Electromagnetic stress-energy formula in wikpedia

[olgerm]

I found equion $T^{\mu\nu} = \frac{1}{\mu_0} \left[ F^{\mu \alpha}F^\nu{}_{\alpha} - \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta} F^{\alpha\beta}\right] \,.$ from wikipedia page https://en.wikipedia.org/wiki/Electromagnetic_stress–energy_tensor .

it's (0,0) component should be electromagnetic energy density which is $\frac{E^2+B^2}{2}$.
But by replacing $\mu$ and $\nu$ with 0 I get
energy density=$T^{0 0}=\sum_{a=0}^D(\eta^{aa}*( (F^{0 a})^2+\frac{1}{4} *\sum_{b=0}^D(\eta^{bb}(F^{ab})^2)))=\sum_{a=0}^D(\eta^{aa}*( (F^{0 a})^2+\frac{1}{4} *\sum_{b=0}^D(\eta^{bb}(F^{ab})^2)))=E^2+\frac{B^2}{2}\neq\frac{E^2+B^2}{2}$
Is it my mistake or is the formula wrong?
i am using sign convention where $\eta^{00}=-1$

 

[olgerm]

Hi.
I just look through and think $E$ cannot come from $F^2$ s . Is it $E^2$?

yes, edited it now.

 

[vanhees71]

I can't make sense of your formulae, but I don't see any obvious mistake in the Wikipedia article.

 

[olgerm]

I can't make sense of your formulae, but I don't see any obvious mistake in the Wikipedia article.

Can you check whether wikipedia equation equals to $\frac{E^2+B^2}{2}$ if you do not understand my equations?

 

[Ibix]

Can you check whether wikipedia equation equals to $\frac{E^2+B^2}{2}$ if you do not understand my equations?

It does. If you download Maxima, a free symbolic maths package, the batch file below will generate all the elements of T if you wish to check.

/* Define the metric */
eta:matrix([-1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]);

/* Define the Faraday tensor with upper indices and generate */
/* mixed and lower index versions                            */
uF:matrix([0,-Ex,-Ey,-Ez],[Ex,0,-Bz,By],[Ey,Bz,0,-Bx],[Ez,-By,Bx,0]);
mF:uF.eta;
lF:eta.uF.eta;

/* Contract the Faraday tensor with itself to get an invariant, F */
F:0;
for alpha:1 thru 4 do block (
  for beta:1 thru 4 do block (
    F:F+uF[alpha,beta]*lF[alpha,beta]
  )
);

/* Calculate the stress-energy tensor */
T:-(1/4)*eta*F;
for mu:1 thru 4 do block (
  for nu:1 thru 4 do block (
    for alpha:1 thru 4 do block (
      T[mu,nu]:T[mu,nu]+uF[mu,alpha]*mF[nu,alpha]
    )
  )
);

/* Display the t-t element of T */
ratsimp(T[1,1]);

Edit: note that the eta after the uF in both lines 7 and 8 should strictly be transpose(eta), but since the metric is always symmetric this has no effect.

 

[olgerm]

It does. If you download Maxima, a free symbolic maths package, the batch file below will generate all the elements of T if you wish to check.

It seams to me that according to that script T[1,1]=(Bx^2+By^2+Bz^2+2*Ex^2+2*Ey^2+2*Ez^2)/2=$E^2+\frac{B ^2}{2}$, but that should be $\frac{E^2+B^2}{2}$. if the formula were correct.
Does it use the same equation that is on my 1. post to generate the matrix?

 

[Ibix]

It seams to me that according to that script T[1,1]=(Bx^2+By^2+Bz^2+2*Ex^2+2*Ey^2+2*Ez^2)/2=$E^2+\frac{B ^2}{2}$, but that should be $\frac{E^2+B^2}{2}$. if the formula were correct.

That's because I somehow posted an incorrect version that calculated $F_{\mu\nu}$ as $\eta_{\mu\rho}F^{\rho\nu}$ instead of $\eta_{\mu\rho}\eta_{\nu\sigma}F^{\rho\sigma}$ (i.e. I calculated the mixed tensor and used it as the lower tensor). Not sure how I managed that. I've corrected the script above (line 8 is the only change) and it gets the correct answer now.