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Homework Statement
The Lorenz gauge ∂Φ/∂t + ∇. A = 0 enables the Maxwell equations (in terms of potentials) to be written as two uncoupled equations;
∂2Φ/∂t2 - ∇2Φ = ρ 1 and
∂2A/∂t2 - ∇2A = j 2
The tensor version using the Lorenz gauge is, i am told,
∂μ∂μ Aα = jα 3
expanded this is: ∂2Φ/∂t2 - ∇2A = jα 4
where Jα is the 4-current; ρ + J
Homework Equations
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If one adds 1 and 2 one gets two terms not included in 4. Namely;
∂2A/∂t2 - ∇2Φ My query is, what happened to these terms when we go to 3 (or 4). Have I misunderstood the transition from 1 and 2 to the tensor forms?
The Attempt at a Solution
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If one tries to cancel these terms, or equate their sum to zero, one sees that the first is a vector and the second a scalar. Secondly; ∂2A/∂t2 = ∂E/∂t which > 0 for a time varying field. And ∇2Φ = ∇. E = div E which also is not zero unless no charges are present, which they can be in 3.
Any enlightenment would be greatly appreciated.