Maxwell stress tensor in different coordinate system

[dapias09]

Hi guys,

I would like to know if the answer given to this thread is correct

https://www.physicsforums.com/showthread.php?t=457405

I got the same doubt, is the expression for the tensor given in cartesian coordinates or is it general to any orthogonal coordinate system?

Thanks in advance

 

[Andy Resnick]

The Maxwell stress-energy tensor can be written in a co-ordinate free form:

4πT$^{ij}$ = F$^{ik}$F$^{j}_{k}$-1/4 η$^{ij}$F$_{ab}$F$^{ab}$

So any coordinate system may be used.

 

[dapias09]

Hi Andy, thanks for your answer.

Well, my punctual question is, can I type

$$ T_{ij} = (E_iE_j - \frac{1}{2}\delta_{ij}E^2) + (B_iB_j - \frac{1}{2}\delta_{ij}B^2)$$

with the dummy indices equal to $x$, $y$, $z$ as well as $r$, $\theta$, $\phi$ , or the indices of any other coordinate system.

I don't know if the expression given is valid only for cartesian coordinates

 

[Meir Achuz]

It would hold in any orthogonal coordinate system, because there are no derivatives involved.

 

[dapias09]

Ok, thank you clem.