Proving Maxwell’s Equations to be Invariant Under a Lorentz Transformation

Author: Michael Wilson

Abstract By expanding Maxwell’s equations in terms of the electric and magnetic fields one can obtain the explicit transformation laws for the two fields. One can plug them into Maxwell’s equations as they are invariant.

Keywords: Invariance, covariant, transformation

Maxwell’s equations can be given the following form:

∂F=4cj (1)

∂F+cyclic (,,)=0 (2)

With F=∂A-∂A. Let us look at the first set: The right-hand side is a vector (this can be proven by looking at the equation for the conservation of charge), therefore the left-hand side must be a vector as well. Being ∂ the covariant components of a dual form, F must be the component of a (0,2) rank tensor. As such under Lorentz transformations, they will transform as:

F ´(x ´)=Ʌ (x) Ʌ (x) F (x)

(where no one can equivalently take Ʌ-1according to how such a map is defined).

Expanding the above equations in terms of the electric and magnetic fields (as derived from the vector potential Asuch that F =∂A−∂A) one can obtain the explicit transformation laws for the two fields. You can directly plug them into Maxwell’s equations by brute force they are left invariant. Another way is to notice that the action is left invariant under the effect of some particular transformations; hence, the equations of motion must be as well.


A statement from Michael:

“I love physics. I have always wanted to be a physicist. My favourite areas of physics are gravity and of course electromagnetism. Maxwell’s equations, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. I chose to write about their invariance because Einstein might not have developed the theory of Special Relativity if the Maxwell Equations were not invariant, as they led him to believe that the notion of absolute space and absolute time (everybody everywhere will agree on the same measurement) was wrong. He had to decide who was right, Newton or Maxwell.

Newton got it right, between the eyes that is. It turned out that time was not absolute. For us believing physicists, the barrier between the past, present, and future, is but a stubborn illusion. All of this thanks to James Clerk Maxwell.”

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