Unimodular Gravity / Relativity  

In General
* Idea: A generalization of general relativity, in which the cosmological constant appears as a single additional variable (not a field), but is then seen to be a constant of the motion.
* Motivation: It provides a possible approach to the problem of time in quantum gravity, and to the cosmological constant problem.
* History: In his paper of 4 November 1915, Einstein proposed the equations Rim = −κ Tim as his new field equations (before proposing the definitive ones in his 25 November 1915 paper); These equations are invariant under unimodular transformations.
@ General references: [in Geroch JMP(70)]; Dieckmann GRG(88); Sorkin pr(88); Brown & York PRD(89); Henneaux & Teitelboim PLB(89); Unruh PRD(89); Abbassi & Abbassi FPL(01)gq/00; Bock IJTP(03)gq [equations for null cones and volume element]; Barceló et al PRD(14)-a1401 [from graviton self-interactions]; Álvarez & González-Martín PRD(15)-a1506 [first-order Lagrangians]; Álvarez et al PRD(16)-a1604 [gauge symmetries]; Oda PRD(16)-a1606 [conformal symmetry].
@ And the cosmological constant: Ng & Van Dam IJMPD(01)ht/99-conf [variable cosmological constant]; Davidson & Rubin CQG(09)-a0905; Totani JCAP(16)-a1512.
@ Arguments against: Kuchař PRD(91).

Versions > s.a. metric decomposition [conformal-traceless form].
* In general: One starts with the same action as in general relativity (but one could use that of some other gravity theory of interest), and adds a condition on the class of metrics one considers in the variation.
* Unit metric determinant version: The metric satisfies the condition det g = 1.
* Fixed volume element version: The volume element defined by the metric matches a preferred one, |det g|1/2 = ω.
* Fixed total volume version: The total volume of the spacetime region matches a fixed value, ∫ d4x |det g(x)|1/2 = V.
* Variation – time as past volume of a point: V(p) is the volume of the past of p; It is strictly increasing along timelike curves iff (M, g) is chronological; If (M, g) is strongly causal, then it is strictly increasing along causal curves; If the latter is true, then (M, g) is causal; If (M, g) is globally hyperbolic, then V(p) is continuous everywhere.
@ Unit determinant version: Buchmüller & Dragon PLB(89); Kreuzer CQG(90); Ng & van Dam PRL(90), JMP(91) [has refs]; Petrov MPLA(91); Ng IJMPD(92); Álvarez & Faedo PRD(07)ht [and coupling of KE and PE to gravity]; Fiol & Garriga JCAP(10)-a0809 [semiclassical]; Jain et al JCAP(12)-a1108, JCAP(12)-a1109.
@ Unit determinant version, canonical: Unruh PRD(89); Unruh & Wald PRD(89); Klusoň PRD(15)-a1409.
@ Fundamental volume element: Anderson & Finkelstein AJP(71)aug; Finkelstein et al JMP(01)gq/00.
@ Time = Volume: Bombelli in(91); Sorkin IJTP(94).
@ Time = Volume, canonical: Bombelli, Couch & Torrence PRD(91).
@ Transverse gravity: López-Villarejo JPCS(10)-a1001, JCAP(11)-a1009 [TransverseDiff gravity]; Álvarez & Vidal PRD(10)-a1001; Álvarez JCAP(12)-a1204 [gauge and equivalence principle]; Álvarez & Herrero-Valea JCAP(13)-a1209 [with external sources].
@ From coupling to 3-form field: Henneaux & Teitelboim PLB(84); Teitelboim PLB(85) [and black holes]; Brooks NPB(94)ht/99.
@ Metric from gauge structure and volume: Wilczek PRL(98)ht.

Quantum Theory
@ General references: Ng & van Dam PRL(90); Gamboa & Mendez NPB(01)ht/00 [3D, strong coupling, path integral]; Calmet EPL(07)ht/05 [and non-commutative spacetime], MPLA(07)-a0704 [and fundamental length]; Smolin PRD(11)-a1008 [and loop quantum gravity]; Eichhorn CQG(13)-a1301 [RG flow and UV fixed point]; Álvarez & Herrero-Valea PRD(13)-a1301 [absence of conformal anomaly]; Saltas PRD(14)-a1410 [UV structure]; Álvarez et al PRD(15)-a1505 [quantum corrections]; Burger et al a1511 [KLT relations].
@ And the cosmological constant: Smolin PRD(09)-a0904; Álvarez et al JHEP(15)-a1505; > s.a. cosmological constant.
@ And general relativity: Álvarez JHEP(05)ht [vs general relativity, quadratic (Fierz-Pauli) regime]; Padilla & Saltas EPJC(14)-a1409 [equivalence to general relativity]; Bufalo et al EPJC(15)-a1505 [two versions, path integral quantization]; Benedetti GRG(16)-a1511 [G is independent of field redefinitions].
@ In quantum cosmology: Daughton et al gq/93-conf [k = 1, initial conditions and unitarity]; Chiou & Geiller PRD(10)-a1007, Bunao a1604 [lqc].
@ Instantons, unitarity: Daughton et al PRD(98)gq.
blue bullet Approaches: see approaches to quantum gravity [volume-preserving diffeomorphisms]; covariant quantum gravity; path-integral formulation.

Related Topics > s.a. BRST symmetry; generalized uncertainty principle; gravitational thermodynamics.
@ Energy-momentum conservation: Tiwari JMP(93), gq/03/JMP; Bock a0704, a0704 [modified, and dark matter / dark energy].
@ Scale invariance: Shaposhnikov & Zenhausern PLB(09)-a0809 [and dark energy]; Singh MPLA(13)-a1205.
@ And cosmology: Álvarez & Herrero-Valea a1302-wd [solutions]; Gao et al JCAP(14)-a1405, Basak et al GRG(16)-a1511 [cosmological perturbations]; Cho & Singh a1412 [and inflation].
@ Other solutions: Chaturvedi et al IJMPD(17)-a1610 [Reissner-Nordström solution].
@ Time, other: Hosoya & Soda PTP(90); Bertolami IJMPD(95); Earman SHPMP(03) [and the cosmological constant, conceptual]; Farajollahi GRG(05)-a0801, IJTP(08) [and observables]; Shlaer a1411 [metric volume element as total derivative].
@ Unimodular f(R) gravity: Eichhorn JHEP(15)-a1501 [renormalization group flow]; Nojiri et al JCAP(16)-a1512; Nojiri et al PRD(16)-a1601 [bounce universe history]; Sáez-Gómez PRD-a1602; Odintsov & Oikonomou ASS(16)-a1602 [unimodular mimetic f(R) gravity, and inflation]; Nojiri et al MPLA(16)-a1605 [Newton's law]; > s.a. early-universe models [bounces].
@ Unimodular f(T) gravity: Nassur et al EPJP-a1602; Bamba et al MPLA(17)-a1605 [inflationary cosmology].
@ And matter: Nishino & Rajpoot PLB(02)ht/01 [and supergravity]; Klinkhamer IJMPD(17)-a1604 [with vacuum-matter energy exchange]; Martin JCAP(17)-a1704 [lepton anomalous magnetic moment].
@ Other variations: Abbassi & Abbassi CQG(08)-a0706, AP(11)-a1003 [density-metric unimodular gravity, vacuum maximally symmetric solutions]; da Rocha et al PRD(10)-a1101 [in teleparallel gravity]; Bradonjić & Stachel EPL(12)-a1110 [unimodular conformal and projective relativity]; Nojiri et al CQG(16)-a1601 [unimodular mimetic gravity]; Barvinsky & Kamenshchik a1705 [Lorentz non-invariant generalization with a locally inert perfect fluid, and dark sector]; Houndjo a1706 [unimodular f(G) gravity]; > s.a. modifications of general relativity.

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