Conformal Gravity  

In General > s.a. 3D gravity; gravity theories; covariant quantum gravity; gauge theory of gravity.
* Idea: A theory of gravity that is invariant under conformal (local scale) transformations; There are several versions in the literature, a popular one being the higher-derivative theory with the Bach equation as the vacuum field equation, and action \[ S = \alpha \int{\rm d}^4x\, |g|^{1/2}\,C_{abcd}\,C^{abcd}\;. \] * Motivation: Initially, the dimensionless coupling constant α, for quantization; Later, used to explain flat galactic rotation curves without dark matter; It can also give rise to a cosmological acceleration.
* And general relativity: One can obtain Einstein gravity from conformal gravity in 4D by introducing a scalar compensator with a vacuum expectation value that spontaneously breaks the conformal invariance and generates the Planck mass, or by compactifying extra dimensions in a higher-dimensional conformal theory of gravity (without the need to introduce the scalar compensator).
* Solutions, and phenomenology: All vacuum solutions of general relativity (e.g., Schwarzschild spacetime) are solutions of conformal gravity, but not the other way around, and not with matter; Linearized theory gives a 4th-order wave equation, \((\partial_t^{\,2} + \nabla^2)^2 \phi = 0\) around Minkowski spacetime.
* Results: One gets an extra attractive effect on matter (from motion in Schwarzschild-like solutions, the Newtonian potential is modified to \(V(r) = -b/r + cr\)), but also an additional repulsive term for light, affecting light deflection (and the latter does not fit observed data); The cosmological \(G_{\rm eff}\) is smaller than the Cavendish one; > s.a. dark matter [alternatives].
@ General references: Boulware et al PRL(83) [zero-energy theorem]; Gorbatenko et al GRG(02)gq/01 [and geometrodynamics]; Gorbatenko & Pushkin GRG(02) [and causality]; Gorbatenko GRG(05) [properties]; Carroll a0705 [and quantum theory]; Mannheim FP(11)-a1101-conf [the case for]; Yoon PRD(13)-a1305 [criticism], Mannheim PRD(16)-a1506 [response]; Lovrekovic PoS-a1505 [canonical charges and asymptotic symmetries].
@ And general relativity: Maldacena a1105; Wheeler PRD(14)-a1310; Ohanian GRG(16)-a1502 [breaking of conformal symmetry]; Anastasiou & Olea PRD(16)-a1608 [equivalence with Einstein gravity with Neumann boundary conditions].
@ Quantum: Wang JPCS(06)gq/05, PTRS(06)gq [canonical, new variables and Immirzi parameter]; Mannheim a0707-proc [no ghosts]; Jizba et al EPJC(15)-a1410 [and inflationary cosmology]; Modesto & Rachwal a1605 [spontaneous breaking of Weyl symmetry and non-singular spacetimes]; Campiglia et al CQG(17)-a1609 [lqg approach, coupled to the Standard Model]; Veraguth & Wang PRD(17)-a1705 [loop quantization]; Rachwal Univ(18)-a1808 [scattering amplitudes and quantum effective action].
@ Barbour's version: Barbour CQG(03)gq/02 [particle motion], Anderson et al CQG(03)gq/02 [geometrodynamics].
> Related topics: see conformal invariance in physics; schwarzschild spacetime; unified theories [Weyl, conformal gravity].

Solutions and Phenomenology > s.a. causality violations [warp drive].
* Newtonian limit: It appears that in this model gravity is attractive on small scales and repulsive on large scales.
@ Solutions: Schmidt AdP(84)gq/01, AN(85)gq/01 [of Bach equation]; Le Brun CMP(91); Edery PRL(99)gq; Dzhunushaliev & Schmidt JMP(00)gq/99 [vacuum]; Bhattacharya et al JCAP(10)-a0910 [Mannheim-Kazanas solution]; Brihaye & Verbin PRD(10)-a0912 [cylindrically-symmetric].
@ With coupled matter: Brihaye & Verbin PRD(09)-a0907 [and scalar-tensor extension], PRD(10)-a0910 [+ gauge theory, spherical symmetry]; Fabbri AFLB-a1101 [Dirac matter], PRD(12)-a1101 [ELKO spinor field].
@ Astrophysics: O'Brien & Mannheim MNRAS(12)-a1107, Mannheim & O'Brien JPCS(13)-a1211 [dwarf galaxy rotation curves]; Yang et al PLB(13)-a1311 [constraints from SNIa and Hubble parameter data]; Varieschi GRG(14)-a1401 [Kerr geometry and geodesic motion]; Bambi et al PRD(17)-a1701 [astrophysical black holes]; Zhang et al EPJC(18)-a1805 [black holes].
@ Galactic rotation curves: Campigotto et al CQG(19)-a1712 [failure]; Li & Modesto a1906 [predicted]; Hobson & Lasenby a2103 [not predicted].
@ Cosmology: Mannheim GRG(90), ap/96 [age of the universe], ap/98, ap/98-proc, gq/99-proc, ApJ(01)ap/99 [cosmic acceleration]; Varieschi GRG(10)-a0809, ISRN-AA(11)-a0812 [kinematical approach]; Mannheim PRD(12) [perturbations]; Nesbet Ent(13)-a1208 [dark matter and dark energy, rev]; Modesto et al a1906 [singularity avoidance]; > s.a. bianchi I models; cosmological-constant problem.
@ Other phenomenology: Barabash & Shtanov PRD(99)ap [Newtonian limit]; Navarro & Van Acoleyen JHEP(05)ht [compactification and general relativity]; Varieschi PRI(12)-a1010 [and the Pioneer anomaly]; Phillips MNRAS(15)-a1502 [attractive and repulsive gravity]; Yang PLB(18)-a1710 [gravitational waves].

@ Spatially conformally invariant theories: Gomes AP(15)-a1310 [canonical description and duality]; > s.a. Shape Dynamics.
@ Other variations: Fabbri PLB(12)-a1101 [with torsion]; Faria AHEP-a1312 [massive conformal gravity]; Dunajski & Tod CMP(14) [self-dual].

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at – modified 28 mar 2021