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Tetrads in General Relativity

XI. Field Equations

Generalized Kronecker-delta's

   In calculations with the Levi-Civita symbol (tensor), it is often convenient to express the result in terms of so-called generalized Kronecker delta’s defined by the determinant:

11.16

\[\delta _{{b_1}{b_2}....{b_p}}^{{a_1}{a_2}...{a_p}}: = p!\delta _{[{b_1}}^{{a_1}}...\delta _{{b_p}]}^{{a_p}}\]

For example, when $p = n$ (the dimension of the vector space), one may derive the compact formula

11.17

\[{\varepsilon ^{{a_1}...{a_n}}}{\varepsilon _{{b_1}...{b_n}}} = - \delta _{{b_1}....{b_n}}^{{a_1}...{a_n}}\]

More generally,

11.18

\[\frac{1}{{(n - p)!}} {\varepsilon _{{a_1}...{a_p}{c_{p + 1}}...{c_n}}} {\varepsilon ^{{b_1}...{b_p}{c_{p + 1}}...{c_n}}} = - \delta _{{a_1}...{a_p}}^{{b_1}...{b_p}}\]

The formulae for $p=2,3$, $n=4$

11.19

\[\frac{1}{{2!}}{\varepsilon _{abcd}} {\varepsilon ^{abmn}} = - \delta _{cd}^{mn}{\qquad}\frac{1}{{1!}} {\varepsilon _{abcd}}{\varepsilon ^{akmn}} = - \delta _{bcd}^{kmn}\]