I Asymmetric Tensor: Overview & Uses

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Is this a homework problem? If so, it should go in the appropriate homework forum, and the homework template should be filled out.

Also, formulas should not be pasted in as images; they should be done using the PF LaTeX feature. Otherwise it's impossible for people to quote your formulas properly when responding.
 
No, this is not a homework, I just learn by myself. thank you for your suggestion.
 
If you generated your equations using LaTeX, simply enclose the code for each equation between $$ delimiters on each side (for standalone equations) or ## delimiters (for "inline" equations, embedded inside a paragraph).
 
We also don't know how to calculate the term, because you don't give us what this omega is. Because omega is antisymmetric, the summation however simplifies, because only ##\frac{1}{2}D(D-1)## components of it are independent.
 
To answer the original question, you have to tell us what you want to achieve. I don't understand what you mean by "how to calculate it". It's just the expression you wrote down, what else should it be?
 

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Thread 'Synchronizing clocks in an inertial frame if light is anisotropic'

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Thread 'Is the variation of the metric ##\delta g_{\mu\nu}## a tensor?'

From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by ##g_{\alpha\beta}## to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation ##\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}## should be a tensor...
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