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 CQG(17)-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**
> s.a. Gravitational Memory [persistent 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];
Pitts FP(18)-a1803.

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**Special Systems and Related Theories**
> 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].

> __Related topics__:
see observables [Einstein-Maxwell theory].

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send feedback and suggestions to bombelli at olemiss.edu – modified 23 jul 2019