A question that has bugged me for ages, and way outside my (limited) physics.

When the nature of space is talked about in the sense of an expanding universe, space expands everywhere but bound entities stay bound, and don’t get bigger. But this of itself doesn’t seem to explain what happens as the space around these objects expands.

So, my question is this.

Consider a single lonely hydrogen atom. Just a proton and electron. It sits all alone in space, as space expands. Over a significant period of time lets say that the size of space doubles along each dimension. So that space the atom occupied is now actually 8 times the volume. But the electron stays bound to the proton via QED. So what happened as the space in and around the atom expanded?

A pair of masses at great enough distance that they are not gravitationally bound simply move further part. However if you were to give each one a nudge in the other’s direction, enough to overcome the expansion of space, they would eventually get within range of one another - and the potential energy between them become real. Given this, it should be possible to wait longer, and have more potential energy added to the pair as space expands. (Or is this exactly cancelled by the extra expanding speed, and thus the extra nudge I need to give the objects?) So we seem to be (maybe) adding energy to the universe in some sense. Now what about bound objects, and my hydrogen atom?

Did the expansion of space cause the electron to gain energy? Could the electron eventually jump an energy level? Would this cause atomic matter spontaneously emit photons as the universe expanded? Is the wave equation for the electron tracing a subtle 3D spiral as space expands within it? Am I so far out of my depth to be asking non-questions?

To avoid this just being a gratuitous bump, and to push the thoughts a bit further - I was trying to think about the question in terms of the Bohr model- which is about where my formal education in this stuff ended. But it isn’t clear that that the Bohr model helps either. It states as an a-priori fact that the electron’s orbit has a circumference of a whole number of wavelengths, but provides no mechanism. So the idea that space is stretching under the orbit either pulls the radius ever so slightly wider - and thus the circumference a teesy bit longer than allowed, or somehow there is an undefined restorative force that keeps the orbit the right length. Then again, what defines length if you can’t measure the start and end points at the same time, and space itself is sliding under you?

The expansion of space doesn’t imply that everything is getting further apart. other forces can counteract the expansion. Objects that are already very far apart will recede from each other, but objects nearby can be kept together by gravity or electromagnetism, or the strong force.

For example, the Milky Way isn’t getting more spread out over time, because gravity between the stars of the Milky Way is strong enough to keep objects at the same distances.

Even the distances between the Milky Way and other nearby galaxies (The Local Group) aren’t necessarily increasing. Some of them are moving towards us and some away from us, but overall, they aren’t really receding much.

Its only when you get beyond the nearby galaxies to greater distances that the expansion of the universe overpowers other effects, afaik.

So the electromagnetic forces that keep the electron orbiting around the nucleus will keep the electron at the same distance for ever, or at least until the protons decay in the very distant future.

The expansion of the universe isn’t a very relevant effect until you get to things in the hundreds of millions of light years apart range.

All true, but the question is what happens as those other forces keep the structures the same size? The world lines of the components are sliding over expanding space in order to keep the sizes the same. It is that sliding world line that bothers me. Just because the other forces overcome the local expansion of space to keep the distance the same doesn’t mean we discount the fact that space expanded locally.

Consider my example of a pair of masses. The gravitational potential energy between the two may be both realisable, and increasing with time. This means the universe has increasing energy content (which it appears is generally acknowledged). So, how does this manifest itself? Can it manifest itself on the atomic scale as well as the extra-galactic scale? Heck, maybe there is a component of the cosmic background attributable to conversion of the expansion of space slowly expanding atoms.

I get that you’re speaking more hypothetically, but in practice I think the expansion of space is totally negligible on the scale of a hydrogen atom. Unless we’re getting into Big Rip scenarios where the rate of expansion keeps increasing forever.

I’m not sure why you think the potential energy increases with time. Either the two masses affect each other gravitationally and thus don’t get farther apart, or they don’t affect each other, get farther apart, and have the same potential energy at the same distance if they ever do get together again.

I don’t really think you are, I reread the rules after I posted that, and it basically says “we seriously frown on any bumps, but if you do, only do it once.”

I’m not sure where the proper physicists are. Maybe I don’t know about an important date in the academic timetable.

I’m not sure either. However two masses in an unexpanding universe will always attract. To avoid this the universe must overcome the acceleration between the two, and that suggests there is a given distance two objects can be which exactly balances. Closer, and they eventually meet (even if in billions of years) - further apart and they never meet, and slowly get further apart. If I have two objects at a greater than this critical distance (as defined for those two objects) and I nudge them enough they will move fast enough that they eventually cross over the (ever moving) critical point, and eventually collide. The energy at their meeting is the gravitational potential energy of the two. If I wait, and the universe expands, can I get more energy due to the greater distance? In a static universe if they were further apart there would be more energy. So, if I let the universe pull them apart is there more?

Indeed, say I have two objects that are just short of the critical distance. In a static universe we can calculate the potential energy outright. In an expanding universe, these two objects may take billions of years to collide, and their world lines will see them traverse a much greater distance than they were apart at the start - and thus they will have fallen towards one another for longer, and further. There should be more energy when they collide than in a static universe. This simplifies things, but is a start on the reasoning.

Similarly with the hydrogen atom. Over a few billion years the space in the atom stretches out. Arguments that it takes such a long time, so it doesn’t matter, make no sense. Current laws of physics don’t have a get out clause about not bothering if it takes too long, so I don’t see why these ones should either. The electron traverses a world line over that time period that does not match the world line it would traverse in a static universe. This should matter. Especially as QCD places some interesting constraints on how it is supposed to move. I feel that although the numbers are mind numbingly small, there is a tiny additional probability that the electron could jump an energy level at any moment, and over the history of the universe these should result in real energy level jumps - perhaps enough that there is a tiny background emission of light. A photon per few billion years per atom.

I think I get what you’re asking, and as far as I know the effect is so miniscule that it is literally impossible to calculate right now. It would require understanding dark energy at a quantum level, and we don’t even understand it at a cosmological level. Even the ridiculously small gravitational force between the proton and electron would override the expansion of space between them, and no one really even knows how that works.

I don’t know enough about cosmology to comment specifically, but I can explain how there is no contradiction necessary here.

Galactic superclusters moving apart from one another can compensate for the increasing potential energy by decreasing their kinetic energy, i.e. slowing down.

That doesn’t imply the universe will stop expanding any more than a rocket traveling faster than escape velocity will return to earth.

I recall there was a debate for a while as to whether there was enough energy for the universe to keep expanding: if not, the universe will collapse again. That debate ended with evidence that the universe is going to expand forever. Presumably that means we’re generally above “escape velocity”.

Indeed, the accelerating expansion of the universe does affect the motion of local, bound systems. The effect is teeny tiny. Some back-of-the-envelope numbers:

galactic cluster scale: 1 part in 10[sup]7[/sup]
galactic scale: 1 part in 10[sup]11[/sup]
solar system scale: 1 part in 10[sup]38[/sup]
atomic scale: 1 part in 10[sup]136[/sup]

If the acceleration is not increasing(*), then the size of the effect will also not increase. In the special case of constant acceleration, the expansion can be thought of as an additional force term in the equations of motion, but this force stays constant. It isn’t adding energy to the local system any more than the force of gravity holding the earth around the sun is. There is energy in the system related to the expansion of the universe, but it’s not an increasing energy. Same thing for hydrogen. If this particular component of the energy tally were to change, the energy levels of the atom would change.

If the acceleration is actually increasing, then this extra force will be increasing with time and will eventually become dominant. In this case, the energy levels of the system would be changing with time toward a scenario where there were no bound states.

Energy accounting gets tricky when you start considering local and non-local effects together. Of note, the dynamics of the universe and of spacetime need not be invariant under time reversal, so the conserved quantity related to time reversal symmetry – namely, energy – needn’t be globally conserved.
(*) this is the case for a cosmological constant or for quintessence models. Although, it hardly matters the model since we have no real idea at this point.

Ignoring dark energy/the cosmological constant/etc for a minute:

If you assume that the mass-energy is completely evenly distributed in the Universe (which is an approximation made in all but the most sophisticated cosmological simulations) then expansion takes place on all scales with the effect becomes less and less on smaller and smaller scales. However the approximation that mass-energy is evenly distributed, whilst being a very good approximation on the largest scales, becomes worse and worse on smaller and smaller scales, so it requires a bit of a closer look to say what the effect of expansion, if any, on smaller scales is.

Expansion is the increase of the distance between co-moving (i.e. otherwise at rest) objects over time, but for a collection of objects existing in a gravitationally bound system, broadly speaking, the distance between them actually tends to decrease over time. There’s two ways of looking at this: we could say that on a smaller scale the denser regions of space in the Universe are actually contracting rather than expanding, which is what many do say, or we could try to subtract out the local gravitational effects on this scale as it’s not clear that objects that are moving towards each other under the influence of gravity are co-moving. We might hope to arrive at some sort of term for expansion which we may expect to act as a small counteracting effect to gravity.

However the first problem with subtracting out the local gravitational effects is that space where mass-energy isn’t evenly distributed is not easy to model, which either leads to approximations being made which are potentially just as unrealistic as totally evenly distributed mass-energy or a very complex and difficult to solve model. The second problem is that there is not necessarily a clear single way to subtract the effects of local gravitational pull out leading to the possibitly of a large degree of arbitrariness in the answer. For these reasons it is still very much an open question how expansion operates on a small scale (and by small I mean small compared to the massive scale of the observable Universe). It’s worth noting that Einstein was one of the first who tried to examine how expansion works on a smaller scale and in his model expansion does not happen at all in the volumes where matter is concentrated, it happens in the spacein between such volumes. However Einstein’s model makes questionable approximations.

Of course though I have specifically ignored dark energy which in the current epoch has the dominant effect on expansion. Most cosmologists suspect (but in truth no-one has any solid evidence one way or the other) that the distribution of dark energy is very even which means that even on tiny scales there could be a small amount of expansion due to dark energy.

Going back to the OPs example of an electron and a proton in otherwise empty space: if space is empty in the most restrictive sense then there is no expansion, so for there to be expansion there must be some sort of dark energy-like effect going on and the exact effect on the proton and the electron will depend entirely on the exact details of the effect. If we assume that the effect takes the form of a positive cosmological constant (which means the spacetime of the Universe is de Sitter space) giving a rate of expansion roughly equal to the rate of expansion in the present day then the answer would be that there would be a mind-bogglingly small tendency for the distance between electron and the proton to increase over time if they weren’t otherwise attracted, however this would be completely dwarfed by the a electromagnetic attraction (and even the tiny gravitational attraction) which would keep them bound together and there would be no noticeable effect on the hydrogen atom.

My understanding was that expansion/contraction is a local phenomenon; any square millimeter of space will expand based on the mass+energy+“quintessence” within it. Even if quintessence is everywhere (and the universe overall is expanding), is it possible that space occupied by an atom is contracting instead?