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 *η*_{ab};
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*(*x*^{a}; *ζ*,
*ζ**), where *ζ* and *ζ** are stereographic
coordinates on S^{2} and parametrize the different
null surfaces through each point (there is an S^{2}'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*'_{ab}
∇_{a}
*Z* ∇_{b}
*Z* = 0 for all (*ζ*, *ζ**), and they
satisfy *G*_{ab} = 0
or Λ*g*_{ab}; 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
[*g*_{ab} 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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