D'Alembertian and wave equation.

[yungman]

I am studying Coulomb and Lorentz gauge. Lorentz gauge help produce wave equation:
$$\nabla^2 V-\mu_0\epsilon_0\frac{\partial^2V}{\partial t^2}=-\frac{\rho}{\epsilon_0},\;and\;\nabla^2 \vec A-\mu_0\epsilon_0\frac{\partial^2\vec A}{\partial t^2}=-\mu_0\vec J$$
Where the 4 dimensional d'Alembertian operator:
$$\square^2=\nabla^2-\mu_0\epsilon_0\frac{\partial^2}{\partial t^2}$$
$$\Rightarrow\;\square^2V=-\frac{\rho}{\epsilon_0},\; and\;\square^2\vec A=-\mu_0\vec J$$

So the wave equations are really 4 dimensional d'Alembertian equations?

 

[vanhees71]

Your equations hold for Lorenz (NOT Lorentz!) gauge but not for Coulomb gauge. Otherwise it's indeed the d'Alembert operator. Note further that $1/(\epsilon_0 \mu_0)=c^2$ is the speed of light squared which is (contrary to the conversion factors $\epsilon_0$ and $\mu_0$) a fundamental constant of nature.

 

[yungman]

Your equations hold for Lorenz (NOT Lorentz!) gauge but not for Coulomb gauge. Otherwise it's indeed the d'Alembert operator. Note further that $1/(\epsilon_0 \mu_0)=c^2$ is the speed of light squared which is (contrary to the conversion factors $\epsilon_0$ and $\mu_0$) a fundamental constant of nature.

Thanks for the reply. I am reading Griffiths p422. It specified Lorentz gauge( that's how Griffiths spell it) put the two in the same footing. Actually Griffiths said Coulomb gauge using $\nabla\cdot\vec A=0$ to simplify $\nabla^2V=-\frac{\rho}{\epsilon_0}$ but make it more complicate for the vector potential $\vec A$. That's the reason EM use Lorentz Gauge. This is all in p421 to 422 of Griffiths.

You cannot combine Coulomb and Lorentz Gauge together as

Coulomb $\Rightarrow\;\nabla\cdot\vec A=0$

Lorentz $\Rightarrow\;\nabla\cdot\vec A=\mu_0\epsilon_0\frac{\partial V}{\partial t}$

 

[WannabeNewton]

It's an extremely common mistake but it should be Lorenz not Lorentz. Yes even Griffiths made that mistake.

 

[PJC01]

Yes, the wave equations shown above can be written in terms of the 4 dimensional d'Alembertian operator, which is a mathematical tool used in the study of waves and wave-like phenomena in four-dimensional spacetime. This operator is a combination of the Laplace operator (which represents the spatial variation of a function) and the second partial derivative with respect to time (which represents the temporal variation of a function). By using the d'Alembertian operator, we can describe the behavior of waves in both space and time, making it a powerful tool in many fields of science, including electromagnetism.

 

[PJC01]

Yes, the wave equations shown here are essentially 4 dimensional d'Alembertian equations. The d'Alembertian operator, denoted by ∇^2 - μ₀ε₀(∂²/∂t²), is often used in physics to describe the behavior of waves in a given system. In this case, the d'Alembertian equations are being used to describe the behavior of the electric and magnetic fields in a system, as represented by the potentials V and A, respectively. The presence of the d'Alembertian operator in these equations indicates that the propagation of these fields follows wave-like behavior. The use of the Coulomb and Lorentz gauges in this context helps to simplify and clarify the equations, making it easier to analyze and understand the behavior of the fields. Overall, these equations provide a powerful tool for understanding the behavior of electromagnetic waves and their interactions with matter.