Galilean Limits of Electromagnetism | Azimuth
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Galilean Limits of Electromagnetism
Maxwell’s equations are invariant under Lorentz transformations. The usual equations of fluid flow are not! Like the rest of Newtonian mechanics, they’re invariant under Galilean transformations like
So, if we simply slap these two theories together, we get a mess! How can we study electrically conductive fluids—like plasma—without bringing special relativity into the game?
We can use a limiting case of Maxwell’s equations where we ignore terms that become tiny when all the particles are moving much slower than light.
There seem to be at least two ways to do this: there’s an ‘electric limit’ of Maxwell’s equations and a ‘magnetic limit’. Both are invariant under Galilean transformations. The original derivation of these limits by Le Bellac and Lévy-Leblond in 1973 used the version of Maxwell’s equations including the electric permittivity and magnetic permeability of the vacuum, whose product is . This is convenient but not necessary, as explained here:
• Jose A. Heras, The Galilean limits of Maxwell’s equations.
In the magnetic limit of Maxwell’s equations, we throw out effects due to time-varying electric fields:
People often use the magnetic limit when studying nonrelativistic electrically conductive fluids. In this situation they often consider a version of the magnetic limit where the charge density is zero, since this is typically close to true in a plasma. However Heras does not do this, nor does the original paper:
• Le Bellac and Levy-Leblond, Galilean electromagnetism.
In the electric limit of Maxwell’s equations, we throw out effects due to time-varying magnetic fields:
It’s fun to compare the magnetic and electric limits.
The magnetic limit has been called ‘pre-Maxwellian’, because it’s like electromagnetism before Maxwell added the extra term that makes a changing electric field create a curl in the magnetic field. Without this term there is no light!
In the electric limit you also can’t have light, because it’s missing the term that makes a changing magnetic field create a curl in the electric field.
In the magnetic limit you can’t have capacitors, because those store energy in the electric field, and in the magnetic limit the energy density is just .
Similarly, in the electric limit you can’t have inductors, because inductors store energy in the magnetic field, and in this limit the energy density is just .
It’s all nicely symmetrical! But still somewhat mysterious to me. All the derivations of these limits that I’ve seen involve too many parameters for my taste, and too much talk. But that’s how I often feel when I’m just starting to study a piece of physics.
Besides the two papers mentioned in my last post, I’ve been looking at this:
• Giovanni Manfredi, Non-relativistic limits of Maxwell’s equations.
There’s a lot I haven’t explained here. I haven’t even said how the electric or magnetic fields transform under Galilean boosts in these limiting theories! I find this subject fairly confusing, and I’d probably have to redo all the calculations to really understand them. As Feynman said, "what I cannot create I do not understand".
Someday I should dig deeper into this subject and explain how the two limits work in a way I find satisfying. I should also draw the connections to this earlier article of mine:
• Magnetohydrodynamics.
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This entry was posted on Saturday, July 18th, 2026 at 4:35 pm and is filed under physics. You can follow any responses to this entry through the RSS 2.0 feed.<br>You can leave a response, or trackback from your own site.
11 Responses to Galilean Limits of Electromagnetism
Michael Weiss says:
18 July, 2026 at 4:52 pm
In the other direction, I assume someone has worked out relativistic equations for fluid flow?
I can imagine that the combination with Maxwell’s equations would be useful when studying some astrophysical phenomena? Maybe the jets near black holes?
(I’m lazy busy, so I didn’t try looking the answers up on Wikipedia.)
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Bob says:
18 July, 2026 at 6:34 pm
Just a quick comment–it is not hard to make a relativistic description of an ideal fluid. But it is tricky to include viscosity, because the coefficient of viscosity is usual defined in a certain rest frame, and the terms that describe dissipation don’t transform in an obvious way under Lorentz transformations.
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Collin Merenoff says:
22 July, 2026 at 8:50 am
Why is that a problem? If the rest frame is the surface of a pseudo-time coordinate that evolves dynamically, is nowhere inertial, and can pass the same spacetime point many times, then all you need to do is replace all the derivatives and integrals with their relativistic equivalents.
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Justin says:
18 July, 2026 at 9:28 pm
That was also my thought on seeing this, but I think that would be relativistic magnetohydrodynamics per e.g. this paper...