Conformal gravity is a generic name for gravity theories which are invariant under conformal transformations

Conformal map

In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

 in the Riemannian geometry

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

 sense; more accurately, they are invariant under Weyl transformations where is the metric tensor

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

 and is a function on spacetime

Spacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

.

Weyl-squared theories

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

Weyl tensor

In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic...

 as the Lagrangian

Lagrangian

The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

where is the Weyl tensor. This is to be contrasted with the usual Einstein–Hilbert action where the Lagrangian is just the Ricci scalar

Scalar curvature

In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...

. The equation of motion upon varying the metric is called the Bach equation,
where is the Ricci tensor

Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...

. 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

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 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

Renormalization

In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....

. 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:


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


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

Dark matter

In astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...

) and the dark energy

Dark energy

In physical cosmology, astronomy and celestial mechanics, dark energy is a hypothetical form of energy that permeates all of space and tends to accelerate the expansion of the universe. Dark energy is the most accepted theory to explain recent observations that the universe appears to be expanding...

 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, although there might be a solution to the ghost problem .

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. The gravity sector can be decoupled in the weak field limit and the resulting quantum field theory can be argued renormalizable in the unbroken SU(2) phase.