Weyl tensor ($C_{\alpha \beta \gamma \delta }$)

The Weyl tensor is the completely trace-free part of the Riemann curvature tensor. In four dimensions,

In four spacetime dimensions, relative to a timelike unit normal $n$, define $B_{ab}=\frac{1}{2}{ϵ}_{acde}{C}^{de}{}_{bf}{n}^c{n}^f$, with the chosen spacetime metric volume tensor. This is the dual tidal part of the Weyl tensor. Normal contractions vanish by curvature and volume-tensor antisymmetry. Its sign depends on the specified orientation and dualization convention.

Relative to a timelike unit normal $n$, the electric part is $E_{ab}=C_{acbd}{n}^c{n}^d$. It describes the tidal component of the Weyl tensor measured by that observer. Antisymmetry in each curvature pair makes it spatial in both slots; pair symmetry and vanishing traces make it symmetric and trace-free.

Every contraction of the Weyl tensor vanishes. Contracting its first and third indices directly cancels the Ricci and scalar-curvature terms; antisymmetry in each pair and symmetry under exchanging the two pairs reduce all other contractions to this one or make them vanish immediately.

Every conformally flat metric in dimension at least four has identically zero Weyl tensor. The Weyl tensor is the obstruction to local conformal flatness in dimensions at least four.