One question in a test I am going to take is: How does the form of the electromagnetic wave equations $$ \Delta \phi - \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = - 4\pi \rho $$ indicate relativistic invariance?

Is there a way to directly conclude this?

  • $\begingroup$ can't you just make lorentz transformation and see wheter the form of the equation changed? $\endgroup$ – Umaxo Oct 14 '20 at 8:58
  • $\begingroup$ Yes, but the exact question is: "What does the form of the equations tell you about their validity regarding relativity." The only answer I can think would be that they can be written in a covariant form.. $\endgroup$ – kaos Oct 14 '20 at 9:01
  • $\begingroup$ Is it not written in a covariant form? On the right hand side is just d'Alembert operator $\endgroup$ – Umaxo Oct 14 '20 at 9:08

Scalars are Lorentz invariant, also the differential operator (d'Alembert).

Edit: The question contains only one component of the EM field not a scalar/anything else. So for the answer see the comments below.

  • $\begingroup$ Does the same apply for the equation for the vector potential? $\endgroup$ – kaos Oct 14 '20 at 10:01
  • $\begingroup$ Well, for a simple vector field: yes. For the EM vector porential: I think only for a Lorentz-invariant gauge. $\endgroup$ – RobertSzili Oct 14 '20 at 10:11
  • 1
    $\begingroup$ Yeah, for the 4-potential only Lorenz-gauge condition gives a manifest Lorentz invariant equation, i.e., the wave equation. $\endgroup$ – daydreamer Oct 14 '20 at 10:30
  • $\begingroup$ The scalar potential is not a Lorentz-scalar, neither is the charge density. The are the time-component of a 4-vector. $\endgroup$ – Frederic Thomas Oct 14 '20 at 11:50
  • $\begingroup$ Sorry, the question was a little bit misleading. $\endgroup$ – RobertSzili Oct 14 '20 at 11:59

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