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?