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 $\ell_{k;i}=?$
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2 Answers
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The legs of the tetrads are just vector fields. Hence the covariant derivative of a tetrad leg is simply
$$\ell_{k;i} = \partial_i \ell_k - \Gamma^j_{ik}\ell_j$$
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$\begingroup$ but in Newman Penrose christoffel symbol doesn't arises directly then how it will reates the Riccis rotation coefficient? $\endgroup$ Commented Oct 7, 2017 at 8:20
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$\begingroup$ Sir in spin coefficients approach what will be the value $\endgroup$ Commented Oct 8, 2017 at 10:07
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In Newman-Penrose (NP) formalism, 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: $$Dl^{a}=(ε+\barε)l^{a}-\barκm^{a}-κ\bar m^{a}$$ $$Δl^{a}=(γ+\barγ)l^{a}-\barτm^{a}-τ\bar m^{a}$$ $$δl^{a}=(\barα+β)l^{a}-\barρm^{a}-σ\bar m^{a}$$ $$\barδl^{a}=(α+\barβ)l^{a}-\barσm^{a}-ρ\bar m^{a}$$
