Two-dimensional higher-derivative gravity and conformal transformations

We consider the lagrangian $L=F(R)$ in classical (=non-quantized) two-dimensional fourth-order gravity and give new relations to Einstein’s theory with a non-minimally coupled scalar field. We distinguish between scale-invariant lagrangians and scale-invariant field equations. $L$ is scale-invariant for $F=c_{1}R\text{sp}k+1$ and a divergence for $F=c_{2}R$. The field equation is scale-invariant not only for the sum of them, but also for $F=R\ln R$. We prove this to be the only exception and show in which sense it is the limit of $\frac{1}{k}R\text{sp}k+1$ as $k\to 0$. More generally: Let $H$ be a divergence and $F$ a scale-invariant lagrangian, then $L=H\ln F$ has a scale-invariant field equation. Further, we comment on the known generalized Birkhoff theorem and exact solutions including black holes.