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.

@ __General references__: Newman & Penrose JMP(62), PRL(65);
Pirani in(65); in Misner et al 73, 870–871;
Penrose & Rindler
84, 86; in Wald 84, §13.2; Law JGP(09)-a0802 [4D
neutral metrics]; Bäckdahl CQG(09)-a0905 [constants
in terms of Geroch-Hansen multipole moments]; Nerozzi a1109 [new approach, in transverse tetrads].

@ __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 GRG(96)gq/95.

@ __For 3D Riemannian manifolds__: Aazami JGP-a1410 [and results on hypersurface-orthogonal vector fields and curvature].

**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*.

– Ψ_{0}:= *C*_{abcd} *l*^{a} *m*^{b} *l*^{c}* m*^{d},
Transverse radiation propagating along
*l*^{a}.

– Ψ_{1}:= *C*_{abcd} *l*^{a} *m*^{b} *l*^{c}* n*^{d},
Longitudinal radiation propagating along *l*^{a};
Pure gauge.

– Ψ_{2}:= *C*_{abcd} *l*^{a}
*m*^{b} *m**^{c}* n*^{d}, "Mass
aspect", the Coulomb
part of the field.

– Ψ_{3}:= *C*_{abcd} *l*^{a}
*n*^{b}* m**^{c}* n*^{d},
Longitudinal radiation propagating along *n*^{a};
Pure gauge.

– Ψ_{4}:= *C*_{abcd} *m**^{a} *n*^{b}* m**^{c}* n*^{d},
Transverse radiation propagating along *n*^{a};
Falls off most slowly.

**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].

**Applications and Variations** > s.a. horizons; perturbations of FLRW models.

@ __Special spacetimes__: Bruni et al AIP(05)gq/04
[in astrophysical relativity]; Wu & Shang CQG(07)
[stationary]; Zhang et al PRD(09)
[stationary electrovacuum]; > s.a. kerr-newman solutions.

@ __Higher dimensions__: Ortaggio et al CQG(07)gq [Ricci identities]; Durkee et al CQG(10)-a1002 [GHP formalism]; García-Parrado Gómez-Lobo &
Martín-García JMP(09), JPCS(11)-a1102 [5D]; Ortaggio et al CQG(13)-a1211 [rev].

@ __Variations__: Law JGP(09)
[four-dimensional neutral metrics].

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2015