Bigravity / Bimetric Theories of Gravity  

In General > s.a. Whitehead Theory [in which one metric is a fixed background].
* History: 1934, Born-Infeld theory can be considered as a form of bimetric theory (according to Moffat); 1973–1974, New version proposed by N Rosen; 1979–1995, J Moffat's non-symmetric gravitation theory; 1992, J Bekenstein's treatment of gravitational lensing and MOND; 1998–2003, Clayton-Moffat scalar-vector-tensor theory.
* Idea: A theory of gravity with two distinct metrics, a gravitational gab which describes the geometry of spacetime and controls the speed of gravity, and a matter metric γab which may be the one coupled to all other fields and controling their propagation; In a scalar-tensor version with a scalar field φ coupled to gravity, it can be given by

γab = gab + (B/c2) ∂aφ ∂bφ ,

where B is a positive number (which will probably be renormalized by quantum effects).
* Dynamics: The action can be of the form S[gab, γab] = M d4x |g|1/2 R[gab] + M d4x |γ|1/2 R[γab] + M d4x V[gab, γab].
* Properties: One of the main differences with general relativity is that it (or some versions of it) does not produce black holes.
Quantum aspects: see Boulware-Deser Ghost; effective action; quantum gravity renormalization.
> Online resources: see Wikipedia page.

Phenomenology > s.a. anomalous acceleration; test-particle motion; variation of constants.
* Variation of c: The different metrics that apply to gravitation and other fields give rise to an effectively changing speed of light; Depending on whether one uses electromagnetic or other (say, gravitational) waves to study the early universe, one gets that the other speed changes in time (the ratio cgrav/cγ is dimensionless and meaningful).
* Bipolar dark matter: A proposal in which two different species of dark matter particles are separately coupled to the two metrics of bigravity and are linked together by an internal vector field; The phenomenology of MOND then results from a mechanism of gravitational polarization.
* Cosmology: If we set the gravitational cgrav to be constant, the electromagnetic cγ will vary, and increase close to the big bang; This solves the horizon problem without using inflation (although the Friedmann equation changes too).
@ General references: Moffat ap/00, IJMPD(03)gq/02 [variable c and supernovas]; Moffat qp/02 [entanglement]; Kopeikin & Ni gq/06-proc [laser ranging delay]; Kopeikin gq/05 [speed of light and gravity]; Clifton et al CQG(10)-a1006 [PPN limit]; Baccetti et al JHEP(12)-a1206 [null-energy-condition violations]; Aoki et al a1602 [relativistic stars].
@ And dark matter: Bañados et al PRD(09)-a0811; Drummond PRD(01)ap/00 [as alternative]; Bernard & Blanchet PRD(15)-a1410; Blanchet & Heisenberg PRD(15)-a1504 [coupling of dark matter]; Bernard et al a1507-proc [rev, dipolar dark matter]; Babichev et al PRD(16)-a1604 [possible particle physics and gravitational signatures].
@ Cosmology: Petit & D'Agostini a0712 [acceleration]; Magueijo et al PRD(10)-a1006 [perturbations, structure formation]; von Strauss et al JCAP(12)-a1111 [general cosmological evolution equations]; Kühnel PRD(13)-a1208 [instability]; Capozziello & Martín-Moruno PLB(13)-a1211 [bounces, turnarounds and singularities]; Galvez Ghersi a1312-MS; De Felice et al JCAP(14)-a1404, Akrami et al PLB(15)-a1503 [viable cosmology]; Schmidt-May a1405-proc [ghost-free]; Mörtsell JCAP(17)-a1701 [graphical approach]; > s.a. cosmological models and perturbations.
@ Gravitational waves: Hazboun et al a1311 [multi-messenger astronomy tests of bimetric gravity]; Cusin et al JCAP(15)-a1412 [cosmological]; Sakakihara & Soda JCAP(15)-a1504 [primordial]; Fasiello & Ribeiro JCAP(15)-a1505 [and bounds on bigravity]; Max et al PRL(17)-a1703 + news [oscillations]; Baker et al a1710 [constraints from GW170817].
@ Other solutions: Bañados et al PRD(11) [black holes and their thermodynamics]; Hassan et al IJMPD(14)-a1407 [properties]; Torsello et al PRD(17)-a1703 [black holes].

References > s.a. 3D gravity; actions for general relativity [bimetric reformulation]; ADM formulation; MOND.
@ General: Rosen AP(63), AP(66) [with flat metric], GRG(73), AP(74), & Rotbart GRG(79); Leiter & Robertson gq/01/GRG [?]; Pitts & Schieve GRG(01)gq ["slightly bimetric"], AIP(01)gq [consistency]; Blas et al PRD(07); Afshar PRD(10)-a0910 [3-dimensional]; Baccetti et al JHEP(12)-a1206 [Gordon and Kerr-Schild Ansätze]; Deser et al PRD(13) [no gauge invariant, partially massless limit]; Akrami et al a1404-GRF; Schmidt-May & von Strauss a1512 [recent progress]; Bonifacio et al a1712 [massless and massive].
@ Hamiltonian / canonical formulation: Soloviev & Chichikina TMP(13)-a1211 [Kuchař's approach]; Klusoň PRD(13)-a1211 [with one particular form of potential], EPJC(13)-a1303 [general theory], EPJC(14)-a1307 [vierbein formulation]; Alexandrov GRG(14)-a1308 [tetrad formulation]; Soloviev a1312-conf, a1505 [and cosmology].
@ Related topics: Damour & Kogan PRD(02) [non-linear, Leff]; Blas et al CQG(06) [solutions, causal structure]; Hohmann & Wohlfarth PRD(09)-a0908 [positive and negative mass], comment Hossenfelder a0909; Deffayet & Jacobson CQG(12)-a1107 [horizon structure]; Talshir a1404 [first-order Lagrangian]; Noller & Melville JCAP(15)-a1408 [coupling to matter]; > s.a. finsler geometry.
@ Bimetric massive gravity: Baccetti et al CQG(13)-a1205, Martín-Moruno et al a1302-MG13 [massive gravity as a suitable limit of bimetric gravity]; Enander & Mörtsell JHEP(13)-a1306 [lensing constraints]; Babichev & Crisostomi PRD(13)-a1307 [spherically symmetric solutions, recovery of general relativity]; Könnig et al JCAP(14)-a1312 [viable cosmological solutions]; Könnig & Amendola PRD(14)-a1402, Könnig et al PRD(14)-a1407 [cosmological models]; Enander & Mörtsell JCAP(15)-a1507 [stars and galaxies]; Cusin et al JCAP(16)-a1512 [general mass term]; > s.a. massive gravity.
@ Multigravity, multimetric gravity: Kogan ap/01-conf [overview]; Hohmann PRD(12)-a1105 [propagation of gravitational waves]; Talshir PRD(13)-a1211 [energy and momentum].
@ Delta-gravity: Alfaro et al CQG(11)-a1009 [quantization]; Alfaro PLB(12) [and accelerated expansion].
@ Other variants: Pitts & Schieve FP(04) [in Minkowski, same light cones]; Hossenfelder PRD(08)-a0807 [with exchange symmetry]; Hohmann & Wohlfarth PRD(10)-a1003 [N copies of the standard model and metrics, and dark energy]; Speziale PRD(10)-a1003 [from a modified Plebański action]; Akrami et al JCAP(13)-a1306 [doubly coupled to matter]; Talshir PRD(14) [energetically stable]; De Felice et al a1702 [with variable graviton mass]; Brax & Valageas a1712 [scalar-bimetric theories].

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