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];
Papadopoulos a1707 [finding integrals and identities];
Gómez & Quiroga a1711 [rev];
Barnich et al a1910 [and the BMS current algebra].

@ __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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