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In Newtonian physics, object A observes the acceleration of object B precisely when A observes a net force acting on B. How should we understand perceived acceleration under the observed expansion of the universe? Can it be attributed to a force? Does this mean that there is no such thing as an "inertial reference frame?"

Thought experiment: Two spherical objects A and B are at rest in empty space and are charged in such a way that the net force between them (gravitational and electromagnetic) is exactly zero. Thus, there is no observed force or acceleration between them, and thus no movement is possible. Since electromagnetic force and gravitational force are both inversely proportional to the distance squared, no matter the separation between the two, their forces acting upon each other will always net to zero.

After a certain period of time, the space between the two objects is measured to have expanded. How do we explain what happened? At each moment there should be no net force observed between the two, yet there also appears to be a relative acceleration between the two. Should A therefore believe that some force was acting upon B?

I am also interested in the question if these two objects should be considered to exist in the same inertial reference frame. Since there is no "real" net-force between the two, perhaps the answer is "yes". Yet, since there is observed acceleration between the two, perhaps the answer is "no". This thought experiment may demonstrate that the existence of an ideal inertial reference frame, though not theoretically impossible, is an experimentally observed impossibility.

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  • $\begingroup$ "Two spherical objects A and B are at rest in empty space and are charged in such a way that the net force between them (gravitational and electromagnetic) is exactly zero." Yes this is correct (with some caveats). See my comments to @Dale. $\endgroup$
    – timm
    Mar 1 at 10:06
  • $\begingroup$ "After a certain period of time, the space between the two objects is measured to have expanded." Correct. Expanding space means that the proper distances between comoving observers (who see the universe isotropic) are increasing proportional to the scale factor. You can imagine such (uncharged) observers between A and B and as they don't feel the electrostatic force their proper distances would increase. In other words they would participate in the expansion of the universe (in contrast to A and B). $\endgroup$
    – timm
    Mar 1 at 10:24
  • $\begingroup$ "I am also interested in the question if these two objects should be considered to exist in the same inertial reference frame." No, not in the same. In FLRW-spacetime you can assume local inertial frames in the vicinity of A and B but not one which covers both. Note this doesn't hold in our inhomogeneous universe. $\endgroup$
    – timm
    Mar 1 at 10:42
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Two spherical objects A and B are at rest in empty space and are charged in such a way that the net force between them (gravitational and electromagnetic) is exactly zero.

Dale's answer seems to suggest that this isn't possible in general relativity because gravity isn't a force, but I think that's wrong, or at least misleading.

If the cosmological constant is zero, there's a family of exact solutions to GR describing black holes at relative rest with their charges and masses proportional such that there is no net relative force between them. There's no reason (I think) that you couldn't replace the black holes with lumps of ordinary matter.

If the cosmological constant is positive, then in addition to the usual attractive force, there is an effective repulsive force proportional to $r$ (not $1/r^2$). It should be possible in principle to counter this by decreasing the charge on the bodies, though not realistically in practice since it would be an unstable equilibrium. I'll assume for the rest of this answer that it is possible.

After a certain period of time, the space between the two objects is measured to have expanded.

No, it won't expand. This is the key misunderstanding. We constructed this system to be static, and static it is. The distance between the objects will not change over time.

The only reason the universe is expanding is that (for unknown reasons, possibly related to inflation) it was expanding in the past, and it wasn't dense enough for gravity to stop and reverse the outward inertia. In recent times, the repulsive force due to the cosmological constant has become large enough to be important, but we countered that too in this system. There's nothing else that would make the distance between the objects increase. The universe doesn't want to expand. There is no physical process that enforces the expansion by inserting space between objects. I wrote another answer about this.

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  • $\begingroup$ I agree with your answer with this exception: "We constructed this system to be static, and static it is. The distance between the objects will not change over time". The proper distance will change over time because $\ddot{a}$ changes over time due to decreasing matter density (we talk about the $\Lambda$ dominated FRW-universe). $\endgroup$
    – timm
    Mar 1 at 8:35
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In the presence of non-uniform gravity spacetime is curved and there are no global inertial frames. Inertial frames only cover regions that are small enough to neglect all tidal effects.

Local inertial reference frames do exist and are easy to identify. They are reference frames where all objects with accelerometers* that read zero move in straight lines with respect to the reference frame.

The local gravitational force, like all inertial forces, is not detected by accelerometers, so in a local inertial frame there is no gravitational force. Thus it is not possible to balance the gravitational force and the electrostatic force as described. In an inertial frame there will only be the electrostatic force, and therefore there is a real net force between the two. This will lead to greater acceleration apart than would normally occur due to the expansion.

Over larger distances spacetime is curved and there is no possibility of using an inertial frame. It is possible to write the equations of motion in some non-inertial frame. When you do so you will obtain some terms that can be interpreted as an inertial force, sometimes called a fictitious force. Generally, we do not bother to do that, and just attribute the so-called Christoffel symbols to the coordinate system rather than a force. But it can be done if desired, and even Einstein took that approach at least once.

*By “accelerometer” I refer to the 6 degree of freedom type that measures acceleration and rotation on all three axes.

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  • $\begingroup$ I am having trouble understanding your comment that it is not possible to balance the gravitational and electrostatic force. What if we change the scenario slightly, so that we are object A and we wish to be in an inertial reference frame with object B. If our ratios of mass to charge are just right, then the equations of Newtonian physics tell us that there should be no net-force acting upon us, and thus no net motion. Thus we observe no force acting upon us, so we are together in the same inertial frame. Yet, as time passes, our distance separates. $\endgroup$
    – Eoin
    Feb 28 at 2:53
  • $\begingroup$ There is no gravitational force in an inertial frame. So there is nothing to balance the electrostatic force. A 0 force cannot balance any non-0 force. Inertial frames are not predicated on no net force between the objects, only on having accelerometers. Ideally the accelerometers would read zero, but even if they don’t you can do a little math and still use them to define an inertial frame. $\endgroup$
    – Dale
    Feb 28 at 2:54
  • $\begingroup$ $\frac{kQ_1Q_2}{r^2}=\frac{m_1 m_2 G}{r^2}$. If $\frac{Q_1Q_2}{m_1m_2}=\frac{G}{k}$ then there is no force between the two. We may initialize them together at rest, they stay at rest (until expansion is observed) and an outside observer would say they exert no net force upon the other. Surely they would also say no net force is observed? $\endgroup$
    – Eoin
    Feb 28 at 3:00
  • $\begingroup$ You are trying to use Newtonian gravity in cosmology. It doesn’t work. They are not compatible. At cosmological scales you have to describe gravity using general relativity and in GR gravity is locally a fictitious force. It doesn’t exist in local inertial frames. Part of your confusion is due to failing to respect that incompatibility. $\endgroup$
    – Dale
    Feb 28 at 3:05
  • $\begingroup$ Thank you for your help clearing my confusion. Are you saying that there is no configuration of charge and mass which would ensure that two objects initiated near each other at rest would remain at rest with respect to each other? $\endgroup$
    – Eoin
    Feb 28 at 3:26
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I don't see this story being fundamentally different than the case of rotating frames in Newtonian mechanics. Newtonian mechanics describes the motion of objects an inertial Newtonian frame. If you have a rotating frame, you have "pseudoforces." These are aspects of the equations of motion which properly describe motion in a rotating frame. It just so happens that, in a rotating frame, those extra terms look like forces, in that they are the multiplication of some value times mass (just like a $F=ma$ term). We call them pseudoforces because they're not really forces, they just look enough like them that we act on them.

Likewise, you can correct for curved spacetime in Newtonian physics with enough transformations. You end up writing out all of the extra terms, and they don't provide you the raw insight that GR would provide, but you can do it. Some of these terms will look like a $ma$ term, and we could call them pseudoforces. Others may not (I haven't done the math myself, but my instinct is that many of the transforms are going to yield things that don't look so tidy).

So the reality is that there must be some corrections to Newtonian physics to account for the curvature of spacetime. Whether you treat them as an ugly mass of correction terms, or if you rephrase it in to General Relativity to get more insight, that's up to you.

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