I have been working through a relativistic gravitation book ("Gravitation and Cosmology" by Stephen Weinberg) and decided to circle back to the early tensor work in chapter two and just work out the basic tensor math to make sure that I have a feel for how it all goes together. Right at the beginning of this I'm in trouble. Or maybe I'm not, but I can't tell. This is also my first attempt at doing anything with LaTeX, so if something isn't correct with my presentation of all of this, please let me know.(adsbygoogle = window.adsbygoogle || []).push({});

Starting with one of the basics - the tensor for electrodynamics, from the text:

[tex]

F_{\gamma\delta}=\eta_{\gamma\alpha}\eta_{\delta\beta}F^{\alpha\beta}

[/tex]

with

[tex]

F^{\alpha\beta} = \left(\begin{array}{cccc}

0 & E_1 & E_2 & E_3\\

-E_1 & 0 & B_3 & -B_2\\

-E_2 & -B_3 & 0 & B_1\\

-E_3 & B_2 & -B_1 & 0\end{array} \right)

[/tex]

and

[tex]

\eta_{\gamma\alpha} = \eta_{\delta\beta} = \left(\begin{array}{cccc}

-1 & 0 & 0 & 0\\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1 \end{array} \right)

[/tex]

If I break down the operation into steps, I can perform two binary operations by creating an interim tensor, [tex]T_\delta^{ \alpha} = \eta_{\delta\beta}F^{\alpha\beta}[/tex] (I already think there's a problem here) and using this in a final operation [tex]F_{\gamma\delta}=\eta_{\gamma\alpha}T_\delta^{ \alpha}[/tex]

We find the components of [tex]T_\delta^{ \alpha}[/tex] by performing the summation over [tex]\beta[/tex]:

[tex]T_0^{ 0} = \eta_{0\beta}F^{0\beta} = 0[/tex]

[tex]T_0^{ 1} = \eta_{0\beta}F^{1\beta} = E_1[/tex]

[tex]T_0^{ 2} = \eta_{0\beta}F^{2\beta} = E_2[/tex]

And so forth until we get all 16 elements of [tex]T_\delta^{ \alpha} [/tex]

When all is done, I have [tex] T_\delta^{ \alpha} = \left(\begin{array}{cccc}

0 & E_1 & E_2 & E_3\\

E_1 & 0 & -B_3 & B_2\\

E_2 & B_3 & 0 & -B_1\\

E_3 & -B_2 & B_1 & 0\end{array} \right) [/tex]

When I take this to the next step, [tex] F_{\gamma\delta}=\eta_{\gamma\alpha}T_\delta^{ \alpha} [/tex] I get

[tex] F_{\gamma\delta} = \left(\begin{array}{cccc}

0 & -E_1 & -E_2 &- E_3\\

E_1 & 0 & B_3 & -B_2\\

E_2 & -B_3 & 0 & B_1\\

E_3 & B_2 & -B_1 & 0\end{array} \right) [/tex]

Now this isn't obviously wrong, it just looks wrong. But maybe it's not. Can someone tell me if the initial formulation is correct (I copied this out of "Gravitation and Cosmology") and if the interim tensor [tex]T_\gamma^{ \alpha}[/tex] is correct, or maybe point out what fundamental error I created here?

thanks to all - Mark

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# Tensor operations, Maxwell's field equations

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