A short note on dynamics and degrees of freedom in $2d$ classical gravity

TL;DR

This paper explores the peculiarities of matter coupled to 2D classical gravity, focusing on degrees of freedom and dynamics, highlighting differences based on Weyl invariance and analyzing scalar field models.

Contribution

It reveals that Weyl invariance in matter actions increases degrees of freedom and shows that scalar field solutions are often trivial or inconsistent in 2D gravity.

Findings

Weyl invariant matter theories have more degrees of freedom.

Scalar field solutions are either trivial or inconsistent.

The study emphasizes standard methods and direct computations.

Abstract

We comment on some peculiarities of matter with and without Weyl invariance coupled to classical $2d$ Einstein-Hilbert gravity for several models, in particular, related to the counting of degrees of freedom and on the dynamics. We find that theories where the matter action is Weyl invariant has generically more degrees of freedom than action without the invariance. This follows from the Weyl invariance of the metric equations of motion independently of the invariance of the action. Then, we study another set of models with scalar fields and show that solutions to the equations of motion are either trivial or inconsistent. To our knowledge, these aspects of classical $2d$ gravity have not been put forward and can be interesting to be remembered when using it as a toy model for $4d$ gravity. The goal of this note is also as a pedagogical exercise: our results follow from standard methods, but we emphasize more direct computations.