Formulations of General Relativity  

As Minkowski Field Theory > s.a. [perturbations]; stress-energy pseudotensors.
* Idea: Recast general relativity as a non-linear theory of the departure of the metric from ab; Works at a linearized level, where one gets a spin-2 field theory, but such theories cannot describe global features such as different spacetime topologes.
@ References: in Weinberg 72; Penrose in(80), in(82); Weinberg & Witten PLB(80); Zel'dovich & Grishchuk SPU(86); Nikolic GRG(99)gq; Straumann ap/00-in; Pitts & Schieve FP(04)gq [causality], FP(03)gq/04 [FRW singularity]; Padmanabhan IJMPD(08)gq/04 [no-go results]; Notte-Cuello & Rodrigues IJMPD(07)mp/06 [Yang-Mills type]; Nieuwenhuizen EPL(07)-a0704; Pitts & Schieve TMP(07) [massive]; in Leclerc CQG(07)gq; Hacyan a0712 [historical]; Baryshev AIP(06), a0809-in [and tests]; Notte-Cuello et al a0907; & Nambu, Feynman, Thirring; > s.a. gauge fixing; phenomenology of gravity.

Metric Variables > s.a. einstein's equation [including approximation schemes]; quasi-local formulation.
* Degrees of freedom: Electric and magnetic parts of the Weyl tensor.
* Metric + connection version: The Palatini formulation of general relativity, or metric-affine gravity.
@ Independent metric and connection: Krasnov ht/06; > s.a. first-order action [Palatini], other types of action.

Connection without Metric > s.a. Gauge Theory of Gravity.
@ Affine connection: Kijowski & Werpachowski RPMP(07)gq/04; Poltorak gq/04-in.
@ GL(4)-invariant form: Floreanini & Percacci CQG(90), CQG(90).
@ Connection + scalar density: Capovilla et al PRL(89), CQG(91); Capovilla & Jacobson MPLA(92)gq; Bengtsson & Peldán PLB(90); Peldán PLB(90) [with matter]; Bengtsson PLB(91); Dadhich et al CQG(91).
@ Two connections: Barbero IJMPD(94)gq/93, PRD(94)gq/93.
@ Two-forms: Plebanski JMP(77); Bengtsson gq/93, CQG(95)gq; Katsuki et al IJMPA(96) [BF theory, quantum gravity]; Pillin CQG(96) [and matter]; Grant gq/97 [self-dual]; Lewandowski & Okolów CQG(00)gq/99; Krasnov a0904 [pedagogical]; > s.a. BF theories, gravity.
@ Other formulations: Floreanini & Percacci CQG(91) [GL(3) connection + other]; Kozameh & Newman GRG(91); Capovilla et al CQG(91), Robinson CQG(96) [sl(2,C)-valued connection, and forms], JMP(03) [generalized forms].

Connection + Spinorial / Vierbein Variables > s.a. initial-value formulation; loop variables; Metric-Affine Gravity.
@ General references: Deser & Isham PRD(76); Dubois-Violette & Madore CMP(87); Abe IJMPA(90) [supersymmetric]; Obukhov & Tertychniy CQG(96); Clayton CQG(97)gq/06 [constraint algebra]; Lusanna & Russo gq/98, gq/98; Aldrovandi & Barbosa gq/02 ["spacetime skeleton"]; Aldrovandi et al GRG(03)gq [gravity as anholonomy], gq/04-in; Kummer & Schütz EPJC(05)gq/04; Estabrook PRD(05)gq/04 [structure of vacuum equations], CQG(06)gq/05 [conservation laws]; Zinoviev ht/05, ht/05 ["dual" formulation]; Cianci et al CQG(05)mp, IJGMP(06)mp; Wang PRD(05)gq; Abou-Zeid & Hull JHEP(06) [chiral expansion]; Itin a0711-in [coframe]; > s.a. canonical general relativity.
@ With fermions: Fatibene et al GRG(98); Canarutto JMP(98) [including degenerate tetrad]; Godina et al GRG(00).
@ With bosons: Fatibene et al GRG(99).

As a Theory of Null Surfaces > s.a. 3D gravity.
* Variables: Surfaces Z(xa; , *), where and * are stereographic coordinates on S2 and parametrize the different null surfaces through each point (there is an S2 worth, but only 9 are independent), and a conformal factor .
* And metric: Can construct explicitly a conformal metric and from the condition that g'ab a Z b Z = 0 for all (, *), and they satisfy Gab = 0 or gab; One naturally gets a complex g', but it is easy to impose reality.
* Drawbacks: The equations are messy; Their nature (hyperbolic, elliptic?) is not known, and only some non-local solutions are known; There is a large, not really understood, gauge freedom.
* Quantization: Might suggest to quantize only , but not really believed.
@ References: Kozameh et al AP(91); Kozameh & Newman GRG(91), in(91), et al JGP(92); Iyer et al JGP(96)gq/95 [holonomies and light cone cut functions]; Frittelli et al JMP(95)gq, JMP(95)gq, JMP(95)gq, JMP(95), PRD(97), JMP(00).

Other Versions > s.a. actions; canonical general relativity; einstein's equation; gravity theories; initial-value formulation; semiclassical gravity.
* Motivation: Look for hyperbolic formulation to show that evolution is well-posed.
* Euclideanized: It has no asymptotically Euclidean nontrivial solution in any dimension, with any topology [@ in Witten CMP(81)].
@ Theory of embeddings: Deser et al PRD(76); Gibbons & Wiltshire NPB(87)ht/01; Paston & Franke TMP(07)-a0711 [canonical]; Faddeev a0906 [action]; > s.a. branes; quantum gravity; spherical symmetry.
@ Group manifold: D'Adda et al AP(85); Regge PRP(86)-in; Nelson & Regge IJMPA(89).
@ Discretized versions: Boström et al gq/93; Regge & Williams JMP(00)gq; Castellani & Pagani AP(02)ht/01; Gambini & Pullin in(03)gq/01; Holfter & Paschke JGP(03)ht/02 [Dirac operator]; > s.a. lattice gravity, regge calculus.
@ Bundle of Frames version: (2 × 2 matrix) Chinea PRL(84); Chinea & Guil JMP(85).
@ Related topics: Essén IJTP(90); Bonanos JMP(91); 't Hooft NPB(91) [chiral]; Cho et al PLB(92) [diffeomorphism gauge theory]; Teitelboim NPB(93) [theory of paths]; Wallner JMP(95); Grant CQG(96)gq/94 [volume-preserving vector fields]; Anderson gq/99 [no need for metric]; Godina et al GRG(00)gq/99 [2-spinor + Dirac fields]; Baulieu ht/00-in [supersymmetric gauge theory]; Kokarev gq/02-GRF [elastic bending of spacetime]; Novello gq/07 [two spinors]; Atkins a0803 [cohomological version, in terms of cochain complex of (n+2)-tensors].
> Other: see 2D gravity; 3D gravity; 3D general relativity; conformal invariance; Ether; gravitational thermodynamics; higher-dimensional gravity; Modified Gravity [MOG]; modified theories; quantum gravity; teleparallel.

Formalism, Techniques > s.a. complex structures; fluids; geometry; hamilton-jacobi theory; numerical general relativity.
@ And global differential geometry: Fischer & Marsden GRG(74); Eguchi, Gilkey & Hanson PRP(80); > s.a. differentiable manifolds.
@ Global methods: Geroch GRG(71), in(71); Penrose 72; > s.a. causality, singularities.
@ Metric from geodesics: Hojman & Rodrigues PLA(91); > s.a. riemann tensor [gab from curvature].
@ Non-holonomic, non-rigid frames for rotating matter: Marklund et al gq/97-MG8.
@ Related topics: Szydlowski et al JMP(96) [with Jacobi metric]; Rocek & van Nieuwenhuizen gq/06 [smoothing, models].

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