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In this question How can a person inside from a veiled and free-falling elevator distinguish whether he is in an inertial or non-inertial frame?, Dale answered:

Within the frame of the free falling elevator the reading on an accelerometer always matches the acceleration with respect to the frame. Therefore the free falling frame is inertial.

In the frame of the ground an accelerometer at rest reads an upward acceleration of g despite having no acceleration with respect to the ground. Therefore the ground’s frame is non-inertial.

Each frame can determine if they are inertial or not by looking at their own accelerometers and their own frame, without reference to any other frame. But the inertial vs non-inertial designation is exactly backwards from what you had indicated. A free falling frame is inertial and the ground frame is accelerating upwards at g.

A free-falling elevator is an inertial reference frame, according to Dale's answer. Now, when the elevator is stationary on Earth's ground (assuming the Earth is a non-inertial reference frame), it is subject to a normal reaction that reacts to the force that the elevator makes on the ground, so this block is being accelerated due to the Earth's reaction (Newton's third law). But now the confusion begins, in this forum this question Does the gravitational force obey Newton's third law of motion? was answered by Rohit Rawat as follows:

Yes, Gravitational force strictly follows Newton’s Third Law of motion. You can think of it since both were formulated and discovered by Sir Isaac Newton.

Newton’s 3rd Law states that for every action force there is an equal and opposite reaction force. This too applies to gravitation.

So if it is assumed that the Earth is a non-inertial frame, and forms an action-reaction pair with the free-falling elevator, why wouldn't a person at rest in outer space say that the elevator is a non-inertial frame if is it being accelerated by gravitational force? Is it possible to extend this situation to a general case where if whenever a frame of reference configures Newton's third law, then it is a non-inertial frame?

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Unfortunately, this is a question of terminology where the terminology has changed over time and become a little inconsistent. When Newton first formulated his laws, little care was taken about things like reference frames. And his personal views were based on absolute time and space, which were quickly discarded by the scientific community.

But as later scientists advanced Newtonian mechanics they started to use rotating and accelerating reference frames. Such frames were called non-inertial because objects do not exhibit inertia in such frames, meaning that force free objects would accelerate as though subject to fictitious forces with no source. Forces were broadly categorized as real or fictitious and inertial frames were ones that lacked fictitious forces.

Without much consideration, gravity was considered a real force. As time went on, accelerometers were developed, and it was discovered that they could measure accelerations due to real forces except for gravity which was undetectable by accelerometers. Einstein, in what he later called his happiest thought, realized that gravity shared this feature with fictitious forces. So gravity itself could be considered a fictitious force and transformed away by choosing an appropriate reference frame.

This produced two conflicting definitions of inertial frames. In the old definition gravity was considered a real force and a lab at rest on the surface of the earth was inertial. In the new definition gravity was considered a fictitious force and a lab on the surface of the earth was non-inertial, accelerating upward at $g$. In both cases there was a gravitational force $mg$ pointing down in the lab and all of the same equations and predictions worked identically. It was only a change in terminology.

The new terminology had some benefits:

  1. it was more practical. Inertial vs. non-inertial could be determined simply using accelerometers. In contrast, the old definition would require you to know the distribution of all of the matter in the universe so that you could correct for gravity.

  2. it led to general relativity. Using this definition a more accurate theory of gravity could be developed. One which was locally compatible with special relativity, and which made many bold predictions that have since been confirmed.

  3. it led to the geometrization of gravity. This explained the equivalence of inertial mass and passive gravitational mass, or at least it built their equality into the theory at a fundamental level.

  4. it allows a conceptually cleaner formulation of Newton’s laws. An object in free fall follows a geodesic. An object acted on by a net force experiences proper acceleration of $\vec F/m$. And momentum is conserved.

Note that in the early days of the development of general relativity, these points did not work for Newtonian gravity. However, later on Cartan invented what he called Newton Cartan gravity, which was a complete geometric reformulation of Newtonian gravity. So now all of these above points can also be applied to Newtonian gravity (including the idea of curved spacetime). As a result there is no longer any incompatibility between this definition of inertial/non-inertial and any major theory of physics.

Unfortunately, because of the large amount of literature predating the adoption of GR and the development of Newton Cartan gravity, there will always be some inconsistency in the use of the terminology. All we can do is explain the modern usage and its benefits, and make people aware of the possibility that the older definition may be used instead on occasion.

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  • $\begingroup$ So, for Newton's spacetime, the answer is: yes, every frame in which Newton's third law is checked is a non-inertial frame. But for Einstein: no, it's relative. For example, the elevator falling in free fall is subject only to gravity, which for GR is a ficticious force, and GR considers that anything accelerated by just a ficticious force is therefore an inertial frame. However, in the case of an elevator on the ground, in addition to gravity, there is a normal force, which is a mechanical force, therefore a real force, so the frame of reference is non-inertial. That's right? $\endgroup$ Dec 29, 2021 at 22:27
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    $\begingroup$ @userN yes, that is essentially correct $\endgroup$
    – Dale
    Dec 30, 2021 at 0:49
  • $\begingroup$ Just correcting a detail about affirmation in Newton's space-time: every frame that is valid Newton's third law, and this reaction of the action-reaction pair generates acceleration in the frame, so this is a non-inertial frame, e.g.: a block that is pushed by the frictional force of a second rough block located below. However, if the third law is verified, without, however, producing an effective acceleration due to the mutual nullification of the forces, then it is an inertial reference, e.g.: a block on the table where the weight force is nullified by the normal force. It's more accurate? $\endgroup$ Dec 30, 2021 at 2:26

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