Canonical General Relativity: Formalisms and Approaches  

In General > s.a. canonical formulation.
* Idea: The standard approaches to 4D canonical general relativity use a 3+1 decomposition of spacetime, where the 3D constant-time manifold is a spacelike hypersurface and the configuration variables on it are a spatial metric, or an orthonormal triad; Other possibilities include using other variables for the same type of spacetime decomposition, different splittings of spacetime, or more covariant formalisms.
@ Modifications: Gomes & Shyam JMP(16)-a1608 [results on spatially covariant generalizations]; Montesinos et al PRD(18)-a1712 [Lorentz-covariant variables].

Other Variables > s.a. 3D general relativity; 3D gravitation; quasilocal gravity.
* Idea: In a slicing (as opposed to threading) approach to canonical gravity, one chooses a (C) 3-manifold Σ which will act as the (unchanging) spatial manifold, and encodes all the information necessary to reconstruct a spacetime metric into a set of fields defined on Σ.
@ General references: Peldán CQG(91)
[non-uniqueness of Hamiltonian]; Tate CQG(92); Lewandowski & Okołów CQG(00)gq/99 [2-form, BF-like]; Farajollahi & Luckock GRG(02)gq/01 [and local observer]; Barbour & Ó Murchadha a1009 [conformal superspace as the configuration space].
@ Embedding variables, reference fluid: Kuchař PRD(92); Braham PRD(94)gq/93 [cylindrical symmetry]; Brown gq/94-proc; Brown & Kuchař PRD(95)gq/94 [and time]; Montani & Zonetti IJMPA(08)-a0807.
@ Dirac eigenvalues: Landi & Rovelli PRL(97)gq/96; Landi LNP(02)gq/99; > s.a. supergravity.
@ Spinorial variables: Grant CQG(99)gq/98 [any dimension].
@ Other variables: Rosas-Rodríguez gq/05 [electric and (complex) magnetic fields]; Katanaev TMP(06)gq [including det g]; Anderson CQG(08) [cyclic variables whose velocities are N, Na, A0]; Reisenberger PRL(08) [null data, Poisson brackets]; Klusoň PRD(12)-a1206 [conformal-traceless decomposition, Hamiltonian analysis]; Parattu et al PRD(13)-a1303 [canonical, thermodynamically conjugate variables].
> Connection dynamics: see connection formulation; loop-variable formulation; Metric-Affine Gravity.

Non-Standard Spacetime Decompositions > s.a. decomposition; quasilocal general relativity [2+2].
* Threading vs slicing: Instead of defining the variables on an abstract 3-manifold, identified with a t = constant slice, one can define them on the world-lines in a spacetime-filling congruence.
* Remark: Both slicing and threading are examples of the congruence point of view, dating back to Ehlers in(59).
@ General references: Cattaneo (in italian); Landau & Lifshitz v2.
@ Gravitomagnetism: Jantzen in-GR12; Jantzen et al AP(92)gq/01.
@ Timelike foliations: Maran gq/03-wd [Ashtekar-like]; Alexandrov & Kádár CQG(05) [lqg-like].
@ Threading: Ehlers in(59); Perjés pr(88), MPLA(93); Jantzen & Carini in(91); Perjés NPB(93); Fodor & Perjés GRG(94); Boersma & Dray JMP(95)gq/94, JMP(95)gq/94, GRG(95)gq/94; Gielen & Wise PRD(12)-a1112 [as spontaneous symmetry breaking]; Wise JPCS(14)-a1310 [general relativity in terms of "observer space"]; Park a1810.

Based on Non-Standard Surfaces > s.a. asymptotic flatness; Linkages.
* Null infinity: Construct a phase space for the radiation degrees of freedom.
@ Null infinity: Ashtekar & Streubel PRS(81); Helfer CMP(95).
@ Null surface: Goldberg FP(84); Goldberg & Soteriou CQG(95).

Covariant Formulation > s.a. hamiltonian dynamics; symplectic structures; multiverse [as space of classical universes].
* Idea: Consider as phase space the space of solutions of the Einstein equation.
* Motivation: Explicit covariance; It makes it easier to relate conserved quantities at spatial and null infinity.
* Limitations: It will probably not tell us anything about the regime in which singularities arise.
* Symplectic structure: Tangent vectors at a point gab of phase space are solutions γab of the linearized equation around gab; The symplectic structure, acting on two such vectors, is

Ω|g(γ,γ') = (1/16πG) εmnpq (γmsn γ'prγ'msn γpr) dS ,

integrated over any Cauchy surface Σ.
@ General references, and histories: Savvidou BJP(05)gq/04-proc; Savvidou CQG(06)gq [Barbero connection]; Gielen & Wise GRG(12)-a1206-GRF [field of local observers]; Hajian & Sheikh-Jabbari PRD(16)-a1512 [conserved charges].
@ Proposals: Segal; Szczyrba CMP(76); Ashtekar & Magnon-Ashtekar CMP(82); Ashtekar et al in(87); Crnković NPB(87); Crnković & Witten in(87); Zuckerman in(87); Ashtekar et al in(91); Frauendiener & Sparling PRS(92) [in terms of soldering form and connection]; Esposito et al NCB(94)gq/95, Dolan & Haugh CQG(97)gq/96 [new variables]; Savvidou CQG(01)gq [and Dirac algebra]; Rovelli gq/02, LNP(03)gq/02; Nester et al a1210-proc [Hamiltonian boundary term]; Cremaschini & Tessarotto APR(16)-a1609, EPJC(17)-a1609 [extended "DeDonder-Weyl" formalism based on a synchronous variational principle]; Castellani & D'Adda a1906 [gravity coupled to p-forms]; Barbero et al a2103 [metric vs tetrad formulations], a2105 [with non-metricity, torsion, and boundaries]; Wieland a2104 [Barnich-Troessart bracket].
@ Special types of spacetimes: Palmer JMP(78) [with symmetries]; Anco & Tung JMP(02)gq/01 [spatially bounded].
@ From a Lagrangian formulation: Friedman & Schutz ApJ(75), ApJ(78).
@ For other theories of gravity: Nester MPLA(91) [general formalism]; Randono CQG(08)-a0805 [Einstein-Cartan].

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at – modified 18 may 2021