Canonical General Relativity: Formalisms and Approaches| 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 (γms ∇n γ'pr − γ'ms ∇n 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].
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