Prove Maxwell Eqs. Covariant: Wave Eqn & 4th-Vector Pot.

[martindrech]

Is it enough to see the covariance of the wave equation the fourth-vector potential (\phi, \bar{A}) satisfy? I mean, is this enough to prove the covariance of Maxwell equations?

The equation would be ∂_{\mu}∂^{\mu}A^{\nu}=\frac{4\pi}{c} J^{\nu}

 

[PeterDonis]

Why can't you just look at Maxwell's equations directly to see that they are covariant?

\partial_{\mu} F^{\nu \rho} + \partial_{\nu} F^{\rho \mu} + \partial^{\rho} F^{\mu \nu} = 0

\nabla_{\mu} F^{\mu \nu} = 4 \pi J^{\nu}

 

[martindrech]

Simply because is easier to look (using fourth-vectors) at the equations for the potentials instead of the equation for the fields.

 

[PeterDonis]

Simply because is easier to look (using fourth-vectors) at the equations for the potentials instead of the equation for the fields.

If the two equations are logically equivalent, yes, you could look at either one. But I don't think the wave equation for the 4-potential is logically equivalent to Maxwell's Equations; Maxwell's Equations imply the wave equation, but I'm not sure the converse is true.