John Carlos Baez on Nostr: Hardcore physics post Maxwell's equations are invariant under Lorentz ...
Hardcore physics post
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
t' = t, 𝐱' = 𝐱 - 𝐯t
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. People use the magnetic limit when studying electrically conductive fluids.
The original derivation of these limits by Le Bellac and Levy-Leblond in 1973 used the version of Maxwell's equations including the electric permittivity ε₀ and magnetic permeability μ₀ of the vacuum, whose product is 1/c². This is convenient but not necessary, as explained here:
• Jose A. Heras, The Galilean limits of Maxwell's equations, https://arxiv.org/abs/1012.1068
An interesting feature of the magnetic limit shown here is that the charge density ρ has to be zero! Luckily this is close to true in a plasma.
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
t' = t, 𝐱' = 𝐱 - 𝐯t
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. People use the magnetic limit when studying electrically conductive fluids.
The original derivation of these limits by Le Bellac and Levy-Leblond in 1973 used the version of Maxwell's equations including the electric permittivity ε₀ and magnetic permeability μ₀ of the vacuum, whose product is 1/c². This is convenient but not necessary, as explained here:
• Jose A. Heras, The Galilean limits of Maxwell's equations, https://arxiv.org/abs/1012.1068
An interesting feature of the magnetic limit shown here is that the charge density ρ has to be zero! Luckily this is close to true in a plasma.