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Consider two flat infinitely wide and high rectangular magnets located a distance $L$ from each other in what is otherwise a vacuum. Visualized below:

enter image description here

If the magnets are aligned properly there will be an attractive force between them. One interpretation of this attractive force is that there is a sea of virtual photons between the magnets and "if the virtual photons have a negative energy, they contribute to the electromagnetic force as an attractive force", so the QED vacuum between the plates has less average energy content than the vacuum external to the plates.

Superficially this sounds similar to the situation with the Casimir effect where the QED Vacuum between the conducting plates ALSO has less average energy than the outside but they are subtly different.

  1. The Casimir Effect is strictly because there is a lack of virtual photons between the plates (any photons whose wave length is too big are excluded)

  2. The Magnets here could be argued to either:

    a. have a surplus of virtual photons with negative energy between them

    b. have a lack of virtual photons of positive energy between them.

Nevertheless by carefully tuning the magnet strength, and distance between the magnets we can make two identical setups, one being a casimir conducting plate setup, the other being this magnet setup where both setups have identically shaped plates an identical distance apart, experiencing identically strong attractive forces (assuming they are held in place and not moving and the magnets have less conductivity than the conducting plates).

So how really are their vacuums different?

One way to shed light on exactly what's happening might be to ask how does light behave between the plates? In the Casimir effect it's predicted that very subtly light might travel faster in an effect called the Scharnhorst effect. Between these two magnets we also have vacuum with lower energy density than the surrounding vacuum so its natural to ask the same question: "is the speed of light between the magnets and normal to the magnets different at all compared to the speed of light in the classical vacuum?" intuitively we would expect "is it faster?"

Ideally I would have the technical skillset to repeat Scharnhorst's derivation of computing $e_0,\mu_0$ between the plates and then checking for myself but I lack the technical knowledge in QFT to do this at this time.

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    $\begingroup$ This should be answerable using the Euler–Heisenberg Lagrangian. $\endgroup$
    – Ghoster
    Commented Jan 15 at 5:50
  • $\begingroup$ There is no negative energy between the magnets. A magnetic field has a positive energy density. In a magnetar that field is amazingly dense. I didn't do the math but Wikipedia claims that it can have 10,000 times the density of lead. $\endgroup$
    – FlatterMann
    Commented Jan 17 at 4:36
  • $\begingroup$ single source of magnetic field itself should probably have positive energy density but the space between the two magnets is a little less clear. The conceptual thing I'm not clear about here is that those two magnets are exchanging a large number of virtual photons of negative energy (this is what it means to have an attractive force) and so if you ask about the average energy of all the virtual photons between the magnets you do end up with a large negative number (the stronger the attraction the more negative that number gets) $\endgroup$
    – Sidharth Ghoshal
    Commented Jan 17 at 18:42
  • $\begingroup$ the logical jump this questions asks is "well okay if the average energy density of the exchanged photons between the magnets is so negative, is it a different kind of negative density than the casimir effect?" that's why i proceed to ask about speed of light between the magnets specifically $\endgroup$
    – Sidharth Ghoshal
    Commented Jan 17 at 18:43
  • $\begingroup$ For all we know energy is an additive property of spacetime. There is no magnetic energy density and spacetime energy density. There is only total energy density. Dark energy density is roughly equivalent to 1e-31g/cm^3. As soon as there is more baryonic matter (or electromagnetic field) than that anywhere, the overall density is positive. If I trust some of the online calculators a magnetic field of roughly 6e-8T without any additional matter is enough to swamp the current cosmic dark energy density. Such a configuration is not stable. It will dissipate and then dark energy takes over. $\endgroup$
    – FlatterMann
    Commented Jan 20 at 4:21

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If you look at the references to the Wikipedia article you cited you will find a paper by Milloni and Svozil called "Impossibility of measuring faster-than-c signalling by the Scharnhorst effect". The paper explains that any alleged FTL effects aren't measurable because any light signal has a spread of positions and velocities governed by the uncertainty principle that would prevent such a measurement. And since there is no physically possible way to measure this alleged FTL motion, there is no reason to say that it's happening. It's just an artefact of a bad description of the system.

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  • $\begingroup$ Thank you for responding. So your argument here is the Scharnhorst effect is false so surely this thought experiment is false too. You are probably right but this answer is less than desirable for a few reasons. Since the two magnetic strengths can be dialed up arbitrarily high without bringing the plates closer together the uncertainty argument shouldn’t be the way to dismiss this. It should be the case that repeating the calculation of Scharnhorst between the magnetic plates simply does NOT result in a change to e_0 and \mu_0 that violates the speed of light. $\endgroup$
    – Sidharth Ghoshal
    Commented Jan 21 at 17:12
  • $\begingroup$ @FlatterMann makes the comment that a free magnetic field has positive energy density (such as the one surrounding a magnetar). The space between two extremely large (or infinite) magnetically attracted plates however has a known large exchange of negative energy virtual photons. So if one wanted to answer the question without going through Scharnhorst's derivation they would have to argue "despite the negative energy virtual photon exchange the energy density of the vacuum between magnets is positive" or "negative energy virtual photon exchange does NOT affect the vacuum energy density" $\endgroup$
    – Sidharth Ghoshal
    Commented Jan 21 at 23:26

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