Maxwell's Equations - Tensor form

[IAN 25]

Homework Statement

The gauge ∂_tχ - A =0 enables Maxwell's equations to be written in terms of A and φ as two uncoupled second order differential equations. However, when the lorentz condition div A = 0 is applied, we are told the equation can be encapsulated as: one tensor equation ∂_μF^μA = j_μ where j is the covariant 4 current. Written out longhand this is:

∂^₂φ/∂t² - ∇²A

which is one coupled second order differential equation in A and φ. What happened to the terms ∂₂A/∂t² -∇²φ. Which were in the uncoupled version?
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Homework Equations

The Attempt at a Solution

 

[Asim Riaz]

The lorentz condition is the divergence of A = 0. This means that the ∇2A term has been removed since it is 0. The ∂2A/∂t2 has been removed because this term is related to ∂2φ/∂t2 through the gauge equation ∂tχ - A = 0. Since ∂2φ/∂t2 = ∂t(∂tχ - A) = ∂2A/∂t2 - ∂tA, we can remove the term ∂2A/∂t2 and end up with the desired equation.