Conformal gravity
Conformal gravity refers to 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}\rightarrow \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. == Weyl-squared theories == The simplest theory in this category has the square of the Weyl tensor as the Lagrangian S = ∫ d 4 x − g C a b c d C a b c d , {\displaystyle {\mathcal {S}}=\int \,\mathrm {d} ^{4}x\,{\sqrt {-g\;}}\,C_{abcd}\,C^{abcd}~,} where C a b c d {\displaystyle \;C_{abcd}\;} is the Weyl tensor.