Weyl-squared Theories
The simplest theory in this category has the square of the Weyl tensor as the Lagrangian
where 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 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.
Read more about this topic: Conformal Gravity
Famous quotes containing the word theories:
“It takes twenty or so years before a mother can know with any certainty how effective her theories have been—and even then there are surprises. The daily newspapers raise the most frightening questions of all for a mother of sons: Could my once sweet babes ever become violent men? Are my sons really who I think they are?”
—Mary Kay Blakely (20th century)