Spin-Coefficient Formalism

NP Formalism (Newman-Penrose) > s.a. Edth Operator; gravitational instanton; tetrads.
* Idea: A formalism that recasts the non-linear, second-order Einstein equation as a set of first-order, mostly linear equations for a set of spin connection coefficients, by a systematic use of null tetrads.
* Advantages: It makes transparent the Petrov type of the space, and the Bianchi identities become particularly simple and tractable.
* Applications: It has become a standard tool for finding exact solutions, solving wave equations in curved backgrounds, studying black-hole perturbations (see Chandrasekhar's book), and analyzing the asymptotic structure of gravitational fields at null infinity.

Individual Coefficients > s.a. weyl tensor.
* Idea: 12 complex quantities that replace the 24 real Ricci rotation coefficients of an orthonormal tetrad when one uses a null one; In an asymptotically flat space time, there is a natural choice of tetrad in the asympototic region for which the Weyl scalars fall off at different powers of r.
– ₀:= C_abcd l^a m^b l^c m^d, Transverse radiation propagating along l^a.
– ₁:= C_abcd l^a m^b l^c n^d, Longitudinal radiation propagating along l^a; Pure gauge.
– ₂:= C_abcd l^a m^b m*^c n^d, "Mass aspect", the Coulomb part of the field.
– ₃:= C_abcd l^a n^b m*^c n^d, Longitudinal radiation propagating along n^a; Pure gauge.
– ₄:= C_abcd m*^a n^b m*^c n^d, Transverse radiation propagating along n^a; Falls off most slowly.

References > s.a. horizons.
@ General: Newman & Penrose JMP(62), PRL(65); Pirani in(65); in Misner et al 73, 870–871; Penrose & Rindler 84, 86; in Wald 84, Sec 13.2; Law a0802 [4D neutral metrics]; Bäckdahl CQG(09)-a0905 [constants in terms of Geroch-Hansen multipole moments].
@ For Riemannian metrics: Goldblatt GRG(94).
@ And Lanczos potential: Andersson & Edgar JMP(00)gq/98.
@ And Sparling forms: Frauendiener GRG(90).
@ Invariants of Riemann spinor: Haddow gq/95.
@ Applications, special spacetimes: Bruni et al gq/04-in [in astrophysical relativity]; Wu & Shang CQG(07) [stationary]; Zhang et al PRD(09) [stationary electrovacuum]; > s.a. kerr-newman.
@ Variations: Ortaggio et al CQG(07)gq [in higher dimensions, Ricci identities]; Law JGP(09) [four-dimensional neutral metrics].

GHP Formalism (Geroch, Held & Penrose)
* Idea: A variation of the Newman-Penrose spin coefficient formalism.
@ General references: Geroch, Held & Penrose JMP(73); Edgar & Ludwig GRG(96), GRG(97), GRG(97)gq [integration], GRG(00); Held GRG(99).
@ Related topics: Ludwig & Edgar CQG(00) [generalized Lie derivative]; Carminati & Vu GRG(01), GRG(03) [Maple package].

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