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* = *α* ∫ d^{4}*x* |*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, (∂_{t}^{2}+∇^{2})^{2} *φ* =
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*_{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 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].

@ __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].

@ __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].

**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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