Conformal gravity

Conformal gravity are gravity theories that are invariant under conformal transformations in the Riemannian geometry sense; more accurately, they are invariant under Weyl transformations $g_{{ab}}\rightarrow \Omega ^{2}(x)g_{{ab}}$ where $g_{ab}$ is the metric tensor and $\Omega (x)$ is a function on spacetime.

Weyl-squared theories

The simplest theory in this category has the square of the Weyl tensor as the Lagrangian

{\mathcal {S}}=\int {\mathrm {d}}^{4}x{\sqrt {-g}}C_{{abcd}}C^{{abcd}},

where $C_{{abcd}}$ is the Weyl tensor. This is to be contrasted with the usual Einstein–Hilbert action where the Lagrangian is just the Ricci scalar. The equation of motion upon varying the metric is called the Bach equation,

2\nabla _{a}\nabla _{d}{{C^{a}}_{{bc}}}^{d}+{{C^{a}}_{{bc}}}^{d}R_{{ad}}=0,

where $R_{ab}$ is the Ricci tensor. Conformally flat metrics are solutions of this equation.

Since these theories lead to fourth order equations for the fluctuations around a fixed background, they are not manifestly unitary. It has therefore been generally believed that they could not be consistently quantized. This is now disputed.^[1]

Four-derivative theories

Conformal gravity is an example of a 4-derivative theory. This means that each term in the wave equation can contain up to 4 derivatives. There are pros and cons of 4-derivative theories. The pros are that the quantized version of the theory is more convergent and renormalisable. The cons are that there may be issues with causality. A simpler example of a 4-derivative wave equation is the scalar 4-derivative wave equation:

\Box ^{2}\Phi =0

The solution for this in a central field of force is:

\Phi (r)=1-{\frac {2m}{r}}+ar+br^{2}

The first two terms are the same as a normal wave equation. Because this equation is a simpler approximation to conformal gravity, m corresponds to the mass of the central source. The last two terms are unique to 4-derivative wave equations. It has been suggested that small values be assigned to them to account for the galactic acceleration constant (also known as dark matter) and the dark energy constant.^[2] The solution equivalent to the Schwarzschild solution in general relativity for a spherical source for conformal gravity has a metric with:

\phi (r)=g^{{00}}=(1-6bc)^{{\frac {1}{2}}}-{\frac {2b}{r}}+cr+{\frac {d}{3}}r^{2}

to show the difference between general relativity. 6bc is very small so can be ignored. The problem is that now c is the total mass-energy of the source, b is the integral of density times distance to source squared. So this is a completely different potential to general relativity and not just a small modification.

The main issue with conformal gravity theories, as well as any theory with higher derivatives, is the typical presence of ghosts, which point to instabilities of the quantum version of the theory, although there might be a solution to the ghost problem.^[3]

An alternative approach is to consider the gravitational constant as a symmetry broken scalar field in which case you would consider a small correction to Newtonian gravity like this (where we consider $\varepsilon$ to be a small correction:

\Box \Phi +\varepsilon ^{2}\Box ^{2}\Phi =0

in which case the general solution is the same as the Newtonian case except there can be an additional term:

\Phi =1-{\frac {2m}{r}}(1+\alpha \sin(r/\varepsilon +\beta ))

where there is an additional component varying sinusoidally over space. The wavelength of this variation could be quite large such as an atomic width. Thus there appears to be several stable potentials around a gravitational force in this model.

Conformal unification to the Standard Model

By adding a suitable gravitational term to the standard model action with gravitational coupling, the theory develops a local conformal (Weyl) invariance in the unitary gauge for the local SU(2). The gauge is fixed by requiring the Higgs scalar to be a constant. This mechanism generates the masses for the vector bosons and matter fields with no physical degrees of freedom for the Higgs.^[4] ^[5]

Notes

↑ Mannheim, Philip D. (2007-07-16). "PASCOS-07, Imperial College London, July 2007". 0707: 2283. Bibcode:2007arXiv0707.2283M. arXiv:0707.2283 . |contribution= ignored (help)
↑ Mannheim, Philip D. (2005-08-01). "Alternatives to Dark Matter and Dark Energy". Prog.Part.Nucl.Phys. 56 (2): 340. Bibcode:2006PrPNP..56..340M. arXiv:astro-ph/0505266 . doi:10.1016/j.ppnp.2005.08.001.
↑ Mannheim, Philip D. (2006-09-06). "Solution to the ghost problem in fourth order derivative theories". Found. Phys. 37 (4–5): 532. Bibcode:2007FoPh...37..532M. arXiv:hep-th/0608154 . doi:10.1007/s10701-007-9119-7.
↑ Montag, J. Lee (1992), "Spontaneously Broken Conformal Symmetry and the Standard Model"
↑ Pawlowski, M.; Raczka, R. (1994), "A Unified Conformal Model for Fundamental Interactions without Dynamical Higgs Field", Foundations of Physics, 24 (9): 1305–1327, Bibcode:1994FoPh...24.1305P, arXiv:hep-th/9407137 , doi:10.1007/BF02148570

References

E.S. Fradkin and A.A. Tseytlin (1985). "Conformal Supergravity". Phys. Rept. 119 (4–5): 233–362. Bibcode:1985PhR...119..233F. doi:10.1016/0370-1573(85)90138-3.
Conformal gravity on arxiv.org

External links

Conformal gravity: New light on dark matter and dark energy talk at the SETI Institute
Falsification of Mannheim's conformal gravity at CERN
Mannheim's rebuttal of above at arXiv.

Theories of gravitation

Standard

Newtonian gravity (NG)	Newton's law of universal gravitation History of gravitational theory
General relativity (GR)	Introduction History Mathematics Resources Tests Post-Newtonian formalism Linearized gravity ADM formalism Gibbons–Hawking–York boundary term

Alternatives to
general relativity

Paradigms	Classical theories of gravitation Quantum gravity Theory of everything
Classical	Einstein–Cartan Bimetric theories Gauge theory gravity Teleparallelism Composite gravity f(R) gravity Massive gravity Modified Newtonian dynamics (MOND) Nonsymmetric gravitation Scalar–tensor theories Brans–Dicke Scalar–tensor–vector Conformal gravity Scalar theories Nordström Whitehead Geometrodynamics Induced gravity Tensor–vector–scalar Chameleon Pressuron
Quantisation	Euclidean quantum gravity Canonical quantum gravity Wheeler–DeWitt equation Loop quantum gravity Spin foam Causal dynamical triangulation Causal sets DGP model Rainbow gravity theory
Unification	Kaluza–Klein theory Dilaton Supergravity
Unification and quantisation	Noncommutative geometry Semiclassical gravity Superfluid vacuum theory Logarithmic BEC vacuum String theory M-theory F-theory Heterotic string theory Type I string theory Type 0 string theory Bosonic string theory Type II string theory Little string theory Twistor theory Twistor string theory
Generalisations / Extensions of GR	Scale relativity Liouville gravity Lovelock theory (2+1)-dimensional topological gravity Gauss–Bonnet gravity Jackiw–Teitelboim gravity

Pre-Newtonian theories and toy models

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