Under Newtonian mechanics, an inertial reference frame is defined as one where Newton's second law applies, so if the Earth weren't rotating by this definition it would be one as things accelerate due to gravity. However, the frame of a falling object which is accelerating relative to Earth is also inertial. Is it the case that Newtonian mechanics classifies both of these as inertial reference frames?
I have read that Special Relativity should not include Earth as an inertial reference frame. What then would be the definition of an inertial reference frame for SR? (Am I correct in thinking this would be the set of frames in which SR holds?)
The Newtonian definition seems to work again with general relativity as gravity is no longer a force, and hence in Earth's reference frame objects accelerate without a force so it is not inertial.
In summary, I am looking for the precise distinction between the definitions of inertial reference frames under these 3 frameworks.