# Inertial reference frames

I'm reviewing physics after ~30yrs of neglect, starting with Halliday & Resnick (and the internet).

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.

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.

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

This is not a feature of Newtonian inertial frames. A frame fixed with respect to the Earth (more precisely, with respect to an idealized non-rotating Earth) is considered inertial in Newtonian physics, even though an accelerometer at rest on the (idealized) Earth measures nonzero acceleration. But the frame of a spaceship traveling in a straight line away from the Earth, in deep space (and far enough away that the effect of the Earth's gravity is negligible), is also inertial in Newtonian physics, even though an accelerometer at rest in this frame measures zero acceleration.

The difference between these inertial frames, in Newtonian physics, is accounted for by the presence of the force of gravity in the first, but not in the second. That force pulls on the accelerometer at rest on the Earth, requiring a counter-force to keep it at rest, and the counter-force is measured by the accelerometer as a nonzero acceleration (i.e., weight). Of course, this raises the question of why the force of gravity itself is not felt as acceleration; that gets into the equivalence principle--see below.
6. A frame is inertial if and only if physical laws are observed to be in their simplest forms.
This was conceptually a feature of Newtonian inertial frames, but of course when relativity was discovered, physicists realized that that conceptual feature was not actually present in Newtonian physics; it only seemed to be because the people who developed Newtonian physics only had data for objects moving very slowly compared to the speed of light.

1. Inertial mass = gravitational mass.

This is true in Newtonian physics, but it's just a postulate adopted because it is experimentally found to be true; there is nothing in the structure of Newtonian physics that explains why it's true. The fact that it's true means that the force of gravity can never be felt, because everything accelerates the same way under it, so an accelerometer moving solely under gravity will have all its parts accelerating the same, so it will read zero.

#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.

No, they're not, because the word "acceleration" has two different meanings, and #3/#4 use one meaning while #5 uses the other. The first meaning (used in #3/#4) is called "coordinate acceleration" in relativity--it means a change in speed relative to some system of coordinates. The second meaning (used in #5) is called "proper acceleration" in relativity--it means what is measured by an (ideal) accelerometer. As should be evident from my discussion of #5 above, these two concepts are different and there is no necessary correlation between them even in Newtonian physics. So you can't derive #5 from #3/#4 in Newtonian physics.

if my accelerometer reads zero. I'm in an inertial frame.

In relativity, yes, this is how inertial frames are defined. But not in Newtonian physics; see above.

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.

Yes, correct; both observers are at rest in a local inertial frame centered on the same event (the event at which they cross paths), and the two local inertial frames (one for each observer) are related by a local Lorentz transformation.

But they can't be in the same inertial frame since each observer measures the other as accelerating.

Is that actually true? Think carefully. Look at it from the Newtonian vantage point of an observer at rest on the Earth: both free-falling observers are at the same altitude, and therefore accelerate downward (relative to Earth--note that this is coordinate acceleration we are talking about) to the same degree, because the Earth's gravity pulls on them both with the same (coordinate) acceleration. What does that imply about the local vantage point?

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

There is no "paradox"; you are simply finding limitations in Newtonian physics. Those limitations are real, and are a major reason why relativity was invented.

Dale
Mentor
2021 Award
I am not sure what paradox you are trying to resolve. Your definitions are excessively complicated, so I suspect that is the source of the confusion.

Try this instead: In both Newtonian physics and relativity an inertial object is one that is not subject to any real forces.

In Newtonian physics there exist reference frames where inertial objects have 0 coordinate acceleration. These are called inertial frames. There are also frames where an inertial object has non 0 coordinate acceleration. In such frames we can introduce fictitious forces to make Newton's 2nd law work. Such forces have the characteristic that they are proportional to the mass and cannot be detected by an accelerometer.

Newtonian gravity, although it shares those two features of fictitious forces, is nonetheless classified as a real force. In relativity, gravity is reclassified as a fictitious force. This allows the generalization that inertial objects have 0 covariant derivative.

PeterDonis, thank you! Your explanation was succinct, complete,and very helpful. It's posts like yours that make this forum such a valuable resource.

I now have a better understanding of the differences between classical and general relativistic inertial frames. After your explanation, it almost seems obvious.

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PeterDonis
Mentor
PeterDonis, thank you!