Derivation of the Electromagnetic Stress-Energy Tensor in Flat Space-time

Question

I am working on deriving the electromagnetic stress energy tensor using the electromagnetic tensor in the $(-, +, +, +)$ sign convention. However, I have hit a snag and cannot figure out where I have gone wrong.

$$ F^{mu alpha}= 
    begin{bmatrix}
    0 & frac{E_{x}}{c} & frac{E_{y}}{c} & frac{E_{z}}{c} 
    -frac{E_{x}}{c} & 0 & B_{z} & -B_{y} 
    -frac{E_{y}}{c} & -B_{z} & 0 &  B_{x} 
    -frac{E_{z}}{c} & B_{y} & -B_{x} & 0 
    end{bmatrix}
$$

$$
F^{mu}_{alpha} = 
    begin{bmatrix}
    0 & frac{E_{x}}{c} & frac{E_{y}}{c} & frac{E_{z}}{c} 
    frac{E_{x}}{c} & 0 & B_{z} & -B_{y} 
    frac{E_{y}}{c} & -B_{z} & 0 &  B_{x} 
    frac{E_{z}}{c} & B_{y} & -B_{x} & 0 
    end{bmatrix}
$$

$$ T^{munu} = frac{1}{mu_0}(F^{mu alpha}F^{v}_{alpha} - frac{1}{4}eta^{munu}F_{alphabeta}F^{alpha beta})$$

Doing matrix multiplication of the matrices $F^{mu alpha}$ and $F^{nu}_{alpha}$ from above gives

$$
F^{mu alpha}F^{nu}_{alpha} =
	begin{bmatrix}
	(frac{E}{c})^{2} & -B_{z}frac{E_{y}}{c} + B_{y}frac{E_{z}}{c}  & frac{E_{x}}{c}B_{z} - frac{E_{z}}{c}B_{x} & -frac{E_{x}}{c}B_{y} + frac{E_{y}}{c}B_{x} 
	B_{z}frac{E_{y}}{c} - B_{y}frac{E_{z}}{c} & -B_{z}^{2} - B_{y}^{2} - (frac{E_{x}}{c})^{2} & -frac{E_{x}}{c}frac{E_{y}}{c} + B_{y}B_{x} & frac{E_{x}}{c}frac{E_{z}}{c} + B_{z}B_{x} 
	-B_{z}frac{E_{x}}{c} + B_{x}frac{E_{z}}{c} & -frac{E_{y}}{c}frac{E_{x}}{c} + B_{x}B_{y} & -(frac{E_{y}}{c})^{2}-B_{z}^{2}-B_{x}^{2} & -frac{E_{y}}{c}frac{E_{z}}{c} + B_{z}B_{y} 
	B_{y}frac{E_{x}}{c} - B_{x}frac{E_{y}}{c} & -frac{E_{z}}{c}frac{E_{x}}{c} + B_{x}B_{z} & -frac{E_{z}}{c}frac{E_{y}}{c} + B_{y}B_{z} & -(frac{E_{z}}{c})^{2}-B_{y}^{2}-B_{x}^{2} 
	end{bmatrix}
$$

Subtracting the $frac{1}{4}eta^{munu}F_{alphabeta}F^{alphabeta}= frac{1}{4}eta^{munu}[2(B^{2} - (frac{E}{c})^{2})]$ and multiplying by $frac{1}{mu_{0}}$ gives

$$ T^{munu}=frac{1}{mu_{0}}  
   begin{bmatrix}
	(frac{E}{c})^{2} + frac{1}{2}(B^{2} - (frac{E}{c})^{2}) & -B_{z}frac{E_{y}}{c} + B_{y}frac{E_{z}}{c}  & frac{E_{x}}{c}B_{z} - frac{E_{z}}{c}B_{x} & -frac{E_{x}}{c}B_{y} + frac{E_{y}}{c}B_{x} 
	B_{z}frac{E_{y}}{c} - B_{y}frac{E_{z}}{c} & -B_{z}^{2} - B_{y}^{2} - (frac{E_{x}}{c})^{2} - frac{1}{2}(B^{2} - (frac{E}{c})^{2}) & -frac{E_{x}}{c}frac{E_{y}}{c} + B_{y}B_{x} & frac{E_{x}}{c}frac{E_{z}}{c} + B_{z}B_{x} 
	-B_{z}frac{E_{x}}{c} + B_{x}frac{E_{z}}{c} & -frac{E_{y}}{c}frac{E_{x}}{c} + B_{x}B_{y} & -(frac{E_{y}}{c})^{2}-B_{z}^{2}-B_{x}^{2} - frac{1}{2}(B^{2} - (frac{E}{c})^{2}) & -frac{E_{y}}{c}frac{E_{z}}{c} + B_{z}B_{y} 
	B_{y}frac{E_{x}}{c} - B_{x}frac{E_{y}}{c} & -frac{E_{z}}{c}frac{E_{x}}{c} + B_{x}B_{z} & -frac{E_{z}}{c}frac{E_{y}}{c} + B_{y}B_{z} & -(frac{E_{z}}{c})^{2}-B_{y}^{2}-B_{x}^{2} - frac{1}{2}(B^{2} - (frac{E}{c})^{2}) 
	end{bmatrix}
$$

However, the textbook definition of the electromagnetic stress energy tensor is:
$$ T^{munu} =
    begin{bmatrix}
    frac{1}{2}(epsilon_{0} |E|^{2} + frac{1}{mu_{0}}|B|^{2}) & frac{S_{x}}{c} & frac{S_{y}}{c} & frac{S_{z}}{c} 
    frac{S_{x}}{c} & -sigma_{xx} & -sigma_{xy}  & -sigma_{xz} 
    frac{S_{y}}{c} & -sigma_{yx} & -sigma_{yy}  & -sigma_{yz} 
frac{S_{z}}{c} & -sigma_{zx} & -sigma_{zy} & -sigma_{zz} 
    end{bmatrix}
$$
with $vec{S} = frac{1}{mu_{0}}(vec{E} times vec{B})$ and $sigma_{ij} = epsilon_{0} E_{i}E_{j} + frac{1}{mu_{0}}B_{i}B_{j} - frac{1}{2}(epsilon_{0} E^{2} + frac{1}{mu_{0}}B^{2})delta_{ij} $

So, I know my $T^{01} = T^{10}$, $T^{02} = T^{20}$, and $T^{03} = T^{30}$ but they do not. They are of opposite signs. What did I do incorrectly?

user4552 · Answer

You could do yourself a big favor by working in units with $c=1$. It's really cumbersome writing all the factors of $c$. You can always reinsert them at the end if you want a result in SI. Or if you want to compare with Wikipedia's equation that's expressed in SI, just drop all the $c$'s from WP's version.

You detected the problem because the final result lacked the proper symmetry. So look at your calculation for the first place where that symmetry is lost. This happens at the very first step, where you calculate $F^{mualpha}F^mu{}_alpha$. This should be the same as $g_{betaalpha}F^{mualpha}F^{nubeta}$, which is manifestly symmetric in $mu$ and $nu$. For example, its 01 component is $F^{02}F^{12}g_{22}+F^{03}F^{13}g_{33}=E_yB_z-E_zB_y$, which is the same as its 10 component. I think the problem is that you say you found this result by matrix multiplication. The ordinary rules of matrix multiplication assume that both matrices are written in mixed upper-lower index form.

Also note that in general there is a distinction between $T^mu{}_nu$ and $T_nu{}^mu$, so you can't just write $T^mu_nu$ without ambiguity. If this was just an issue with mathjax, the syntax that works is this: T^mu{}_nu.

Jay · Answer

Ok...I have done a little more work and I think I have it.
If 
$$ F^{mu alpha}= 
    begin{bmatrix}
    0 & frac{E_{x}}{c} & frac{E_{y}}{c} & frac{E_{z}}{c} 
    -frac{E_{x}}{c} & 0 & B_{z} & -B_{y} 
    -frac{E_{y}}{c} & -B_{z} & 0 &  B_{x} 
    -frac{E_{z}}{c} & B_{y} & -B_{x} & 0 
    end{bmatrix}
$$
and
$$ T^{mv} = frac{1}{mu_0}(F^{mu alpha}F^{nu}{}_{alpha} - frac{1}{4}eta^{munu}F_{alphabeta}F^{alpha beta})$$
then letting c = 1
$$
F^{mu alpha}F^{v}{}_{alpha} = eta_{betaalpha}F^{mualpha}F^{nubeta}
$$
If we let $mu = 0,1,2,3$, $nu = 0, 1, 2, 3$, and $alpha=beta$ summing over repeated indexes gives

for $mu = 0$ and $nu = 0$

$$
eta_{beta alpha}F^{0alpha}F^{0beta} = eta_{00}F^{00}F^{00}+eta_{11}F^{01}F^{01}+ eta_{22}F^{02}F^{02} + eta_{33}F^{03}F^{03} = E_{x}^{2} + E_{y}^{2} + E_{z}^{2}
$$

for $mu = 1$ and $nu = 0$

$$
eta_{beta alpha}F^{1alpha}F^{0beta} = eta_{00}F^{10}F^{00}+eta_{11}F^{11}F^{01}+ eta_{22}F^{12}F^{02} + eta_{33}F^{13}F^{03} = B_{z}E_{y} - B_{y}E_{z}
$$

and so on...

Is this correct? When looking at the indexes for the metric will $alpha=beta$ always be true when working in general relativity?

Derivation of the Electromagnetic Stress-Energy Tensor in Flat Space-time

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