Brans-Dicke Theory of Gravity  

In General > s.a. gravitational constant; scalar-tensor theories; types of higher-order theories [equivalence].
* Idea: A scalar-tensor theory of gravity containing an arbitrary "Dicke coupling constant" ω and, in the generalized form, a mass m; It amounts to effectively introducing a dynamics for the gravitational coupling G.
* In the Jordan frame: The action, if K (K0) is the trace of the extrinsic curvature induced by g (η) on ∂M and including a cosmological constant, is

SJ[g, Φ, ...] = \(1\over16\pi\) M dv [Φ (R − 2Λ) − ω gab Φ−1aΦ ∇bΦ] + 2 MK− Φ0K0) ds + Smatter ;

The field equations, in units in which G = 1, are

\(\square\)Φ = (\(8\pi\over3\)+ 2ω) T ,     Gab = (8π Φ−1)Tab + (ωΦ−2) (Φ,aΦ,b − \(1\over2\)gab Φ,mΦ,m) + Φ−1;abgab \(\square\)Φ) .

* In the Einstein frame: The action is

SE[g, Φ, ...] = \(1\over16\pi\) M dv [R − (ω + 3/2) Φ,a Φ,a] + M dv exp{−2Φ}\(\cal L\)m ;

The field equations are

Gab = 8π (Φ,aΦ,b − \(1\over2\)gab gmn Φ,m Φ,n) .

* History: Perfectly viable in 1971, it was neglected for some years, then revived since the 1980s because it (with other matter fields) arises in Kaluza-Klein theories, and as an effective theory in the low-energy limit of string theory; 2016, There is still interest in the theory, despite the fact that Solar System tests impose strong constraints on it, rendering it indistinguishable from general relativity.
@ General references: Brans & Dicke PR(61); Dicke PR(62); in Dicke 64; Jackiw NC(68); Miyazaki gq/00/PRD [and general relativity]; Jackiw & Pi a1511-fs [reasons for current interest]; Fabris et al a1603-fs [new perspectives]; Brando et al IJMPD(19)-a1810 [and general relativity].
@ Summaries: in Misner et al 73, 1070ff; Will in(79).
@ Jordan vs Einstein frame: Bhadra et al MPLA(07); Artymowski et al PRD(13)-a1309 [in quantization à la lqc]; Gionti PRD-a2003 [canonical analysis].
@ Related topics: Chernodub & Niemi PRD(08)-a0709 [and electroweak theory]; Faraoni CQG(09)-a0906 [mass and range of scalar field]; Lobo IJGMP(18)-a1610 [non-metricity]; > s.a. brans-dicke phenomenology [including solutions and cosmology]; gravitational thermodynamics; spherical general relativity.
> Online resources: see Wikipedia page.

Variations and Generalizations > s.a. 3D gravity; quantum cosmology [third quantization]; quantum cosmology models; quintessence.
@ ω → ∞ limit: Banerjee & Sen PRD(97); Faraoni PLA(98)gq, PRD(99)gq; Quirós gq/99/PLA; Bhadra & Nandi PRD(01)gq/04 [scalar field]; Chauvineau CQG(03).
@ ω → 0 limit: Punzi et al PLB(08)-a0804 [geometrical structure].
@ Massive: Alsing et al PRD(12)-a1112 [radiation from compact binaries].
@ 3-dimensional: Alonso et al PRD(03); do Prado et al a1012 [and cosmological acceleration].
@ Quantum theory: Cho CQG(97) [and equivalence principle]; Zhu et al MPLA(98) [quantum cosmology and cosmological constant]; Pimentel & Mora PLA(01)gq/00 [3rd quantization]; Ohkuwa NCB(03) [WKB time]; Pal PRD(16)-a1608 [canonical quantization, FRW model, phase transition]; Frion & Almeida PRD(19)-a1810 [affine quantisation]; > s.a. renormalization of quantum gravity.
@ Lqg / lqc quantization: Zhang & Ma JPCS(12); Zhang at al PRD(13)-a1211, Jin et al JCAP(19)-a1808; Song et al a2004 [alternative Hamiltonian constraint]
@ Generalizations, extensions: Chakraborty & Ghosh PS(03); Catena PRD(07)ht/06 [minimal supersymmetric extension]; Amendola et al PRD(08)-a0801 [with Gauss-Bonnet term, and Solar System tests]; Smolyakov IJMPA(10)-a0907 [non-linear, Born-Infeld-like]; Qiang et al PLB(09) [5D, and cosmology]; Ponce de León CQG(10)-a0912, JCAP(10) [5D, effective 4D theory and cosmological acceleration]; De Felice & Tsujikawa JCAP(10)-a1005; Wu & Wang PRD(12) [with torsion, in Riemann-Cartan spacetime]; Rasouli et al CQG(14)-a1405 [in arbitray dimensions]; Caramês et al EPJC(14)-a1409 [Brans-Dicke-Rastall theory, with violation of conservation laws]; > s.a. hořava gravity; lovelock gravity.
@ Related topics: Arazi & Simeone MPLA(00)gq [linearized, strings]; Rador PLB(07) [Brans-Dicke type higher-order gravity]; Chang-Young et al CQG(12)-a1105 [as an entropic phenomenon]; Roshan & Shojai CQG(11)-a1106 [post-Newtonian limit]; Almeida et al PRD(14)-a1311 [geometrical scalar-tensor theory]; Kofinas AP-a1510 [complete theory].

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