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Here's what I understand to be standard Newtonian/classical inertial frames:

1. There exists a set of reference frames, called inertial frames, in which mass, time, force, acceleration, etc. are (Galilean) invariant but position, velocity, translational and angular momentum, work, kinetic energy, etc. aren't.

2. Measurements in one inertial frame can be converted into measurements in another frame by a Galilean transformation of coordinates.

3. All inertial frames are in relative rectilinear motion.

4. Any two frames are inertial if and only if they measure the same accelerations for all particles.

5. A frame is inertial if and only if a perfect accelerometer at rest measures no acceleration.

Also:

6. A frame is inertial if and only if physical laws are observed to be in their simplest forms.

When I got to the Equivalence principle:

1. Inertial mass = gravitational mass.

2. No experiment can determine whether you're in a free falling elevator or an elevator floating out in space. It is understood that the free falling elevator is just above the Earth's atmosphere and that the elevator in space is far from any gravitational source. Also, the free falling elevator is small enough, and falls for a short enough time period, that tidal deviations are below the level of experimental detection.

I had the following problem: I realized that #5 above was not true in the classical sense. Given two observers, each with a damped-spring accelerometer, one observer is in free fall above Earth's atmosphere and the other is floating out in space, then each observer will see that both accelerometers measure zero acceleration, even though they are accelerating with respect to each other. Each observer knows that his zero acceleration measurement proves that he is in an inertial frame, but also can see that the other observer's accelerometer aslo measures zero acceleration, that that therefore the other observer must also be in an inertial frame. And yet each observer clearly measures, in thier own inertial frame, that the other observer is accelerating.

I had assumed I would be able to do classical physics without having to worry about relativity, and that I could do relativity when I was ready. I guess was wrong.

I tried several things to get around the problem:

1. #5 above isn't really Newtonian, it must actually be an idea from relativity. This doesn't work since it can clearly be seen that #3-#5 above are all more or less equivalent.

2. I reasoned that the speeds and gravitaional fields involved were not relativistic, that somehow then the problem should go away and that the frames involved should be able to be described as noninertial in the classical sense. But this leads nowhere, if my accelerometer reads zero. I'm in an inertial frame.

3. I tried to convince myself that the 2 observers can't read each other's accelerometer, i.e., a signal sent between them would somehow be messed up by some "relativistic effect," but I knew this was rediculous since the speeds and field strengths involved weren't relativistic.

I gave up and opened my old general relativity book (Gravitation by MTW). The 1st chapter describes inertial motion:

Following a geodesic (free fall, orbit, etc.) = natural (weightless) motion of a particle = local inertial (i.e., Lorentz) frame = particles move at constant speed in straight lines. Physics (physical laws) is simple when viewed locally.

So I tried to revise my definition of an inertial frame:

1. For any particle moving in Nature, there exists a set of local reference frames, called inertial frames, in which mass, time, force, acceleration, etc. are (Lorentzian) invariant but position, velocity, translational and angular momentum, work, kinetic energy, etc. aren't.

2. Measurements in one local inertial frame can be converted into measurements in another local frame by a Lorentzian transformation of coordinates (Galilean at low speed).

3. All local inertial frames are in relative rectilinear motion.

4. Any two local frames are inertial if and only if they measure the same accelerations for all particles.

5. A local frame is inertial if and only if a perfect accelerometer at rest measures no acceleration.

Also:

6. A local frame is inertial if and only if physical laws are observed to be in their simplest forms.

This seemed simple enough and I hoped to continue with classical physics, but I was bothered that "local" is opposed to "global" and that global should refer to the shape of the whole universe, not merely the difference between free falling near the Earth and floating relatively nearby in space, which seems "local" with respect to the entire universe. Then I realized I still had the above problem. If one obserever is free falling above Earth's atmoshpere and the other observer is in low Earth orbit, and if the free falling and orbiting observers pass through the same event, i.e., the same point in space at the same time, then, at that event, i.e., that point on the spacetime manifold, both observers should have the SAME set of local inertial frames, since they are passing through the same point in space at the same time and both of their accelerometers read zero. But they can't be in the same inertial frame since each observer measures the other as accelerating.

So it seems I can't avoid certain general-relativistic ideas, even when trying to avoid them and just stick to classical physics.

Any help with the resolution of this paradox will be greatly appreciated.

Thanks in advance.

P.S. I would like to avoid the mathematical formalism of special or general relativity, if possible, in the resolution of this problem. Intuitively, I strongly suspect it isn't needed, as these ideas are fundamental to an understanding of inertial frames, and as such should be relatively simple to explain.