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I just can't wrap my head around how non-inverse-square-laws are supposed to work for forces. Why doesn't this (and MOND) violate a fundamental law? I thought t

I just can't wrap my head around how non-inverse-square-laws are supposed to work for forces. Why doesn't this (and MOND) violate a fundamental law? I thought the point of inverse square laws was that we have a point source, area increases with r^2, energy is radiated symmetrically in space, and thus energy at any given point has to drop off with 1/r^2 in order for total energy to remain constant. So how is a non inverse square law supposed to work? How can we not be violating something fundamental like conservation of energy, number of dimensions in space, spherical symmetry, causality, etc.?

dotancohen 4y ago
Maybe things in _our_ perceivable universe follow the inverse-square-law because they exist in the 3 dimensions familiar to us, as you succinctly explain. But maybe the Strong force propagates in less, or more, or different, dimensions? Does the Strong force propagate at C? Can we test that? It might give some hint.
joe__f 4y ago
If you have an electric dipole with a positive and negative electric charge near to each other then the force drops off like 1/r^3, faster than 1/r^2. This result is for a single quark in isolation; in reality quarks come in pairs or triplets, and they each have different colour charges which cancel out the long distance behaviour of the strong force
- dataflow 4y ago
  > If you have an electric dipole [...] the force drops off like 1/r^3 > This result is for a single quark in isolation Sorry, I'm confused. Doesn't this confirm what I'm saying? For multiple particles I get how it could be different from 1/r^2 (though only less than 1/r^2, not more!), but as you say, this is about a single quark in isolation, which is neither multiple particles nor less than 1/r^2, so the problem is still there right? (The fact that quarks aren't ever found in isolation in practice seems irrelevant to me, unless the claim is "quarks absolutely cannot be found in isolation due to conservation of energy", which I've never read.) And what about something like MOND? I see the same problem, and it's about gravity!
  - jcranmer 4y ago
    > though only less than 1/r^2, not more! The electric field of a wire drops off like 1/r, which is larger than 1/r^2.
    - dataflow 4y ago
      A wire doesn't look like a point source from far away, unlike a dipole. Cut off a segment of that wire (so it's not infinite-length) and it won't be 1/r at large distances.
  - joe__f 4y ago
    The dipole example is meant to illustrate the fact that if you have more than one particle then you can have faster decay as you move away from zero. In the dipole example you also have opposite electric charges, which is analogous to the colour charge in the case with the quarks. The strong force has a property called color confinement which means that quarks cannot exist in isolation under normal circumstances. This is a property of the SU(3) gauge theory that describes the strong interaction, as well as something which we observe experimentally. It's not a direct consequence of conservation of energy, although energy will be conserved. MOND is a totally different situation. For the strong force we write down a totally solid theory which is built from all the necessary symmetries and agrees well with current experimental data. In MOND, people take Newton's second law and change it to try and fit the data better. My understanding is that this doesn't work very well, and also I see no reason why it should conserve energy a priori (i.e., maybe it does but this is not clear to me without seeing a proof). Conservation of energy when gravity is taken into consideration is more subtle, so MOND may not need to conserve energy.
atty 4y ago
You can think of it, at a very high level, being the difference between a force mediator that has no self interaction (photon) and a force mediator that does have self-interaction (gluon). Gluons interact with each other, whereas photons don’t. Even this by itself isn’t naively enough to get all the strong force’s interesting behavior. But the math works out in such a way that when you start to pull two strongly bound particles apart, the color field forms a flux tube of gluons between the two particles, and the force required to pull them apart further remains approximately constant (in other words the energy stored per unit length in the flux tube is ~constant). This only happens because of the self interaction. It doesn’t violate energy conservation at all! For the trivial reason that field itself is not the entity using energy to displace two bound particles ;)
- silicon2401 4y ago
  Thanks for posting this comment, it was really entertaining to see how quickly it got wildly over my head and started sounding like sci-fi gibberish > You can think of it, at a very high level, being the difference between a force mediator that has no self interaction (photon) Don't really know what a force mediator is but I can somewhat imagine, and photons not interacting with themselves I think makes sense > a force mediator that does have self-interaction (gluon) I'm guessing these particles interact with themselves and maybe they're called gluons because they stick together or something, like glue > Gluons interact with each other, whereas photons don’t. I think I'm making progress > Even this by itself isn’t naively enough to get all the strong force’s interesting behavior. But the math works out in such a way that when you start to pull two strongly bound particles apart, Good so far > the color field forms a flux tube of gluons between the two particles, lost me, you just switched lanes into back to the future speak
  - dataflow 4y ago
    >> the color field forms a flux tube of gluons between the two particles, > lost me, you just switched lanes into back to the future speak The way I interpreted this was that the interaction happens over a line beam (rather than a spherical surface) so it doesn't drop off as 1/r^2, but as 1 (i.e. constant). Which raised more questions for me (and made me wonder what the MOND case is), but I'm still digesting the comment.
jfengel 4y ago
A completely different way to think about it: it doesn't surprise you that spring forces get stronger as you stretch them. You expect it because there is material there. In the case of strong force, you get virtual particles appearing when you separate particles. So the strength increases with distance. It doesn't become infinite because particles also attract their "opposites", and so the result is zero from a distance. (Like the way single charged particles usually are found in uncharged clumps.) (Opposite is a harder concept in strong force, which has more symmetries than the electrical one.)
- dataflow 4y ago
  I think I could see how this could work, but I'm not sure. So it sounds to me like we have a situation where either (a) quark isolation would violate conservation of energy, or (b) isolated quarks (assuming we could at hypothetically have them in theory, even if not in practice) wouldn't actually exert this constant strong force on each other (at least until they're close together and the "link" can get established, so to speak), and instead only exert some sub-1/r^2 force on each other before then. Am I correct to guess the situation is more like (b)?
  - addaon 4y ago
    Are you confusing force and energy? Force without acceleration does not change energy.
    - dataflow 4y ago
      I don't believe I am. Unless you're saying two particles placed some distance apart in otherwise empty space could exert a force on each other and somehow still not move?
      - addaon 4y ago
        If you have two particles in empty space with an (attractive) force between them… how did they get apart to begin with? This potential energy was introduced to the system when they were separated, and will be converted to kinetic energy as they attract. Color confinement is basically a way of saying that the potential energy needed to separate color-charged particles stays high with increasing distance, until you put in enough energy to produce new (color-neutral) particle pairs that locally bind to the particles you were separating.
pdonis 4y ago
> I thought the point of inverse square laws was that we have a point source, area increases with r^2, energy is radiated symmetrically in space, and thus energy at any given point has to drop off with 1/r^2 in order for total energy to remain constant. This explains an inverse square law for intensity of radiation, but why would you expect it to apply to forces in general?
tlb 4y ago
It can only work between pairs of particles that are bound together. Otherwise, the total forces between every pair of quarks (N^2 pairs) in the universe would be preposterously large.

dotancohen 4y ago

Maybe things in _our_ perceivable universe follow the inverse-square-law because they exist in the 3 dimensions familiar to us, as you succinctly explain. But maybe the Strong force propagates in less, or more, or different, dimensions? Does the Strong force propagate at C? Can we test that? It might give some hint.

joe__f 4y ago

If you have an electric dipole with a positive and negative electric charge near to each other then the force drops off like 1/r^3, faster than 1/r^2. This result is for a single quark in isolation; in reality quarks come in pairs or triplets, and they each have different colour charges which cancel out the long distance behaviour of the strong force

dataflow 4y ago
> If you have an electric dipole [...] the force drops off like 1/r^3 > This result is for a single quark in isolation Sorry, I'm confused. Doesn't this confirm what I'm saying? For multiple particles I get how it could be different from 1/r^2 (though only less than 1/r^2, not more!), but as you say, this is about a single quark in isolation, which is neither multiple particles nor less than 1/r^2, so the problem is still there right? (The fact that quarks aren't ever found in isolation in practice seems irrelevant to me, unless the claim is "quarks absolutely cannot be found in isolation due to conservation of energy", which I've never read.) And what about something like MOND? I see the same problem, and it's about gravity!
- jcranmer 4y ago
  > though only less than 1/r^2, not more! The electric field of a wire drops off like 1/r, which is larger than 1/r^2.
  - dataflow 4y ago
    A wire doesn't look like a point source from far away, unlike a dipole. Cut off a segment of that wire (so it's not infinite-length) and it won't be 1/r at large distances.
- joe__f 4y ago
  The dipole example is meant to illustrate the fact that if you have more than one particle then you can have faster decay as you move away from zero. In the dipole example you also have opposite electric charges, which is analogous to the colour charge in the case with the quarks. The strong force has a property called color confinement which means that quarks cannot exist in isolation under normal circumstances. This is a property of the SU(3) gauge theory that describes the strong interaction, as well as something which we observe experimentally. It's not a direct consequence of conservation of energy, although energy will be conserved. MOND is a totally different situation. For the strong force we write down a totally solid theory which is built from all the necessary symmetries and agrees well with current experimental data. In MOND, people take Newton's second law and change it to try and fit the data better. My understanding is that this doesn't work very well, and also I see no reason why it should conserve energy a priori (i.e., maybe it does but this is not clear to me without seeing a proof). Conservation of energy when gravity is taken into consideration is more subtle, so MOND may not need to conserve energy.

atty 4y ago

You can think of it, at a very high level, being the difference between a force mediator that has no self interaction (photon) and a force mediator that does have self-interaction (gluon). Gluons interact with each other, whereas photons don’t. Even this by itself isn’t naively enough to get all the strong force’s interesting behavior. But the math works out in such a way that when you start to pull two strongly bound particles apart, the color field forms a flux tube of gluons between the two particles, and the force required to pull them apart further remains approximately constant (in other words the energy stored per unit length in the flux tube is ~constant). This only happens because of the self interaction. It doesn’t violate energy conservation at all! For the trivial reason that field itself is not the entity using energy to displace two bound particles ;)

silicon2401 4y ago
Thanks for posting this comment, it was really entertaining to see how quickly it got wildly over my head and started sounding like sci-fi gibberish > You can think of it, at a very high level, being the difference between a force mediator that has no self interaction (photon) Don't really know what a force mediator is but I can somewhat imagine, and photons not interacting with themselves I think makes sense > a force mediator that does have self-interaction (gluon) I'm guessing these particles interact with themselves and maybe they're called gluons because they stick together or something, like glue > Gluons interact with each other, whereas photons don’t. I think I'm making progress > Even this by itself isn’t naively enough to get all the strong force’s interesting behavior. But the math works out in such a way that when you start to pull two strongly bound particles apart, Good so far > the color field forms a flux tube of gluons between the two particles, lost me, you just switched lanes into back to the future speak
- dataflow 4y ago
  >> the color field forms a flux tube of gluons between the two particles, > lost me, you just switched lanes into back to the future speak The way I interpreted this was that the interaction happens over a line beam (rather than a spherical surface) so it doesn't drop off as 1/r^2, but as 1 (i.e. constant). Which raised more questions for me (and made me wonder what the MOND case is), but I'm still digesting the comment.

jfengel 4y ago

A completely different way to think about it: it doesn't surprise you that spring forces get stronger as you stretch them. You expect it because there is material there. In the case of strong force, you get virtual particles appearing when you separate particles. So the strength increases with distance. It doesn't become infinite because particles also attract their "opposites", and so the result is zero from a distance. (Like the way single charged particles usually are found in uncharged clumps.) (Opposite is a harder concept in strong force, which has more symmetries than the electrical one.)

dataflow 4y ago
I think I could see how this could work, but I'm not sure. So it sounds to me like we have a situation where either (a) quark isolation would violate conservation of energy, or (b) isolated quarks (assuming we could at hypothetically have them in theory, even if not in practice) wouldn't actually exert this constant strong force on each other (at least until they're close together and the "link" can get established, so to speak), and instead only exert some sub-1/r^2 force on each other before then. Am I correct to guess the situation is more like (b)?
- addaon 4y ago
  Are you confusing force and energy? Force without acceleration does not change energy.
  - dataflow 4y ago
    I don't believe I am. Unless you're saying two particles placed some distance apart in otherwise empty space could exert a force on each other and somehow still not move?
    - addaon 4y ago
      If you have two particles in empty space with an (attractive) force between them… how did they get apart to begin with? This potential energy was introduced to the system when they were separated, and will be converted to kinetic energy as they attract. Color confinement is basically a way of saying that the potential energy needed to separate color-charged particles stays high with increasing distance, until you put in enough energy to produce new (color-neutral) particle pairs that locally bind to the particles you were separating.

pdonis 4y ago

> I thought the point of inverse square laws was that we have a point source, area increases with r^2, energy is radiated symmetrically in space, and thus energy at any given point has to drop off with 1/r^2 in order for total energy to remain constant. This explains an inverse square law for intensity of radiation, but why would you expect it to apply to forces in general?

tlb 4y ago

It can only work between pairs of particles that are bound together. Otherwise, the total forces between every pair of quarks (N^2 pairs) in the universe would be preposterously large.