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
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