Two-dimensional higher-derivative gravity and conformal transformations

TL;DR

This paper explores two-dimensional higher-derivative gravity, revealing new relations to Einstein's theory, characterizing scale-invariant Lagrangians and field equations, and discussing exact solutions like black holes.

Contribution

It identifies specific conditions for scale-invariance in two-dimensional higher-derivative gravity and introduces new relations to Einstein's theory with non-minimally coupled scalar fields.

Findings

Scale-invariant Lagrangians are characterized by specific power laws and logarithmic forms.

The field equation is scale-invariant for certain Lagrangians, with a unique exception involving R ln R.

Discussion of exact solutions including black holes and the generalized Birkhoff theorem.

Abstract

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 $F = c_1 R\sp {k+1}$ 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 $\frac{1}{k} R\sp{k+1}$ 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.