Formulations of General Relativity  

As Minkowski Space Field Theory > s.a. perturbations; stress-energy pseudotensors.
* Spin-2 field in Minkowski: Recast general relativity as a non-linear theory of the departure of the metric from ηab; This works at a linearized level, where one gets a spin-2 field theory, but such theories cannot describe global features such as different spacetime topologies.
@ References: Ogievetsky & Polubarinov AP(65); in Weinberg 72; Penrose in(80), in(82); Weinberg & Witten PLB(80); Castagnino & Chimento GRG(80)-a1206; Zel'dovich & Grishchuk SPU(86); Nikolić GRG(99)gq; Straumann ap/00-conf; Trenčevski IJTP(11)gq/04 [2-form field and non-linear connection]; Pitts & Schieve FP(04)gq [causality], FP(03)gq/04 [FLRW 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-proc [and tests]; Notte-Cuello et al JPM(10)-a0907; & Nambu, Feynman, Thirring; Deser GRG(10) [and self-interactions]; Rodrigues RPMP(12)-a1109-conf [intro, and legitimate energy-momentum tensor]; Scharf a1208; > s.a. gauge fixing; phenomenology of gravity; theories of gravity.

Metric and Other Variables > s.a. einstein's equation [including approximation schemes]; quasi-local formulation.
* Possibilities: General relativity has three fully equivalent representations as a theory of metric curvature, and torsion or non-metricity of a connection.
* Metric variables: The degrees of freedom are the electric and magnetic parts of the Weyl tensor.
* Metric + connection version: The Palatini formulation of general relativity, or metric-affine gravity.
@ References: Krasnov ht/06 [metric and connection]; Tamanini PRD(12)-a1205 [metric + two affine connections]; Harada a2001 [?]; Jiménez et al a1903 [three alternative formulations].
> Other variables: see 1st-order action [Palatini]; affine connections [non-metricity]; other types of action; teleparallel theories [torsion].

As a Theory of Null Surfaces > s.a. 3D general relativity.
* Variables: Surfaces Z(xa; ζ, ζ*), where ζ and ζ* are stereographic coordinates on S2 and parametrize the different null surfaces through each point (there is an S2's worth, but only 9 are independent), and a conformal factor Ω.
* And metric: One can construct explicitly a conformal metric and Ω from the condition that g'aba Zb Z = 0 for all (ζ, ζ*), and they satisfy Gab = 0 or Λgab; 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 Ω, 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); Bordcoch et al a1201.

Other Versions > s.a. actions; canonical [including ADM] and initial-value formulation; einstein's equation; gravity theories; semiclassical gravity.
* Motivation: Look for hyperbolic formulation to show that evolution is well-posed.
* Euclideanized: It has no asymptotically Euclidean non-trivial solution in any dimension, with any topology [@ in Witten CMP(81)].
@ Theory of embeddings: Deser et al PRD(76); Regge & Teitelboim in(77); Gibbons & Wiltshire NPB(87)ht/01; Paston & Franke TMP(07)-a0711 [canonical]; Faddeev a0906 [action]; Paston & Semenova IJTP(10)-a1003 [canonical, constraint algebra]; Paston TMP(11)-a1111; Willison a1311 [Cauchy problem]; Sheykin & Paston AIP(14)-a1402; Paston et al G&C(17)-a1705 [canonical]; > s.a. branes; friedmann cosmology; quantum gravity; spherical symmetry.
@ Group manifold: D'Adda et al AP(85); Regge PRP(86)-proc; 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]; Huang et al a2006 [on graphs]; > s.a. lattice gravity; regge calculus.
@ Elastic / plastic deformations: Kokarev gq/02-GRF [elastic bending of spacetime]; Fernández & Rodrigues 10 [distorted Lorentz vacuum]; > s.a. spacetime structure.
@ As a theory of paths: Teitelboim NPB(93), ZNA(97); Padmanabhan IJMPD(19)-a1908; > s.a. lines; loop variables; Paths.
@ Other variables: Jadczyk IJTP(79) [conformal structure + scalar density]; Grant CQG(96)gq/94 [volume-preserving vector fields]; Godina et al GRG(00)gq/99 [2-spinor + Dirac fields]; Novello JCAP(07)gq [two spinors]; Barnett a1412 [gravitational field tensor].
@ Related topics: Essén IJTP(90) [conformally invariant scalar gauge field theory]; Bonanos JMP(91) [matrix-valued differential forms]; 't Hooft NPB(91) [chiral]; Wallner JMP(95); Anderson gq/99 [no need for metric]; Atkins a0803 [cohomological version, in terms of cochain complex of (n+2)-tensors]; Maharana a1004 ['t Hooft's chiral alternative to the vierbein]; Gomes et al CQG(11)-a1010 [as a 3D conformally invariant theory]; Obukhov & Hehl PPN(14) [fundamental spinors]; Vey CQG(15)-a1404 [n-plectic vielbein gravity]; Adamo IJMPD(15)-a1505 [as a 2D CFT]; Hehl et al IJMPD(16)-a1607-conf [pre-metric formulation, and electromagnetism]; Herfray a1807-th [chiral, twistor, 3-forms].
> Other: see 2D gravity; 3D gravity; 3D general relativity; Cartan Geometry; conformal invariance; Ether; Faddeev Formulation; gravitational thermodynamics; higher-dimensional gravity; Modified Gravity [MOG]; modified theories; Observers; quantum gravity; Shape Dynamics; simplex; unified theories; Weyl Space.

Formalism, Techniques > s.a. emergent gravity [analog models]; complex structures; geometry; hamilton-jacobi theory.
@ 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].
@ Related topics: Szydłowski et al JMP(96) [with Jacobi metric]; Marklund et al gq/97-MG8 [non-holonomic, non-rigid frames for rotating matter]; Roček & van Nieuwenhuizen gq/06 [smoothing, models]; Boroojerdian IJTP(13)-a1211 [z-graded tangent bundle and geometrization of mass]; Struckmeier PRD(15)-a1411 [as en extended canonical gauge theory]; Dray 14 [differential forms]; Hilditch a1509 [dual foliation formulations]; Hardy a1608 [operational formulation]; Donoghue et al a1702 [as a quantum effective field theory]; > s.a. category theory in physics [stacks]; fluids; numerical general relativity.

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