Conformal Gravity |

**In General** > s.a. 3D gravity; gravity theories;
covariant quantum 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 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];
Campigotto et al a1712 [failure to describe galactic rotation curves];
Zhang et al EPJC(18)-a1805 [black holes].

@ __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];
> 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].

**Variations**

@ __Spatially conformally invariant theories__:
Gomes AP-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].

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