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 ([itex]\phi[/itex], [itex]\bar{A}[/itex]) satisfy? I mean, is this enough to prove the covariance of Maxwell equations?

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

[itex] [/itex]

 

[PeterDonis]

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

[tex]\partial_{\mu} F^{\nu \rho} + \partial_{\nu} F^{\rho \mu} + \partial^{\rho} F^{\mu \nu} = 0[/tex]

[tex]\nabla_{\mu} F^{\mu \nu} = 4 \pi J^{\nu}[/tex]

 

[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.

 

[sardar]

- \frac{1}{c^2} \frac{\partial^2 A^{\nu}}{\partial t^2}

In order to prove the covariance of Maxwell's equations, it is not enough to simply show that the wave equation and the fourth-vector potential satisfy the same covariance transformation. The wave equation and the fourth-vector potential are just one small part of a larger set of equations known as Maxwell's equations, which describe the fundamental laws of electromagnetism.

To truly prove the covariance of Maxwell's equations, one must show that all four equations (Gauss's law, Gauss's law for magnetism, Faraday's law, and Ampere's law) are covariant under Lorentz transformations. This means that the equations must have the same form and hold true in all inertial reference frames.

Furthermore, the wave equation and fourth-vector potential are not independent of the other equations in Maxwell's equations. They are derived from the other three equations and must also be covariant in order for the entire set of equations to be covariant.

In summary, while the covariance of the wave equation and fourth-vector potential is an important aspect of proving the covariance of Maxwell's equations, it is not enough on its own. The entire set of Maxwell's equations must be shown to be covariant under Lorentz transformations in order for the covariance to be fully proven.