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 a b → Ω 2 ( x ) g a b {displaystyle g_{ab} ightarrow Omega ^{2}(x)g_{ab}} where g a b {displaystyle g_{ab}} is the metric tensor and Ω ( x ) {displaystyle Omega (x)} is a function on spacetime. 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 a b → Ω 2 ( x ) g a b {displaystyle g_{ab} ightarrow Omega ^{2}(x)g_{ab}} where g a b {displaystyle g_{ab}} is the metric tensor and Ω ( x ) {displaystyle Omega (x)} is a function on spacetime. The simplest theory in this category has the square of the Weyl tensor as the Lagrangian where C a b c d {displaystyle 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, where R a b {displaystyle 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. 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: