= Weyl tensor
{c}
{title2=$C_{\alpha\beta\gamma\delta}$}
{wiki}
The Weyl tensor is the completely trace-free part of the <Riemann curvature tensor>. In four dimensions,
$$
\begin{aligned}
C_{\alpha\beta\gamma\delta}
={}&R_{\alpha\beta\gamma\delta}
-\frac12(
g_{\alpha\gamma}R_{\beta\delta}
+g_{\beta\delta}R_{\alpha\gamma}\\
&\qquad
-g_{\alpha\delta}R_{\beta\gamma}
-g_{\beta\gamma}R_{\alpha\delta})
+\frac R6(
g_{\alpha\gamma}g_{\beta\delta}
-g_{\alpha\delta}g_{\beta\gamma}).
\end{aligned}
$$
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