Observables in Classical Gravity  

In General > s.a. 3D gravity; canonical general relativity [diffeomorphisms]; parametrized theories; unimodular relativity.
* Physical observables: Diffeomorphism-invariant functions of the dynamical field variables (configuration space or phase space).
* Rem: In an abstract sense, observables are as defined above; But this may not be very useful physically, so (as emphasized by Ehlers) keep in mind that the real observables are those related to proper times and lengths, and things one can actually measure.
* Requirement on the set of observables: Observables should be "local" (space or spacetime integrals of quantities expressed in terms of the metric field and its derivatives) and the set should be "large" (it should separate the points of physical phase space).
* Problem: Gravity does not admit any local (gauge invariant) observables.
* Chaos and observables: Because general relativity is likely chaotic, observables have to be generalized to non-differentiable or even discontinuous ones, which do not admit a standard quantization; Dittrich et al propose to use a polymer-type spacetime topology so that enough observables become continuous.
* How to find observables: One way to define observables is to deparametrize the theory and obtain a preferred time; But this cannot be done in the full theory, only in some sectors; Another approach is that of relational observables, proposed by Rovelli and generalized to field theory by Dittrich; Given a function f on phase space and a "clock" function T, the observable F gives the value of f when T has some given value τ.
@ Reviews: Giesel IJMPA(08)-proc.
@ General references: Bergmann RMP(61), in(65); Ohta & Kimura NCB(91); Rovelli CQG(91); Torre CQG(91), gq/94-conf; Anderson gq/92; Efroimsky & Lazarian CQG(93); Torre PRD(93)gq [no known ones for closed spacetimes]; Lusanna & Pauri GRG(06)gq/04 [Bergmann vs Dirac observables]; Pons et al MPLA(09)-a0902, CQG(10)-a1001-proc; Anderson Sigma(14)-a1312 [and beables, in classical and quantum gravity]; Dittrich et al PLB(17)-a1602 [continuity of observables, chaos and spacetime topology]; Pitts a1609 [and change, for equivalent theories].
@ And gauge, diffeomorphism invariance: Husain in(94) [and diffeomorphism invariance]; Pons CQG(01)gq [and gauge]; Khavkine CQG(15)-a1503 & CQG+ [generalized local gauge-invariant observables]; Donnelly & Giddings PRD(16)-a1507 [diffeomorphism-invariant observables, non-local algebra].
@ Related topics: Lusanna gq/02-conf [and non-inertial observers], gq/03-conf [and Einstein's hole argument].
> Related topics: see Event; gauge transformations; riemann tensor [invariants]; time in gravitation.

Types of Observables
@ Relational: Westman & Sonego FP(08)-a0708, AP(09)-a0711 [and symmetries, coordinates]; Girelli & Poulin PRD(08) [and deformed symmetries]; Tambornino Sigma(12)-a1109 [rev]; Dapor et al PRD(13)-a1305 [general formulation]; > s.a. FLRW quantum cosmology.
@ Evolving constants, t-dependent invariants: Page & Wootters PRD(83); Anderson gq/05; Pons & Salisbury PRD(05)gq; Gambini et al PRD(09)-a0809 [and relational time].
@ Partial observables: Rovelli PRD(02)gq/01; Dittrich GRG(07)gq/04, CQG(06)gq/05.
@ Dirac eigenvalues: Landi & Rovelli PRL(97)gq/96; Abdalla et al PLB(02)gq [and symplectic form]; Dittrich & Tambornino CQG(07)gq/06 [approximation scheme].
@ Other observables: Smolin PRD(94)gq/93; De Pietri & Rovelli CQG(95)gq/94; Rovelli PRD(02)gq/01 [GPS-type]; Szabados CQG(06)gq/05 [2-surfaces]; Khavkine PRD(12) [astrometric observables, time delay]; Requardt a1206 [spontaneously broken diffeomorphism group and non-diffeomorphism-invariant observables]; Duch et al JHEP(14)-a1403, JHEP(15)-a1503 [related to geometry]; Anderson a1604 [constrained theories and A-observables].
@ Related topics: Feng & Huang IJTP(97) ["Dirac condition does not apply"??]; Marolf CQG(15)-a1508 [non-locality, microcausality and chaos].

Special Systems > s.a. Lemaître-Tolman-Bondi Spacetimes; minisuperspace; spin-foam models.
@ Types of spacetimes: Neville CQG(93) [plane waves]; Husain PRD(94)gq, PRD(97)gq [2-Killing vector field reduction]; Salisbury et al GRG(08)gq/05 [Bianchi I + scalar, time-dependent invariants].
@ Gravitating bodies: Damour et al PRD(00)gq/99 [2-body]; Rovelli gq/01 [general relativity + particles].


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