# Conformal Gravity

CONFORMAL SYMMETRY:
Conformal Gravity is based on a local symmetry principle,
conformal stretching of the space-time metric, which is observed
by the strong, weak, and electro-magnetic forces, but not
by Einstein's General Relativity.
Conformal symmetry eliminates all terms in the action
except conformally invariant terms multiplied by
dimensionless coupling constants.
Thus unlike most alternative gravity theories,
the options are limited.
The Ricci scalar used in GR is not allowed.
Instead, the Weyl tensor (traceless component of Riemann)
is contracted with itself to obtain a unique
conformally invariant gravitational action.

The conformal gravity field equations are 4th order:
4 alp W_{mu nu} = T_{mu nu},
where W_{mu nu} is the Bach tensor, and where the stress-energy
tensor T_{mu nu} must be conformally invariant and traceless.

VACUUM SOLUTIONS:
In spherical symmetry a conformal transformation brings the
metric into a standard form:
g_{mu nu} = diag[ B(r), -1/B(r), -r^2, -r^2 sin^2(theta) ].
There are 4 field equations.
Spherical symmetry implies
W^theta_theta = W^phi_phi.
Vanishing trace implies
W^0_0 + W^r_r + W^theta_theta + W^phi_phi = 0.
The field equation W^0_0 - W^r_r = 0
gives a 4-th order Poisson equation:
( r B(r) )'''' = r f(r),
with source
f(r) = (3 / 4 alp B) (T^0_0-T^r_r)
Outside a compact source, integrating 4th-order (W^0_0-W^r_r) = 0 gives
B(r) = a - ( 2 b / r ) + g r - k r^2.
with 4 integration constants a, b, g, k.
The 3rd-order equation W^r_r=0 then gives
a^2 = 1 - 6 g b.
Solar system fits with b = M, provided g b << 1e-12.

Linear potential g r augments inward acceleration in galaxy outskirts, in the form
g r = ( g0 + g_sun (M/Msun) ) r.
Flat rotation curves arise if galaxies end when M/r ~ g r.
The g0 term fits rising rotation curves in dwarf galaxies.
The g_sun term fits flat rotation curves in typical large galaxies.

DYNAMICAL MASSES
Fermion and electro-weak guage fields are conformally invariant,
provided intrinsic masses are all 0, with mass terms
then being generated dynamically.

A scalar (Higgs) field S(x) can be included in the
lagrangian by conformally coupling it to the metric,
S^mu S_mu - R S^2 / 12.
In 4 spacetime dimensions the scalar field potential
is required to be quartic,
V(S) = lam S^4.
Neglecting spacetime gradients S^mu_mu gives spontaneous symmetry breaking,
S^2 => S0^2 = - ( R / 24 lam ),
generating fermion mass
m = mu S0,
and vacuum energy density
lam S0^4 = ( R / 24 lam )^2 .

CONFORMAL COSMOLOGY:

W_{mu nu} vanishes for Robertson-Walker metric, leading to
familiar cosmological models but with a modified Friedman equation:
(H(x)/H0)^2 = eps ( Or x^4 + Om x^3 + Ol ) + ( 1 - eps (Or + Om + Ol) ) x^2.
Here x = (1+z) = R0/R is the expansion factor,
Or/Om/Ol are current (x=1) energy densities of radiation/matter/vacuum,
in current critical density units
rho_crit = 3 H0^2 / 8 pi G,
and the "normal" gravity effects are scaled by a factor
eps = ( Geff / G ) = -( 3 / 4 pi G S0^2 ).
The effective gravitational constant Geff is negative,
so that matter, radiation and vacuum energy are all repulsive.
Radiation dominates early on, preventing a big bang singularity.
Matter is always sub-dominant.
The scalar field S0 dominates at late times, driving
exponential inflation with (H(x)/H0)^2 => eps Ol, so that the
effective vacuum energy density parameter lives in the range
0 < eps Ol (H0/H(x))^2 < 1,
compatible with the "observed" value 0.7.

PROBLEMS WITH CONFORMAL GRAVITY:
Light rays seem to bend the wrong way. (Solved by Lim & Wang 2016 ?)
No matter-dominated era.
Structure formation not yet studied.
CMB power spectrum not yet analysed.
Unclear if Nucleosynthesis works.

LINKS TO CONFORMAL GRAVITY LITERATURE

** Overview: **
Mannheim (2006) PrPNP 56, 340.
Alternatives to dark matter and dark energy.

** Metrics: **
Lim (2016) PRD 93, 4045
Charged C-metric in conformal gravity.
Mannheim (2016) PRD 93, 8501
Comment on "Problems with Mannheim's conformal gravity program".
Phillips (2015) MNRAS 448, 681
Attraction and Repulsion in Conformal Gravity.
Deliduman, Kasikci, Yapiskan (2015) JCAP? arXiv:1511.07731
Flat Galactic Rotation Curves from Geometry in Weyl Gravity.
Brihaye, Hartmann, Tojiev (2013) PRD 88, 104006.
AdS solitons with conformal scalar hair.
Lu, Wang (2013) PLB 718, 1536.
Exact Greenʼs function and Fermi surfaces from conformal gravity.
Liu, Lu (2013) JHEP 02, 139.
Charged rotating AdS black hole and
its thermodynamics in conformal gravity.
Liu, Pang, Pope (2013) PRD 87, 4013.
Black holes in six-dimensional conformal gravity.
Liu, Lu, Pope, Vazquez-Poritz (2013) CQGra 30, 5015.
Not conformally Einstein metrics in conformal gravity.
Yoon (2013) PRD 88, 7504.
Problems with Mannheim's conformal gravity program.
Grumiller, Irakleidou, Lovrekovic, McNees (2013) 1310.0819
Conformal gravity holography in four dimensions.
Fabbri (2013) JMP 54, 2501.
Metric solutions in torsionless gauge for vacuum conformal gravity.
Payandeh, Fathi (2012) IJTP 51, 2227.
Spherical Solutions due to the Exterior Geometry
of a Charged Weyl Black Hole.
Said, Sultana, Adami (2012) 85, 4054.
Exact static cylindrical solution to conformal Weyl gravity.
Varieschi, Burstein (2012) 1208.3706.
Conformal Gravity and the Alcubierre Warp Drive Metric.
Lu, Pang, Pope (2011) PRD 84, 4001.
Conformal gravity and extensions of critical gravity.
Verbin, Brihaye (2011) GReGr 43, 284.
Exact string-like solutions in conformal gravity.
Brihaye, Verbin (2010) PRD 81, 4041.
Spherical non-Abelian solutions in conformal gravity.
Brihaye, Verbin (2010) PRD 81, 4022.
Cylindrically symmetric solutions in conformal gravity.
Brihaye, Verbin (2009) PRD 80, 4048.
Spherical structures in conformal gravity
and its scalar-tensor extension.
Mannheim (2007) PRD 75, 4006.
Schwarzschild limit of conformal gravity
in the presence of macroscopic scalar fields.
Flanagan (2006) PRD 74, 3002.
Fourth order Weyl gravity.
Mannheim (2006) PrPNP 56, 340.
Alternatives to dark matter and dark energy.
Esteban, Kazanas (2001) GReGr 33, 1281.
Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity.
Dzhunushaliev, Schmidt (2000) JMP 41, 3007.
New vacuum solutions of conformal Weyl gravity.
Barabash, Shtanov (1999) PRD 60, 4008.
Newtonian limit of conformal gravity.
Edery, Paranjape (1999) GReGr 31, 1031.
Causal Structure of Vacuum Solutions to Conformal (Weyl) Gravity
Spyrou, Kazanas, Esteban (1997) CQGra 14, 2663.
Steadily rotating perfect-fluid gravitating prolate spheroids
in Weyl gravity.
Mannheim (1996) FoPh 26, 1683.
Local and global gravity.
Perlick, Xu (1995) ApJ 449, 47.
Matching Exterior to Interior Solutions in Weyl Gravity:
Comment on ``Exact Vacuum Solution to Conformal Weyl Gravity
and Galactic Rotation Curves''.
Mannheim, Kazanas (1994) GReGr 26, 337.
Newtonian limit of conformal gravity
and the lack of necessity of the second order Poisson equation.
Mannheim (1993) GReGr 25, 697.
Dynamical mass and geodesic motion.
Kazanas, Mannheim (1991) ApJS 76, 431.
General structure of the gravitational equations of motion
in conformal Weyl gravity.
Mannheim, Kazanas (1991) PRD 44, 417.
Solutions to the Reissner-Nordström, Kerr, and Kerr-Newman
problems in fourth-order conformal Weyl gravity.
Mannheim, Kazanas (1989) ApJ 342, 635.
Exact vacuum solution to conformal Weyl gravity
and galactic rotation curves.
Riegert (1984) PRL 53, 315.
Birkhoff's theorem in conformal gravity.

** Wormholes: **
Varieschi, Ault (2016) IJMPD 25 1650064
Wormhole geometries in fourth-order conformal Weyl gravity
Oliva, Tempo, Troncoso (2009) IJMPA 24, 1528.
Static Wormholes in Vacuum for Conformal Gravity.
Lobo (2008) CQGra 25, 5006.
General class of wormhole geometries in conformal Weyl gravity.

** Quantum Gravity: **
Mannheim (2017) PPNP 94, 125
Mass generation, the cosmological constant problem,
conformal symmetry, and the Higgs boson.
Mannheim (2016) IJMPD 25, 50003
Conformal invariance and the metrication of the fundamental forces
Mannheim 2015 arXiv:1512.04915.
Antilinearity Rather than Hermiticity
as a Guiding Principle for Quantum Theory.
Mannheim 2015 arXiv:1506.01399.
Living Without Supersymmetry --
the Conformal Alternative and a Dynamical Higgs Boson.
Mannheim 2015 arXiv:1506.04120.
Colloquium on the 2013 Nobel Prize in Physics
Awarded to Francois Englert and Peter Higgs.
't Hooft (2015) arXiv1511.04427
Singularities, horizons, firewalls, and local conformal symmetry.
't Hooft (2015) IJMPD 24, 43001
Local conformal symmetry: The missing symmetry component
for space and time
Edery, Graham (2015) JPhCS 615, 2005
The Coleman-Weinberg mechanism in a conformal (Weyl) invariant theory:
application to a magnetic monopole
Edery, Nakayama (2015) MPLA 30, 50152
Generating Einstein gravity, cosmological constant and Higgs mass
from restricted Weyl invariance
Edery, Nakayama (2015) PRD 90, 3007
Restricted Weyl invariance in four-dimensional curved spacetime
Alishahia, Faraji Astaneh, Mozaffar (2013) 1311.4329
Holographic Entanglement Entropy for 4D Conformal Gravity.
Edery, Graham (2013) 1310.7878
Radiatively induced symmetry breaking and
the conformally coupled magnetic monopole in AdS space
Nesbet (2013) 1304.4650
Conformal Higgs model: Charged gauge fields
can produce a 125GeV resonance
Mannheim (2013) ForPh 61, 140.
Astrophysical evidence for the non-Hermitian
but PT-symmetric Hamiltonian of conformal gravity.
Mannheim (2012) FoPh 42, 388.
Making the Case for Conformal Gravity.
Fabbri (2012) 1205.5386.
Conformal Gravity with Electrodynamics for Fermion Fields
and their Symmetry Breaking Mechanism.
Fabbri (2012) GReGr 44, 3127.
Conformal Standard Model.
Maldacena (2011) 1105.5632.
Einstein Gravity from Conformal Gravity.
't Hooft (2011) FoPh 41, 1829.
A Class of Elementary Particle Models
Without Any Adjustable Real Parameters.
Mannheim (2011) MPLA 26, 2375.
Intrinsically Quantum-Mechanical Gravity
and the Cosmological Constant Problem.
Nesbet (2010) 1009.1372.
The Higgs scalar field with no massive Higgs particle.
Edery, Fabbri, Paranjape (1999) CaJPh 87, 251.
Gravitationally coupled magnetic monopole and conformal symmetry breaking.
Nesbet (2010) 1004.5097.
Conformal Higgs model of dark energy.
Mannheim (2009) ASPC 413, 279.
Conformal Gravity Challenges String Theory.
Bender, Mannheim (2008) JPhA 41, 4018.
Giving up the ghost.
Bender, Mannheim (2008) PRD 78, 5022.
Exactly solvable PT-symmetric Hamiltonian
having no Hermitian counterpart.
Bender, Mannheim (2008) PRL 100, 402.
No-Ghost Theorem for the Fourth-Order Derivative
Pais-Uhlenbeck Oscillator Model.
Mannheim (2007) FoPh 37, 532.
Solution to the Ghost Problem in Fourth Order Derivative Theories.
Navarro, van Acoleyen (2005) JHEP 08, 019.
Compactifications of conformal gravity.
Mannheim, Davidson (2000) hep-th/0001115
Fourth Order Theories Without Ghosts
Riegert (1984) PhLA 195, 110.
The particle content of linearized conformal gravity.
Zee (1983) AnPhy 151, 431.
Einstein gravity emerging from quantum Weyl gravity.
Kaku (1983) PRD 27, 2819.
Strong-coupling approach to the quantization of conformal gravity.

** Solar System: **
Varieschi (2014) Gen.Rel.Grav. 46, 1714.
Kerr metric, geodesic motion, and Flyby Anomaly
in fourth-order Conformal Gravity
Said, Sultana, Adami (2013) PRD 88, 7504.
Gravitomagnetic effects in conformal gravity.
Sultana, Kazanas, Said (2012) PRD 86, 4008.
Conformal Weyl gravity and perihelion precession.
Varieschi (2012) PRI 2012, 1.
Conformal Cosmology and the Pioneer Anomaly.
Anderson, Laing, Lau, Liu, Nieto, Turyschev (2002) PRD 65, 2004.
Study of the anomalous acceleration of Pioneer 10 and 11
Wood, Moreau (2001) gr-qc/0102056.
Solutions of Conformal Gravity with Dynamical Mass Generation
in the Solar System.
Wood, Nemiroff (1991) ApJ 369, 54.
Constraints on Weyl gravity on subgalactic distance scales.

** Galaxy Dynamics: **
Phillips, P.R. (2018) MNRAS in press.
Schwarzschild and linear potentials in Mannheim's model of conformal gravity.
O'Brien, et al. (2018) ApJ 852, 6.
Alternative Gravity Rotation Curves for the LITTLE THINGS Survey.
O'Brien, Chiarelli, Mannheim (2017) arXiv:1704.03921
Universal Properties of Centripetal Accelerations in Spiral Galaxies.
Shenavar (2016) AP&SS 361 378.
Motion of particles in solar and galactic systems
by using Neumann boundary condition.
Horne (2016) MNRAS 458, 4122
Conformal Gravity Rotation Curves with a Conformal Higgs Halo.
Scholz (2016) Foundations of Physics 46, 176
MOND-Like Acceleration in Integrable Weyl Geometric Gravity.
Deliduman, Kasikci, Yapiskan (2015) JCAP? arXiv:1511.07731
Flat Galactic Rotation Curves from Geometry in Weyl Gravity.
Nesbet (2015) EPL 109, 59001
Dark galactic halos without dark matter.
Nesbet (2014) arXiv:1410.8076
Conformal gravity in the Schwarzschild metric.
Mannheim, O'Brien (2013) JPhCS 437m 2002.
Galactic rotation curves in conformal gravity.
Mannheim, O'Brien (2012) PRD 85, 4020.
Fitting galactic rotation curves with conformal gravity
and a global quadratic potential
O'Brien, Mannheim (2012) MNRAS 421, 1273.
Fitting dwarf galaxy rotation curves with conformal gravity.
Mannheim, O'Brien (2011) PRL 106, 1101.
Impact of a Global Quadratic Potential
on Galactic Rotation Curves.
Nesbet (2011) 1109.3626
Proposed explanation of galactic halos.
Mannheim (1997) ApJ 479, 659.
Are Galactic Rotation Curves Really Flat?
Mannheim (1993) ApJ 419, 150.
Linear Potentials and Galactic Rotation Curves

** Gravitational Waves: **
Caprini, Hölscher, Schwarz (2018) arXiv:18040187,
Astrophysical Gravitational Waves in Conformal Gravity.
Fabbri, Paranjape (2011) JMPD 20, 1941.
Conformal Gravity and Gravitational Waves.
Fabbri, Paranjape (2011) PRD 83, 4046.
Monochromatic plane-fronted waves in conformal gravity
are pure gauge.
Bouchami, Paranjape (2008) PRD 78, 4022.
Spontaneous breaking of conformal invariance, solitons,
and gravitational waves in theories of conformally
invariant gravitation.

** Lensing: **
Mureika, Varieschi (2016) arXiv:1611.00399
Black hole shadows in fourth-order conformal Weyl gravity.
Lim, Wang (2017) PRD 95, 4004.
Exact gravitational lensing in conformal gravity and
Schwarzschild-de Sitter spacetime.
Hoseini, et al. (2016) PRD94, 44021
Analytic treatment of complete geodesics in a static
cylindrically symmetric conformal spacetime.
Hoseini, Saffari, Soroushfar (2016) arXiv:1606.06558
Study of the geodesic equations of a spherical symmetric
spacetime in conformal gravity
Hoesini, Saffari, Soroushfar (2016) arXiv:1606.06545
Geodesic Motion in the Spacetime Of a SU(2)-Colored (A)dS
Black Hole in Conformal Gravity
Potapov, Izmailov, Nandi (2016) PRD 93, 4070
Mass decomposition of SLACS lens galaxies in Weyl conformal gravity
Cutajar, Zarb Adami (2014) MNRAS 441, 1291.
Strong lensing as a test for conformal Weyl gravity.
Sultana (2013) PRD 88, 2003.
Contribution of the cosmological constant to the bending
of light in Kerr-de Sitter spacetime.
Sultana (2013) JCAP 04, 48.
Deflection of light to second order in conformal Weyl gravity.
Cattani, Scalia, Laserra, Bochicchio, Nandi (2013) PRD 87, 7503.
Correct light deflection in Weyl conformal gravity.
Villanueva, Olivares (2013) JCAP 06, 040.
On the null trajectories in conformal Weyl gravity
Sultana, Kazanas (2012) PRD 85, 1502.
Bending of light in modified gravity at large distances.
Bhattacharya, Isaev, Scalia, Cattani, Nandi (2010) JCAP 09, 004.
Light bending in the galactic halo by Rindler-Ishak method.
Sultana, Kazanas (2010) PRD 81, 7502.
Bending of light in conformal Weyl gravity.
Ishak, Rindler, Dossett, Moldenhauer, Allison
(2008) MNRAS 388, 1279.
A new independent limit on the cosmological constant/dark energy
from the relativistic bending of light by Galaxies and clusters
of Galaxies.
Ishak (2008) PRD 85, 3005.
Light deflection, lensing, and time delays from gravitational
potentials and Fermat's principle in the presence of a
cosmological constant.
Rindler, Ishak (2007) PRD 76, 3006,
Contribution of the cosmological constant
to the relativistic bending of light revisited.
Pireaux (2004) QCGra 21, 4317.
Light deflection in Weyl gravity:
constraints on the linear parameter
Pireaux (2004) CQGra 21, 1897.
Light deflection in Weyl gravity:
critical distances for photon paths.
Edery, Methot, Paranjape (2001) GReGr 33, 2075.
Gauge Choice and Geodetic Deflection in Conformal Gravity.
Edery, Paranjape (1999) AIPC 493, 275.
Transformation of vacuum solutions of conformal gravity to flat space.
Edery (1999) PRL 83, 3990.
The Bright Side of Dark Matter.
Edery, Paranjape (1998) PRD 58, 4011.
Classical tests for Weyl gravity: Deflection of light and time delay.

** Galaxy Clusters: **
Diaferio, Ostorero (2009) MNRAS 393, 215.
X-ray clusters of galaxies in conformal gravity.
Horne (2006) MNRAS 369, 1667.
X-ray gas in the galaxy cluster Abell 2029:
conformal gravity versus dark matter.
Mannheim (1995) astro-ph/9504022.
Linear Potentials in Galaxies and Clusters of Galaxies.

** Conformal Cosmology: **
Amarasinghe, Phelps, Mannheim (2018) arXiv:1805.06807
Cosmological perturbations in conformal gravity II.
Roberts, Horne, Hodson, Leggat (2018) arXiv:1711.10369
Tests of LambdaCDM and Conformal Gravity using GRB and
Quasars as Standard Candles out to z~8.
Oda (2016) MPLA 31, 1650218.
Cosmology in Weyl transverse gravity.
Myung, Park (2016) EPJC 76, 79.
Scale-invariant power spectra from a Weyl-invariant
scalar-tensor theory
Oda (2015) arXiv:1505.06760
Conformal Higgs Gravity.
Variesche (2014) Galax 2, 577.
Astrophysical Tests of Kinematical Conformal Cosmology
in Fourth-Order ConformalWeyl Gravity
Bars, Steinhardt, Turok (2014) PhRvD 89, 3515.
Local conformal symmetry in physics and cosmology.
Bars, Steinhardt, Turok (2014) PhRvD 89, 1302.
Sailing through the big crunch-big bang transition.
Bars, Steinhardt, Turok (2013) PhLB 726, 50.
Cyclic cosmology, conformal symmetry and the metastability of the Higgs.
Yoon (2013) 1309.1990
Conformally Coupled Induced Gravity as an Infrared Fixed Point.
Yoon (2013) 1308.4952
The Dark Energy Regulated by Emergent Conformal Symmetry.
Yang, Chen, Xhao, Li, Liu (2013) 1311.2800
Test of conformal gravity with astrophysical observations.
Bars, Shih-Hung, Steinhardt, Turok (2012) PhRvD 86, 3542.
Complete set of homogeneous isotropic analytic solutions in
scalar-tensor cosmology with radiation and curvature.
Bars, Shih-Hung, Steinhardt, Turok (2012) PhLB 715, 278.
Antigravity and the big crunch/big bang transition.
Mannheim (2012) PRD 85, 12, 4008.
Cosmological perturbations in conformal gravity.
Diaferio, Ostorero, Cardone (2011) JCAP 10, 8.
Gamma-ray bursts as cosmological probes:
ΛCDM vs. conformal gravity.
Nesbet (2011) MPLA 26, 893.
Cosmological Implications of Conformal Field Theory.
Phillips (2011) MNRAS 417, 2276.
Modified conformal cosmology with
a positive effective gravitational constant.
Mannheim (2011) GReGr 43, 703.
Comprehensive solution to the cosmological constant,
zero-point energy, and quantum gravity problems.
Varieschi (2010) GReGr 42, 929.
A kinematical approach to conformal cosmology.
Mannheim (2008) 0809.1200.
Dynamical symmetry breaking and the cosmological constant problem.
Nesbet (2008) 0811.4161
Higgs mass determined by cosmological parameters.
Edery, Fabbri, Paranjape (2006) CQGra 23, 6409.
Spontaneous breaking of conformal invariance in theories of
conformally coupled matter and Weyl gravity.
Mannheim (2003) AIPC 672, 47.
Options for cosmology at redshifts above one
Mannheim (2003) JMPD 12, 893.
How Recent is Cosmic Acceleration?
Behnke, Blaschke, Pervushin, Proskurin (2002) PhLB 530, 20.
Description of supernova data in conformal cosmology
without cosmological constant.
Mannheim (1998) PRD 58, 3511.
Implications of cosmic repulsion for gravitational theory.
Elizondo, Yepes (1994) ApJ 428, 17.
Can conformal Weyl gravity be considered
a viable cosmological theory?
Mannheim (1992) ApJ 391, 429.
Conformal gravity and the flatness problem.
Mannheim (1990) GReGr 22, 289.
Conformal cosmology with no cosmological constant.

** BBN: **
Kaplinghat, Steigman, Walker (2000) PRD 61, 103507.
Nucleosynthesis in power-law cosmologies.
Sethi, Batra, Lohiya (1999) PRD 60, 108301.
Comment on ``Observational constraints on power-law cosmologies''
Lohiya, Batra, Mahajan, Mukherjee (1999) arXiv:nucl.th 2022
Nucleosynhthesis in a Simmering Universe.
Knox, Kosowsky (1993)
Primordial nucleosynthesis in conformal Weyl gravity.
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Keith Horne (kdh1@st-andrews.ac.uk)