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Conformal gravity

Conformal gravity is a generic name for gravity theories which are invariant under conformal transformations in the Riemannian geometry sense; more accurately, they are invariant under Weyl transformations

g_{ab}rightarrowOmega^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}^4x 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,

2nabla_anabla_d{{C^a}_{bc}}^d+{{C^a}_{bc}}^dR_{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

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 is in a central field of force is:

Phi(r)= 1 -2m/r +ar +br^2

The first two terms are the same as a normal wave equation. Since this equation is a simpler approximation to Conformal gravity then m corresponds to mass of the central source. The last two terms are unique to 4-derivative wave equations. It has been suggested to assign small values to them to account for the galactic acceleration constant (also known as dark matter) and the dark energy constant

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.