according to the geodesic equation, which is an equation for their world-lines. This is obviously a coupled system - the metric determines the geodesics along which matter moves, but the matter also determines the metric.

*"Non-inertial frames" may perfectly well be studied already in special relativity - special relativity is just about matter moving on geodesics in the flat Minkowski metric, and ignoring the backreaction of matter on the geometry, which corresponds to "turning off gravity". A non-inertial frame where is simply a coordinate system in which the Minkwoski metric does not take the standard form of $\mathrm{diag}(-1,1,1,1)$ (or signs switched), i.e. one that cannot be reached by a Lorentz transformation from an inertial frame. Whether you want to consider the metric a "field" in this setting is a matter of choice, it is certainly non-dynamical and not a field in the usual sense of a field theory.
