# Covariant derivatives of null tetrads

I am trying to understand the Newman Penrose null tetrads and facing some problems. Given $\ell_k$ is a null tetrad in Newman-Penrose formalism, what is $\ell_{k;i}=?$

$$\ell_{k;i} = \partial_i \ell_k - \Gamma^j_{ik}\ell_j$$
In Newman-Penrose (NP) formalmism, instead of using the standard four connections $$∇_{u}$$, we consider instead four locally-defined directional covariant derivatives on the flow of the tetrad, which are historically denoted by $$(D,Δ,δ,\barδ)$$. directional covariant derivatives of tetrad vector $$l_{u}$$ are: $$Dl^{a}=(ε+\barε)l^{a}-\barκm^{a}-κ\bar m^{a}$$ $$Δl^{a}=(γ+\barγ)l^{a}-\barτm^{a}-τ\bar m^{a}$$ $$δl^{a}=(\barα+β)l^{a}-\barρm^{a}-σ\bar m^{a}$$ $$\barδl^{a}=(α+\barβ)l^{a}-\barσm^{a}-ρ\bar m^{a}$$