Time in Gravity  

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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