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*, *N*^{a}, *A*_{0}];
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"].

**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; Makes it easier to relate conserved quantities at
spatial and null infinity.

* __Limitations__: Will
probably not tell us anything about the regime in which we run into singularities.

* __Symplectic structure__:
Tangent vectors at a point *g*_{ab}
of phase space are solutions *γ*_{ab}
of the linearized equation around *g*_{ab};
The symplectic structure, acting on two such vectors, is

Ω|_{g}(*γ*,*γ*')
= (1/16π*G*) ∫ *ε*^{mnp}_{q}
(*γ*_{ms} ∇_{n}
*γ*'_{pr} − *γ*'_{ms}
∇_{n}*γ*_{pr})
d*S* ,

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].

@ __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].

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