Tensor of electromagnetic field

[Petar Mali]

$$F_{\mu\nu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}$$

$$F_{\mu\nu}=-F_{\nu\mu}$$

$$F_{ii}\equiv 0$$

$$F_{11}=F_{22}=F_{33}=F_{44}=0$$
where

$$A_{\mu}=(-\vec{A},\frac{1}{c}\varphi)$$$$(F_{\mu\nu})=\left(\begin{array}{cccc}<br /> 0& -B_z&B_y& -\frac{1}{c}E_x\\<br /> B_z&0&-B_x& -\frac{1}{c}E_y \\<br /> -B_y&B_x&0&-\frac{1}{c}E_z\\<br /> \frac{1}{c}E_x& \frac{1}{c}E_y & \frac{1}{c}E_z & 0\\<br /> \end{array} \right)$$$$F^{\mu\nu}=g^{\mu\rho}g^{\nu\sigma}F_{\rho\sigma}$$$$(F^{\mu\nu})=\left(\begin{array}{cccc}<br /> 0& -B_z&B_y& \frac{1}{c}E_x\\<br /> B_z&0&-B_x& \frac{1}{c}E_y \\<br /> -B_y&B_x&0&\frac{1}{c}E_z\\<br /> -\frac{1}{c}E_x& -\frac{1}{c}E_y & -\frac{1}{c}E_z & 0\\<br /> \end{array} \right)$$

How do I know that $$rotA_{\mu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}$$ is electromagnetic field tensor?

 

[tiny-tim]

Hi Petar!

How do I know that $$rotA_{\mu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}$$ is electromagnetic field tensor?

I don't understand your question …

isn't A_µ defined as the potential of the electromagnetic field tensor (in which case that has to be rotA)?

 

[Petar Mali]

Well its all ok for me except why I say that $$F_{\mu\nu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}$$

is EM field tensor. Why not

$$F_{\mu\nu}=\frac{\partial A_{\mu}}{\partial x^{\nu}}-\frac{\partial A_{\nu}}{\partial x^{\mu}}$$

for example?

Or some other functions?

It's like a postulate. EM field tensor is $$F_{\mu\nu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}$$

Let's form a matrix. No problem. Components of that matrix are electric field components and magnetic field components. Ok. That have sence. But how I know to start with $$F_{\mu\nu}$$. Do you know perhaps history of this problem.

 

[tiny-tim]

Why not

$$F_{\mu\nu}=\frac{\partial A_{\mu}}{\partial x^{\nu}}-\frac{\partial A_{\nu}}{\partial x^{\mu}}$$

for example?

That's minus the electromagnetic field tensor, so yes, it'll also be an electromagnetic field tensor.

Let's form a matrix. No problem. Components of that matrix are electric field components and magnetic field components. Ok. That have sence. But how I know to start with $$F_{\mu\nu}$$. Do you know perhaps history of this problem.

As you say, E and B are the 6 components of F.

F has to be a tensor because experiment tells us that is the way E and B transform in different frames.