Unimodular Gravity / Relativity| 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].
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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