Unimodular Gravity / Relativity  

In General
* Idea: A generalization of general relativity (or other theories of gravity), 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; It is generally assumed to be equivalent to general relativity, but the quantum theories may be inequivalent.
* 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]; Davidson & Rubin CQG(09)-a0905; Totani JCAP(16)-a1512.
@ Arguments against: Kuchař PRD(91).

Versions Based on General Relativity > s.a. metric decomposition [conformal-traceless form].
* In general: One starts with the same action as in general relativity, 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; Jora a2004 [gauge condition revisited].
@ 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.
@ Other: Wilczek PRL(98)ht [metric from gauge structure and volume]; Jiroušek & Vikman JCAP(19)-a1811 [Weyl-invariant, generally-covariant form formulated as vector-tensor theory with higher derivatives]; Hammer et al a2001 [Pontryagin density of gauge field as volume element].

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, JHEP(15)-a1505 [quantum corrections]; Burger et al a1511 [KLT relations]; de León et al PRD(18)-a1710 [path integral, one-loop effective action]; Yamashita PRD(20)-a2003 [connection representation, Hamiltonian analysis]; de Brito & Pereira a2007 [beyond perturbation theory].
@ And the cosmological constant: Smolin PRD(09)-a0904; Álvarez et al JHEP(15)-a1505; Percacci FP(18)-a1712-proc; > 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]; Herrero-Valea JHEP(18)-a1806 [equivalence of the S-matrix around flat space-time]; > s.a. matter below.
@ In quantum cosmology: Daughton et al gq/93-conf [k = 1, initial conditions and unitarity]; Chiou & Geiller PRD(10)-a1007, Bunao CQG(17)-a1604 [lqc]; Riahi Gal(18)-a1801 [wavepacket evolution].
@ Related topics: Daughton et al PRD(98)gq [instantons, unitarity]; Álvarez et al EPJC(16)-a1605 [tree level scattering amplitudes]; de Brito et al PRD-a2012 [asymptotically safe].
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]; García-Aspeitia et al PRD(19)-a1903, PDU-a1912 [acceleration]; Corral et al PRD(20)-a2005 [acceleration and energy diffusion]; Kamenshchik et al JETPL(20)-a2003 [generalized, Friedmann and Kantowski-Sachs spacetimes].
@ And matter: Klinkhamer IJMPD(17)-a1604 [with vacuum-matter energy exchange]; Martín JCAP(17)-a1704 [lepton anomalous magnetic moment]; González-Martín & Martin JCAP(18)-a1711 [scattering of massive scalar particles]; Herrero-Valea & Santos-García JHEP(20)-a2006 [non-minimal coupling].
@ Other solutions: Chaturvedi et al IJMPD(17)-a1610 [Reissner-Nordström solution]; Astorga-Moreno et al JCAP(19)-a1905 [stellar objects].
@ 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].

Versions Based on Other Gravity Theories
@ 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(16)-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(16)-a1602; Bamba et al MPLA(17)-a1605 [inflationary cosmology].
@ Unimodular supergravity: Nishino & Rajpoot PLB(02)ht/01; Anero et al JHEP(20)-a1911 [off-shell N = 1, d = 4].
@ Unimodular versions of other generalized theories: 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]; Houndjo EPJC(17)-a1706 [unimodular f(G) gravity]; Rajabi & Nozari PRD(17)-a1710 [unimodular f(R,T) gravity]; Bonder & Corral PRD(18)-a1802 [unimodular Einstein-Cartan gravity].
@ And cosmology: Alexander & Carballo-Rubio PRD(20)-a1810 [cosmological constant]; Barvinsky & Kolganov PRD(19)-a1908 [inflation].
@ Other variations: Abbassi & Abbassi CQG(08)-a0706, AP(11)-a1003 [density-metric unimodular gravity, vacuum maximally symmetric solutions]; Barvinsky & Kamenshchik PLB(17)-a1705 [Lorentz non-invariant generalization, and dark sector]; Daouda et al IJMPD(19)-a1802 [without energy-momentum conservation]; Böhmer & Carloni PRD(18)-a1806 [with dynamical volume form]; Álvarez et al CQG(20)-a1806 [ghost-free massive deformation]; Barvinsky et al PRD(21)-a2011 [equivalence to k-essence theory]; Jiroušek et al JCAP(21)-a2011 [with Newton's constant and Planck's constant as global degrees of freedom]; Barbero et al PRD(21)-a2101 [parametrized unimodular extension of the Holst action]; > s.a. modifications of general relativity.

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