Conformal gravity refers to gravity theories that are invariant underconformal transformations in theRiemannian geometry sense; more accurately, they are invariant underWeyl transformations${\displaystyle g_{ab}\to {\mathrm{\Omega }}^2(x)g_{ab}}$ where${\displaystyle g_{ab}}$ is themetric tensor and${\displaystyle \mathrm{\Omega }(x)}$ is a function onspacetime.

The simplest theory in this category has the square of theWeyl tensor as theLagrangian

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

where${\displaystyle C_{abcd}}$ is the Weyl tensor. This is to be contrasted with the usualEinstein–Hilbert action where the Lagrangian is just theRicci scalar. The equation of motion upon varying the metric is called theBach tensor,

${\displaystyle 2{\mathrm{\partial }}_a{\mathrm{\partial }}_d{{C^a}_{bc}}^d+R_{ad}{{C^a}_{bc}}^d=0,}$

where${\displaystyle R_{ab}}$ is theRicci tensor. Conformally flat metrics are solutions of this equation.

Since these theories lead tofourth-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]

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

${\displaystyle {◻}^2\mathrm{\Phi }=0}$

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

${\displaystyle \mathrm{\Phi }(r)=1-\frac{2m}{r}+ar+b{r}^2}$

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 thegalactic acceleration constant (also known asdark matter) and thedark energy constant.^[2] The solution equivalent to theSchwarzschild solution ingeneral relativity for a spherical source for conformal gravity has a metric with:

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

to show the difference between general relativity. 6bc is very small, and so can be ignored. The problem is that now c is the totalmass-energy of the source, and b is theintegral of density, times the distance to source, squared. So this is a completely different potential fromgeneral 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 ofghosts, which point to instabilities of thequantum 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 asymmetry brokenscalar field, in which case you would consider a small correction toNewtonian gravity like this (where we consider${\displaystyle \epsilon }$ to be a small correction):

${\displaystyle ◻\mathrm{\Phi }+{\epsilon }^2{◻}^2\mathrm{\Phi }=0}$

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

${\displaystyle \mathrm{\Phi }=1-\frac{2m}{r}\left(1+\alpha \sin \left(\frac{r}{\epsilon }+\beta \right)\right)}$

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

By adding a suitable gravitational term to theStandard Model action incurvedspacetime, the theory develops a local conformal (Weyl) invariance. The conformal gauge is fixed by choosing a reference mass scale based on the gravitational constant. This approach generates the masses for thevector bosons and matter fields similar to theHiggs mechanism without traditional spontaneous symmetry breaking.^[4]

• Conformal supergravity
• Hoyle–Narlikar theory of gravity

• E.S. Fradkin and A.A. Tseytlin (1985). "Conformal Supergravity".Phys. Rep.119 (4–5):233–362.Bibcode:1985PhR...119..233F.doi:10.1016/0370-1573(85)90138-3.
• Falsification of Mannheim's conformal gravity at CERN
• Mannheim's rebuttal of above at arXiv.

• Conformal geometry
• Lagrangian mechanics
• Spacetime
• Theories of gravity

• Articles with short description
• Short description matches Wikidata