- #1

olgerm

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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##

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##

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