Gauss' Law as a derivative of the electromagnetic field tensor

[mjordan2nd]

Homework Statement

Prove $$\nabla \bullet E =4 \pi \rho$$ from $$\partial_{\beta}F^{\alpha \beta}=4 \pi J^{\alpha}$$ where $$J^{\alpha}=(\rho, J^{1}, J^{2}, J^{3})$$.

Homework Equations

We are given that $$F_{\alpha \beta}$$ is

0~~~~E_x~~~E_y~~~E_z
-E_x~~~0~~~~-B_z~~B_y
-E_y~~B_z~~~~0~~~-B_x
-E_z~~-B_y~~~B_x~~~0

(Sorry, don't know how to do matrices.)

Raising the indices I should get

0~~~-E_x~~-E_y~~-E_z
E_x~~~0~~~~-B_z~~B_y
E_y~~B_z~~~~0~~~-B_x
E_z~~-B_y~~~B_x~~~0

The Attempt at a Solution

$$\partial_{\beta}F^{\alpha \beta}=4 \pi J^{\alpha}=>-\partial_{i}F^{0i}=\rho=-\partial_{i}E_{i}$$. I don't know why I keep getting that pesky negative sign! Can anyone point me in the right direction?

 

[mjordan2nd]

For whatever reason I can't edit the latex, but it should say it as follows:

(interestingly enough after this post the op corrected itself)

Homework Statement

Prove $$\nabla \bullet E =4 \pi \rho$$ from $$\partial_{\beta}F^{\alpha \beta}=4 \pi J^{\alpha}$$ where $$J^{\alpha}=(\rho, J^{1}, J^{2}, J^{3})$$.

Homework Equations

We are given that $$F_{\alpha \beta}$$ is

0~~~~E_x~~~E_y~~~E_z
-E_x~~~0~~~~-B_z~~B_y
-E_y~~B_z~~~~0~~~-B_x
-E_z~~-B_y~~~B_x~~~0

(Sorry, don't know how to do matrices.)

Raising the indices I should get

0~~~-E_x~~-E_y~~-E_z
E_x~~~0~~~~-B_z~~B_y
E_y~~B_z~~~~0~~~-B_x
E_z~~-B_y~~~B_x~~~0

The Attempt at a Solution

$$\partial_{\beta}F^{\alpha \beta}=4 \pi J^{\alpha}=>-\partial_{i}F^{0i}=\rho=-\partial_{i}E^{i}$$. I don't know why I keep getting that pesky negative sign! Can anyone point me in the right direction?

 

[mjordan2nd]

Bleh, never mind, I got it. Just got the rows and columns confused somehow... Thanks anyway, sorry for wasting your time.