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].
@ 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]; > s.a. kerr-newman.
@ Variations: Ortaggio et al CQG(07)gq [in higher dimensions, Ricci identities].

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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Send feedback and suggestions to bombelli at olemiss.edu – Modified 29 jun 2008