A little thought experiment

I got into a long and protracted argument with someone on an online forum recently over a seemingly minor point and got accused of not knowing basic physics.  It would be funny if the guy weren’t so rude and dismissive about it.  At any rate, part of the argument came out of the following thought experiment I concocted (actually the version I present is a much better version than the one that sparked the argument).  In it, I devise a situation (albeit highly contrived) in which it is impossible to tell whether a certain object is attracted to a certain type of box due to gravitation or electrostatics.  This highly contrived story takes place in a universe in which we have massless, charged particles (something we don’t presently see in nature) and assumes that the given materials are all that you have in your possession (actually, in theory, you wouldn’t even need massless, charged particles as long as it was plausible that you could have a small black hole inside the box).


You are in possession of a neutral object of some mass like, for instance, a piece of Scotch tape or a balloon or something of that nature.  You are also in possession of  special type of box that happens to be both massless and neutral.  In the box is either a strong gravitational field source (a neutral, massive object) or a strong electromagnetic field source (a massless, charged object).  You do not know which.  You begin to bring the object (i.e. the balloon or tape or whatever) closer to the box.  At some point it becomes attracted to the box.  It is your job to determine whether the attraction is due to gravity or electrostatics.

If it is an electrostatic attraction that occurs, this is due to polarization which means the sign of the charge producing it doesn’t matter (this is a simple experiment/demo I do with Scotch tape in one of my classes).  Since you can’t see inside the box you have no idea how much charge or mass is present (so, for example, the box might even contain a small black hole) which means you can’t determine what the source is based on the rate at which the object was attracted to the box.  Therefore, you have absolutely no way to tell whether it was attracted due to gravity or electromagnetism!

Are there holes in the argument?  Remember that this is all you have access to (i.e. you can’t go running out to get an electrically charged object to see if it is repelled).


25 thoughts on “A little thought experiment

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  1. Yes, big hole. Force due to gravity goes like 1/r^2, dipole monopole interactions go like 1/r^3, and induced dipole monopole interactions scale like 1/r^5. Therefore you just need to know the force at two points. If you insist on locality, you just need to know the derivative of the force.

  2. But what experiment could you perform to determine the derivative of the force? I’m thinking empirically here (and I am insisting on locality, i.e. no measuring the force at two points). I’m not saying you’re wrong I just really want to know.

  3. Throw in another hole – when you let the balloon fall it will generate electromagnetic waves. Observing how those waves propagate around the box will also distinguish the two systems.

  4. I’ve got a better question – how do you measure the presence of a force in a purely local fashion? At some point you’re going to have to measure an acceleration or measure the deformation of some macroscopic object. The moment you do that you can, in principal, detect the difference.

  5. I can even go so far as to give you a 1 dimensional macroscopic object – just watch the deformation over time. I admit, that I’m basically cheating here and using the extent of the object to measure the force at multiple points, but I don’t know of any way to actually measure any force at one point without resorting to F=ma or comparing it with some reference force (eg a spring).

    Even then, it’s somewhat disingenuous to only allow a single measurement. Because it should be obvious that since E&M and gravity are both forces we have to have some way to define the difference. That difference is usually defined in terms of the behavior of multiple test objects placed in different conditions.

  6. I suppose you’re right about measuring force locally. Honestly I was just parroting Einstein’s thought experiment. But anyway, there doesn’t seem to be a purely mechanical way to tell. Obviously you can measure the presence of the E&M field which would give it away.

    This all came out of an argument over the nature of spacetime, by the way, which is why I was focused on a mechanical method for measurement. It was argued that the curvature of spacetime was solely due to gravitational fields. I argued otherwise and was accused of having less knowledge than an undergraduate (which is pretty ripe considering I have former students who managed to get into Yale and Dartmouth for grad school and who took GR, QM, Modern Physics, and a host of other courses from me).

    Anyway, my point was that you could theoretically build up something that looks an awful lot like GR but with with charge instead of mass. Allowing for massless charges would allow for a metric describing the curvature of spacetime solely in terms of charge. The obvious difference is that electrostatic forces are both repulsive and attractive while gravity appears to only be attractive (I say “appears” since we had to reintroduce the cosmological constant into GR about a decade ago thanks to observational evidence and thus it might be that there is some kind of gravitational repulsiveness depending on how that is interpreted – dark matter, dark energy, all sorts of wild and crazy things).

  7. I see what you’re getting at – and I figured out the answer to my question. To measure the force at a point you “imagine” repeating the measurement with smaller and smaller apparatuses and taking a limit that way. QM puts a lower limit on that process in reality, but if you’re sticking with the classical realm there’s no problem with it.

    You are, of course, correct in your description. In fact this is one of the sources of why I find gravity kind of odd: it presents itself like any other gauge theory in terms of the having a connection, curvature, and metric, but it’s the only one where we essentially insist on a Lagrangian that’s linear in the curvature and a metric that’s dynamical. In every other gauge theory we leave the metric static, putting the dynamics in the connection, and use a Lagrangian that’s second order in the curvature. I know about f(R) theories for gravity (as in, I know they exist), and I know a couple of the justifications – we want a 2nd order Lagrangian that reproduces regular gravity at low curvature – but I still find it to be an odd duck.

  8. Yeah, gravity is a seriously odd duck. I’m glad you see what I’m getting at, even if it is whacked or even wrong. The folks I argued with on the forum didn’t bother even trying and, despite one of them having helped me with a fairly complex presentation recently, accused me of knowing less than an undergraduate – and in a rather derogatory manner at that. What is it with people these days? People can’t have a debate without tossing out personal insults? Well, anyway, intriguing stuff to say the least (gravity I mean, not self-righteous scientists and mathematicians). 🙂

  9. Take your piece of scotch close enough to your box so that your feel the attraction. Then cut it into two parts (is this allowed ?), one close to the box, one far from the box. If the attraction is electrostatic, your piece of scotch was polarized and you should now have one positively charged and one negatively charged half. One is then attracted by the box, and the other repelled.

    On the contrary if the box contains a black hole, both halves should be attracted.

    Is that cheating for you ?

  10. I guess that’s allowed, but what if you have something that you can’t break into pieces? At any rate, it’s an intriguing problem – even if there are holes in it, it’s not necessarily as simple as it first appears.

  11. Are there different objects to choose from? If we are allowed to have another object of different mass, we can check whether or not its acceleration towards the box depends on its mass.

  12. Hi Ian,

    I think there might be a bit of a conflict in your setup. How exactly do you have a massless box? I understand that this may simply be meant as an idealisation, but the fact that massless particles are constrained to move at the speed of light seems to break the setup a little, and if you ignore this, then you can’t use relativity. This may sound unimportant, but I guess you are aiming at making some kind of equivalence principle for electromagnetism, in which case it would seem to matter. If of course the box is massive, then we could use the relationship between inertial mass and gravitational mass to measure to measure it’s gravitational field (by throwing it).

  13. Hi Joe,

    Yes, it is an idealization (and actually one that is fairly common in some thought experiments). The box was supposed to mask what was inside it. So maybe it wasn’t a box. The idea was, though, that it was preventing us from seeing or directly touching the field source on the inside. As I mentioned above, my idea, at its core, is that in a universe in which mass doesn’t exist but charge does, one should be able to build up a theory quite similar (though clearly not the same) to GR using charge and E&M fields. In fact I intend to try it to see what such a universe would look like. The point is that it might open up some new questions (or new approaches to old questions) in this universe.

    By the way, I noticed you’re lecturing at Bob Coecke’s QICS categories school. I didn’t know you were interested in categories. I just gave a presentation on categorical quantum channels at the APS March Meeting. I’ll send you my LaTeX presentation file if you’re interested.


  14. Yes, I guessed it was something like that. Still, the massless of the box makes me uneasy. Does fixing the mass of the box change anything? You could have a fixed mass box and fill it up with photons to up the gravitational mass, or you could reduce some of the mass of the box, or you could replace some amount of the mass of the neutral box with charged particles. I would think that gets the situation you want without having to bend the laws of physics.

    Actually, it is something that I used to think about when I was in school, though from a slightly different perspective: Imagine all massive particles have a charge, and fix all the charges to be positive. Now you have something similar to anti-gravitation. The big difference is of course magnetism, which starts you wondering if there should be a gravitational equivalent of magnetism.

  15. Oh, and yes, I am lecturing at QICS. I’ve read up a bit on categories, but I don’t really have a huge research interest in the area. I’m there to talk about MBQC (blind quantum computing and QMIP=MIP*). That said, some of the categorical formulations of quantum computing are quite neat, and I was planning on proving some of the UBQC results using Bob’s diagrammatic framework.

  16. Joe,

    Tom Moore (who is a cosmologist, textbook writer, and one of the deepest thinkers I know) has a very convincing argument for the fact that magnetic fields are nothing more than electric fields viewed from a frame that is in relative motion to those electric fields. In fact his argument can be extended and used to explain why magnetic monopoles have never been observed in nature.

    I sent Bob my slides from the APS meeting. I guess I have more work to do on it but I’m primarily interested in categories for my work in foundations these days.


  17. Actually, I do know of the reference frame viewpoint you mention. So I guess I’m just wondering if you can get a nice symmetry between mass and charge in this type of set up.

  18. That’s kind of what I’d like to know too. The problem is that if one assumes that the metric encodes the gravitational field then it becomes meaningless to talk about viewing the gravitational field from a different frame. That’s why I want to first see if you can create a toy universe with charge but no mass. What would magnetism look like in such a universe if it now became the case that it was meaningless to talk about viewing the electrical field from a different frame?

  19. Well, I would think the thing to do is simply to consider only special relativity. Forget about the equivalence principle. Then take plane old Newtonian gravity, and look at effect of the Lorentz transformed gravitational field.

  20. Actually, I was thinking about going all the way back to some sort of pseudo-Newtonian thing, e.g. what would things like momentum look like in such a world? How would it be defined?

  21. The massless box is not a problem because he’s simply using it as a way to define a “no go zone”

    Also, cutting the object in half won’t produce a charge separation unless the object is a conductor. For an insulator cutting the balloon in half will just give you two polarized halves.

    Last, I haven’t heard his argument, but I’d be willing to bet that he’s wrong. While you can always make a change of reference frame to reduce the force on any particular charge to a purely electric one (the rest frame of the charge, natch), you cannot always change to a reference frame where there is no magnetic field at a given point (get the hence to a copy of Jackson). You can even prove that a moving electric field has to produce a magnetic field just by combining electrostatics with special relativity (see Purcell’s freshman level textbook, for instance), not all magnetic fields arise from any identifiable motion of a charge. I’m speaking, in particular, of the magnetic field produced by the spin of a fundamental particle, eg an electron. Put an electron in any superposition of up and down spins and it will produce a dipole magnetic field with a particular orientation in space – this arises from the fact that any spin state corresponds to spin up along some particular axis. I understand that quantum field theory introduces some complications to that picture, but I’m not familiar with what they are.

  22. Actually, my argument is precisely (it seems I like that word today) based on a point particle as you describe it. In fact it seems as if you’ve made half of my argument for me – quantum spin naturally leads to a dipole magnetic field, hence monopoles are highly unlikely.

    The additional piece is once again related to an interpretation. We have been conditioned over nearly a century to assume, because of the peculiar behavior of quantum spin and the fact that point charges like electrons have no physical size, that quantum spin isn’t really motion of any sort. But it does impart an angular momentum (i.e. J = L + S) so, if we wanted to be consistent, we shouldn’t assume this. I see no reason not to assume that it appears to represent some kind of motion. In that case the fact that it always produces a dipole magnetic field makes complete sense since it is analogous to the classical case – the moment you introduce an axis about which to rotate something you are immediately presented with two choices regarding which direction you wish to rotate it and thus are able to distinguish two poles (note that if you only had one way to spin something around an axis you could not distinguish the poles since to distinguish them you need to compare the spinning object to some other spinning object).

    In any case, my point is that it’s still relative motion (if we interpret it that way) that produces the magnetic field. In other words, if quantum spin didn’t exist, point particles wouldn’t have magnetic fields. To me, that’s a relativistic idea.

  23. What does this have to do with magnetic monopoles? A magnetic monopole would produce an electric dipole field, naturally, so I don’t see how one is related to another.

    Ok, here’s the problem I see with the angular momentum/spin thing. Let me ask a question – in SR E = gamma * m * c^2. Does that mean that a translating object is more massive? If so, what’s the purpose of distinguishing between energy and mass at all? The point I’m trying to get at is that while rest mass is a category of energy, we should not regard all energy as mass if mass is to remain an independently useful concept. So, too, the distinction between spin and orbital angular momentum. Just like an object at rest can have energy, and thus rest mass, an object that is not rotating can have angular momentum.

    That’s the simplistic case, the more detailed case involves saying that the fermion fields transform in a representation of the Lorentz group that doesn’t permit excitations of the field to have zero angular momentum. In more detail, with fewer words:
    SL(1,3) ~ SO(4) complexified
    SO(4) ~ SU(2) X SU(2)
    => particles must transform under some combinations of the irreducible representations of two SU(2) groups. IIRC electrons are in a (0) X (1/2) + (1/2) X (0).

    So your two choices analogy is wrong because it is, in principal, possible to imagine particles transforming in different representations that allow for more than two possible options: massive vector (spin 1), spin 3/2, scalar (spin 0), etc. While we have yet to observe any fundamental particles with more than 2 spin degrees of freedom (spin 1/2 and massless spin 1), that doesn’t preclude them yet.

    Let’s go back to asking questions since I think that will be more illuminating than making assertions. How do you define something to be physically rotating? The presence or absence of angular momentum seems to be insufficient, even though angular momentum is the generator of rotations, because that would just make the terms synonymous and any possible distinction irrelevant. I would put forth the following definition of whether or not a fundamental particle is rotating: a particle is not rotating if all of the momentum density flux in the field is irrotational. Now, I do not actually know the answer to this of the top of my head, but I believe that even an electron at rest with spin up satisfies this criterion.

    What’s far more interesting, and I imagine has already been done, is to ask what happens to the electron’s spin in a rotating reference frame? How does it behave? Is it possible to have an electron in a zero angular momentum state through the proper choice of rotating frame? That would be another way to define whether or not spin corresponds to some kind of rotational motion. My gut instinct is that it won’t work out because rotating frames are defined in the vector representation, and are thus spin 1. Thus in a rotating frame the electron will appear to have an integer amount of orbital angular momentum, but it’s spin cannot be made to vanish, though the direction it points may change. I haven’t done the math on that one, so I can’t say for sure.

    Another interesting question is to say: what about the electric and magnetic fields? See, if you take an electric monopole and stick a magnetic dipole in the center there is a rotational flux of momentum in the electromagnetic field. This is one of those ones that I’ve assumed that QFT with renormalization addresses, but it would still be an interesting exercise to see done.


  24. Interesting topic..
    Gravitation satisfies equivalence principle, but not electromagnetism (or any other non-gravitational forces).
    Thus since inertial mass is equal to gravitational mass, acceleration of the point particle due to gravity is irrelevant to its mass. But acceleration due to electromagnetic force is inversely proportional to its mass. So if we know the mass of the particle we can find out whether its acceleration is a function of its mass. If so, then it is a non-gravitational force.
    Am I correct?

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