Effective action for scalar fields in two-dimensional gravity

TL;DR

This paper derives an effective action for scalar fields in two-dimensional gravity models, eliminating geometric variables and highlighting boundary contributions, with applications to spherically reduced gravity solutions.

Contribution

It provides a general method to obtain an effective scalar field action in 2D gravity by exactly solving geometric equations, resulting in a boundary-term-based action applicable to various models.

Findings

Effective action is a boundary term in Minkowski space.

The method applies to both open and closed universes.

Reproduces known solutions like Fisher and Roberts in spherically reduced gravity.

Abstract

We consider a general two-dimensional gravity model minimally or nonminimally coupled to a scalar field. The canonical form of the model is elucidated, and a general solution of the equations of motion in the massless case is reviewed. In the presence of a scalar field all geometric fields (zweibein and Lorentz connection) are excluded from the model by solving exactly their Hamiltonian equations of motion. In this way the effective equations of motion and the corresponding effective action for a scalar field are obtained. It is written in a Minkowskian space-time and does not include any geometric variables. The effective action arises as a boundary term and is nontrivial both for open and closed universes. The reason is that unphysical degrees of freedom cannot be compactly supported because they must satisfy the constraint equation. As an example we consider spherically reduced gravity minimally coupled to a massless scalar field. The effective action is used to reproduce the Fisher and Roberts solutions.