Spin-Coefficient Formalism| 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.
– Ψ0:=
Cabcd la
mb lc
md,
Transverse radiation propagating along
la.
– Ψ1:=
Cabcd
la
mb
lc
nd, Longitudinal radiation
propagating along la;
Pure gauge.
– Ψ2:=
Cabcd
la
mb
m*c
nd,
"Mass aspect", the Coulomb part of the field.
– Ψ3:=
Cabcd
la
nb
m*c
nd,
Longitudinal radiation propagating along
na; Pure gauge.
– Ψ4:=
Cabcd m*a
nb m*c
nd, Transverse radiation
propagating along na;
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].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 31 oct 2019