Coulomb Gauge invariance, properties of Lambda

[tylerscott]

Homework Statement

A gauge transformation is defined so as to leave the fields invariant. The gauge transformations are such that [itex]\vec{A}=\vec{A'}+\nabla\Lambda[/itex] and [itex]\Phi=\Phi'-\frac{\partial\Lambda}{\partial t}[/itex]. Consider the Coulomb Gauge [itex]\nabla\cdot\vec{A}=0[/itex]. Find out what properties the function [itex]\Lambda[/itex] must satisfy in order for the Coulomb Gauge to be satisfied.

Homework Equations

Lorentz Gauge Condition
[itex]\nabla\vec{A}+\frac{1}{c^{2}}\frac{\partial\Phi}{\partial t}=0[/itex]

The Attempt at a Solution

[itex]\nabla\cdot\vec{A}=\nabla\left(\vec{A'}+\nabla\Lambda\right)=0[/itex]
[itex]\nabla\cdot\vec{A'}+\nabla^{2}\Lambda=0[/itex]
so
[itex]\nabla\cdot\vec{A'}+\nabla^{2}\Lambda+\frac{1}{c^{2}}\frac{\partial }{\partial t}(\Phi'-\frac{\partial\Lambda}{\partial t})=0[/itex]
and
[itex]\nabla\cdot\vec{A'}+\frac{1}{c^{2}}\frac{\partial\Phi'}{\partial t}=\frac{1}{c^{2}}\frac{\partial^{2}\Lambda}{\partial t^{2}}-\nabla^{2}\Lambda[/itex]

So in order for these to be invariant, [itex]\frac{1}{c^{2}}\frac{\partial^{2}\Lambda}{\partial t^{2}}-\nabla^{2}\Lambda=0[/itex]

So [itex]\Lambda[/itex] must satisfy wave equation.

Now, I feel like I'm missing some of the properties of the gauge, so any help into more insight into [itex]\Lambda[/itex] would be appreciated.

 

[BruceW]

You shouldn't be solving for the Lorentz gauge. The question only mentions the Coulomb gauge. That is the one you need to satisfy. The question says:
[tex]\vec{A}=\vec{A'}+\nabla\Lambda[/tex]
So from here, if you want both A and A' to satisfy the Coulomb gauge, then what does that tell you about lambda? (I'll admit, the question wasn't very specific about what it was asking for).

 

[tylerscott]

I don't think I understand what that tells me about lambda... More insight please? Haha

 

[BruceW]

If a general vector V is going to satisfy the Coulomb gauge, then you need:
[tex]\nabla \cdot \vec{V} = 0 [/tex]
right, and the question is saying that it wants both A and A' to satisfy the Coulomb gauge. Also they give you the equation:
[tex]\vec{A}=\vec{A'}+\nabla\Lambda[/tex]
So using the given information, can you get an equation that contains only lambda (not A or A').

 

[paranormal]

Also, I'm not sure if the properties of \Lambda have to do with Coulomb's Law (electric force between two charges), but if they do, then I would say that \Lambda must satisfy the condition that the potential between two charges is inversely proportional to the distance between them, as this is a fundamental property of Coulomb's Law. Additionally, \Lambda must also satisfy the Lorentz gauge condition, which ensures that the equations of motion are invariant under Lorentz transformations.