Wednesday, January 3, 2018

general relativity - Square bracket notation for anti-symmetric part of a tensor


I know that $A_{[a} B_{b]} = \frac{1}{2!}(A_{a}B_{b} - A_{b}B_{a})$


But how can write $E_{[a} F_{bc]}$ like the above?


Can you provide a reference where this notational matter is discussed?



Answer



I use following method:


$T_{[ijk]}$


Set indices into determinant


$\left| \begin{matrix} i & j & k \\i & j & k \\i & j & k \end{matrix} \right| = ijk + jki + kij - ikj - jik - kji.$



Next, apply 6 indices and sign into $T$, so we get


$T_{[ijk]} = \dfrac{1}{3!} (T_{ijk}+T_{jki}+T_{kij}-T_{ikj}-T_{jik}-T_{kji}).$


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