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In Classical General Relativity > s.a. canonical
general relativity; parametrized theories.
* Multi-fingered nature of time:
In general relativity, there is no single naturally-defined time function, but an
infinity of them; In asymptotically flat spaces, there is an asymptotic time
translation group which is a symmetry, for a given spacelike hypersurface, and any
asymptotic time translation can be extended in infinitely many ways to the interior;
However, this causes no problems in the classical theory; In the symplectic formulation,
each extension gives rise to a different Hamiltonian, which generates a different
canonical transformation.
* Frozen formalism: An expression
that refers to the fact that, for a compact spacelike hypersurface, the Hamiltonian
of general relativity vanishes on the constraints; This does not happen in the
asymptotically flat case.
* Arrow of time: One can physically
be associated with gravitational clustering (> see arrow
of time; gravitational phenomenology).
@ References: Rosen FP(91);
Rugh & Zinkernagel SHPMP(09)-a0805 [cosmic time];
Yahalom IJMPD(09) [existence of time from gravity];
Ludwin & Horwitz MPLA(11) [and covariant dynamics];
Ciufolini EJPW-a1306-ch [time travel, clock puzzles, and tests];
Anderson NYAS(14)-a1306 [Machian strategy];
Anderson a1809 [proposed solution to the problem of time].
@ In cosmology: Balbi a1304-proc;
Anderson a1403 [slightly inhomogeneous, problem of time];
Rugh & Zinkernagel a1603-in [limits of the concept of time].
> Effects: see chaos
in the metric; doppler effect; tests
of general relativity [time dilation].
> Related topics:
see dynamical wave-function collapse.
Time Functions > s.a. gauge choice;
Paneitz Operator and unimodular
gravity [spacetime volume].
* Cosmological time function:
The function τ(q):=
supp < q
d(p, q); It is called regular iff τ(q) <
∞ for all q and τ → 0 along every past-inextendible causal curve;
If τ is regular, (M, g) has several pleasant properties.
* York time: The parameter
T = (1/12π G) K, proportional to the trace of the
extrinsic curvature of a spatial hypersurface; This leads to using spatial slices
of constant mean curvature, whose importance has been known at least since York's
solution of the initial-value problem of general relativity.
* Epoch function: A scalar
field P on spacetime, constructed from
Rabcd and its covariant
derivatives, which reflects the Weyl curvature and is monotonically increasing
along almost all timelike trajectories for non conformally flat spacetimes.
@ General references:
Bernal & Sánchez CMP(05),
LMP(06)
[smoothness in globally hyperbolic spacetimes];
Farajollahi IJTP(07)-a0801;
Müller a0904 [on globally hyperbolic manifolds];
Rennie & Whale a1412 [and the Lorentzian distance].
@ Cosmic time: Hawking PRS(68);
Seifert GRG(77);
Qadir & Wheeler in(85);
Andersson et al CQG(98)gq/97;
Wegener FP(04).
@ York time: in Choquet-Bruhat & York in(80);
in Roser & Valentini CQG(14)-a1406;
Roser GRG(16)-a1407
[extension of solutions from T < 0 to T > 0],
a1511 [cosmological model, physical Hamiltonian density];
Roser & Valentini GRG(17)-a1606 [and cosmology: inflation and perturbations];
Roser a1609-PhD.
@ Epoch function: Pelavas & Lake PRD(00)gq/98 [and Weyl tensor/entropy].
@ From special lapse / shift:
Maia gq/96;
Gyngazov et al GRG(98)gq [and Higgs].
@ For 2+1 gravity: Benedetti & Guadagnini NPB(01)gq/00.
@ For specific types of spacetimes: Abreu & Visser PRD(10)-a1004 [spherically symmetric, Kodama time].
@ For gravity coupled to fluids:
Salopek & Stewart PRD(93) [and fluids];
Cianfrani et al a0904-proc [entropy].
@ Other matter coupled to gravity: Nakonieczna & Lewandowski PRD(15)-a1508 [scalar field].
@ Thermal time: Borghi FP(16)-a1807 [clocks and physical time];
> s.a. gravitational thermodynamics;
Tolman-Ehrenfest Effect.
@ Related topics:
Kummer & Basri IJTP(69) [initial surface];
Tiemblo & Tresguerres gq/96 [from gauge fixing],
GRG(98) [Frobenius foliation],
GRG(02);
Pulido et al GRG(01)gq/00;
Thiemann ap/06 [from phantom/K-essence Lagrangian];
Minguzzi CMP(10) [time functions and utility theory];
Ribeiro a1010 [for diamond-shaped regions / Alexandrov sets];
Chruściel et al AHP(16)-a1301 [differentiability of volume time functions];
Minguzzi CQG(16)-a1601,
JPCS(18)-a1711 [smooth steep time functions, existence and recovery of spacetime structure];
Ita et al CQG(21)-a1707 [spatial volume, Intrinsic Time Gravity].
In Semiclassical Gravity
* Results: Tomonaga-Schwinger time
does not exist on Riem(M), but it does on Riem(M)/Diff(M);
However, different foliations give rise to unitarily non-equivalent theories.
@ References: Halliwell PRD(87);
Brout & Venturi PRD(89);
Venturi CQG(90);
Kiefer in(94)gq;
Salopek ap/94,
PRD(95)ap,
ap/95-proc,
ap/95-proc,
ap/95-proc [Hamilton-Jacobi];
Giulini & Kiefer CQG(95)gq/94 [Tomonaga-Schwinger].
General References > s.a. causality and
causality conditions; cosmology;
time in quantum gravity.
@ Timeless gravity: Barbour gq/03-proc [in shape dynamics];
Shyam & Ramachandra a1209 [phase-space reformulation of Barbour's theory].
@ Clocks:
Teyssandier & Tucker CQG(96) [def];
Goy gq/97,
gq/97-conf [synchronization].
@ Relation to quantum theory: Kitada & Fletcher
Ap(96)gq/01;
Macías & Camacho PLB(08) [incompatibility].
@ Initial singularity: Lévy-Leblond AJP(90)feb [beginning of time];
Minguzzi IJMPD(09)-a0901-FQXi [and global existence of time].
@ Two-time physics:
Bars & Kounnas PLB(97);
Bars AIP(02)ht/01;
Nieto GRG(07)ht/05 [and Ashtekar variables];
Bars PRD(08)-a0804;
Piceno et al EPJP(16)-a1512 [fundamental constraints];
> s.a. modified general relativity; time.
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