In the Newman–Penrose (NP) formalism of general relativity, Weyl scalars refer to a set of five complex scalars { Ψ 0 , Ψ 1 , Ψ 2 , Ψ 3 , Ψ 4 } {\displaystyle \{\Psi _{0},\Psi _{1},\Psi _{2},\Psi _{3},\Psi _{4}\}} which encode the ten independent components of the Weyl tensor of a four-dimensional spacetime.
Definitions Given a complex null tetrad { l a , n a , m a , m ¯ a } {\displaystyle \{l^{a},n^{a},m^{a},{\bar {m}}^{a}\}} and with the convention { ( − , + , + , + ) ; l a n a = − 1 , m a m ¯ a = 1 } {\displaystyle \{(-,+,+,+);l^{a}n_{a}=-1\,,m^{a}{\bar {m}}_{a}=1\}} , the Weyl-NP scalars are defined by
Ψ 0 := C α β γ δ l α m β l γ m δ , {\displaystyle \Psi _{0}:=C_{\alpha \beta \gamma \delta }l^{\alpha }m^{\beta }l^{\gamma }m^{\delta }\ ,}
Ψ 1 := C α β γ δ l α n β l γ m δ , {\displaystyle \Psi _{1}:=C_{\alpha \beta \gamma \delta }l^{\alpha }n^{\beta }l^{\gamma }m^{\delta }\ ,}
Ψ 2 := C α β γ δ l α m β m ¯ γ n δ , {\displaystyle \Psi _{2}:=C_{\alpha \beta \gamma \delta }l^{\alpha }m^{\beta }{\bar {m}}^{\gamma }n^{\delta }\ ,}
Ψ 3 := C α β γ δ l α n β m ¯ γ n δ , {\displaystyle \Psi _{3}:=C_{\alpha \beta \gamma \delta }l^{\alpha }n^{\beta }{\bar {m}}^{\gamma }n^{\delta }\ ,}
Ψ 4 := C α β γ δ n α m ¯ β n γ m ¯ δ . {\displaystyle \Psi _{4}:=C_{\alpha \beta \gamma \delta }n^{\alpha }{\bar {m}}^{\beta }n^{\gamma }{\bar {m}}^{\delta }\ .}
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