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kent davidge

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- Thread starter kent davidge
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kent davidge

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I don't understand what you mean by "the retarded scalar potential satisfies the Loren(t)z gauge condition". A scalar potential alone cannot fulfill a gauge condition (except in the temporal gauge condition, ##\Phi=0##, but than in general you cannot in addition fulfill the Lorenz gauge condition if charges and currents are present).

If you impose the Lorenz gauge condition the four-potential separate component wise to wave equations,

$$\Delta A^{\mu}=\frac{1}{c} j^{\mu},$$

and the usually needed solution is the retarded one, using the corresponding Green's function of the D'Alembert operator. The solution then reads

$$A^{\mu}(x)=\int_{\mathbb{R}^4} \mathrm{d}^4 x' G_{\text{ret}}(x-x') \frac{1}{c} j^{\mu}(x').$$

It's important to check that this solution fulfills the gauge condition,

$$\partial_{\mu} A^{\mu}=0,$$

since otherwise these solutions do not lead to a valid solution of Maxwell's equations.

For details, see my SRT FAQ:

https://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

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