Under Newtonian mechanics not all inertial reference frames are moving at constant velocity relative to each other?

Question

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.

Answer 1

In Newtonian mechanics, the frame of a falling object, let's say an elevator, is not inertial:

If a researcher in the elevator (height $h(t)$) drops a steel ball with mass $m$, then there is a gravitational force on the ball:

$$ F_g = G\frac{M_{\rm Earth}m}{(R_{\rm Earth}+h(t))^2} $$

The researcher, using:

$$ f = ma $$

predicts a downward acceleration of:

$$ a(t) = \frac f m = G\frac{M_{\rm Earth}}{(R_{\rm Earth}+h(t))^2} $$

but measures:

$$ a(t) = 0 $$

in violation of Newton's Second Law.

Answer 2

Inertial frame is frame attached to free body. This is true for all 3 frameworks.

In Newtonian mechanics, the falling object in gravitational field is certainly not free, as it is acted on by gravitation. So frame attached to falling object is not inertial (the problem is little deeper, because the object can be acted on by forces that add up to zero and then frame of this object will be inertial, even though it is not free. This is explained better in JEB answer).

In special theory of relativity gravitation is quite problematic. That is because you need to generalize Newton gravitational force to 4D spacetime. There are some foundational problems to include gravity in STR, that is why Einstein developed quite different theory and did not simply generalize the gravitational force. You can read about Schild's argument in MTW, albeit I have a feeling this book is too advanced for you. The point is, STR cannot deal with gravity.

In general relativity, the spacetime is curved and falling object is free, because there is no force of gravity in this framework. Thus the frame is inertial. But because spacetime is curved, the frame is defined only in small neighborhood of your falling object.