Two-dimensional higher-derivative gravity and conformal transformations

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.

16 pages, latex, no figures, [email protected], Class. Quant. Grav. to appear

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• The tetralogy of Birkhoff theorems
• The classical solutions of two-dimensional gravity
• Cosmology of the Jackiw-Teitelboim model
• Conformal Vacuum and Fluctuation-Dissipation in de-Sitter Universe and Black Hole Spacetimes
• Two-dimensional black holes in accelerated frames: Spacetime structure
• A two-dimensional representation of four-dimensional gravitational waves
• Symmetric Variations of the Metric and Extrema of the Action for Pure Gravity
• Birkhoffs Theorem and Lie Symmetry Analysis