% Bernd Berg Oct 04 2018.
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\centerline{\bf Special and General Relativity (PHZ 4601/5606) Fall 2018 }
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\centerline{\bf Set 6}
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16. Four-dimensional Levi-Civita tensor.
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\noindent (1) The tensor $\epsilon^{\alpha\beta\gamma\delta}$ is zero
unless $\alpha\beta\gamma\delta$ is a permutation of the numbers
$1234$. Therefore,
$$\epsilon_{\alpha\beta\gamma\delta}=(-1)^3\epsilon^{\alpha\beta\gamma
  \delta} = - \epsilon^{\alpha\beta\gamma\delta}\,. $$
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\noindent (2) We have $\epsilon_{\alpha\beta\gamma_1\delta_1}
\epsilon^{\alpha\beta\gamma_2\delta_2}=0$ unless either $\gamma_1=
\gamma_2$, $\delta_1=\delta_2$ or $\gamma_1=\delta_2$, $\delta_1=
\gamma_2$ or (otherwise one of the already used numbers will be 
repeated by $\alpha$ or $\beta$ of the sums). Therefore,
$$\epsilon_{\alpha\beta\gamma_1\delta_1} 
  \epsilon^{\alpha\beta\gamma_2\delta_2} = 
  a\,\delta_{\gamma_1}^{\ \gamma_2}\,\delta_{\delta_1}^{\ \delta_2}+
  b\,\delta_{\gamma_1}^{\ \delta_2}\,\delta_{\delta_1}^{\ \gamma_2}$$
holds. With no summation in the permutation indices $\pi_1$ to $\pi_4$
the constants follow from
\begin{eqnarray} \nonumber
 \epsilon_{\alpha\beta\pi_2\pi_3}\epsilon^{\alpha\beta\pi_2\pi_3} =
 \epsilon_{\pi_4\pi_1\pi_2\pi_3}\epsilon^{\pi_4\pi_1\pi_2\pi_3} +
 \epsilon_{\pi_1\pi_4\pi_2\pi_3}\epsilon^{\pi_1\pi_4\pi_2\pi_3} =
 -2 = a\,, \\ \nonumber
 \epsilon_{\alpha\beta\pi_2\pi_3}\epsilon^{\alpha\beta\pi_3\pi_2} =
 \epsilon_{\pi_4\pi_1\pi_2\pi_3}\epsilon^{\pi_4\pi_1\pi_3\pi_2} +
 \epsilon_{\pi_1\pi_4\pi_2\pi_3}\epsilon^{\pi_1\pi_4\pi_3\pi_2} =
 +2 = b\,. 
\end{eqnarray}
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\noindent (3) We have $\epsilon_{\alpha\beta_1\gamma_1\delta_1}
\epsilon^{\alpha\beta_2\gamma_2\delta_2}=0$ unless $\beta_2\gamma_2
\delta_2$ is a permutation of $\beta_1\gamma_1\delta_1$. There are
six such permutations, so that the results is a sum of the form
\begin{eqnarray} \nonumber
   \epsilon_{\alpha\beta_1\gamma_1\delta_1} 
   \epsilon^{\alpha\beta_2\gamma_2\delta_2}\, &=&
    a_1\,\delta_{\beta_1}^{~\beta_2}\delta_{\gamma_1}^{~\gamma_2}
         \delta_{\delta_1}^{~\delta_2}
  + a_2\,\delta_{\beta_1}^{~\gamma_2}\delta_{\gamma1}^{~\delta_2}
         \delta_{\delta_1}^{~\beta_2}
  + a_3\,\delta_{\beta_1}^{~\delta_2}\delta_{\gamma_1}^{~\beta_2}
         \delta_{\delta_1}^{~\gamma_2} \\ \nonumber &+&
    b_1\,\delta_{\beta_1}^{~\beta_2}\delta_{\gamma_1}^{~\delta_2}
         \delta_{\delta_1}^{~\gamma_2}
  + b_2\,\delta_{\beta_1}^{~\delta_2}\delta_{\gamma_1}^{~\gamma_2}
         \delta_{\delta_1}^{~\beta_2}
  + b_3\,\delta_{\beta_1}^{~\gamma_2}\delta_{\gamma_1}^{~\beta_2}
         \delta_{\delta_1}^{~\delta_2} \,,
\end{eqnarray}
where it follows from (no summation)
$\epsilon_{\pi_1\pi_2\pi_3\pi_4}\epsilon^{\pi_1\pi_2\pi_3\pi_4}=-1$
that $a_i=-1$ and $b_i=1$ holds for $i=1,2,3$.
  
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