Reduced Massive Gravity with Two St\"uckelberg Fields

TL;DR

This paper analyzes a simplified model of non-linear massive gravity with two St"uckelberg fields, deriving conditions for degrees of freedom and gauge symmetries, revealing differences between 1+1 and 3+1 dimensions.

Contribution

It provides an analytic expression for the kinetic matrix determinant and performs a full constraint analysis for the two-field case in massive gravity.

Findings

In 1+1 dimensions, the theory has gauge symmetry eliminating scalar degrees of freedom.

In 3+1 dimensions, both scalar fields generally propagate, indicating physical degrees of freedom.

The determinant of the kinetic matrix vanishes identically in 1+1 dimensions, but not in 3+1 dimensions.

Abstract

We consider the non-linear massive gravity as a theory of a number of St\"uckelberg scalar fields minimally coupled to the Einstein-Hilbert gravity and argue that the counting of degrees of freedom can be done for scalar theory and gravity separately. In this paper we investigate the system with only two St\"uckelberg scalar fields. In this case we find the analytic expression for the determinant of the kinetic matrix of the scalar field Lagrangian and perform the full constraint analysis. In 1+1 space-time dimensions the theory corresponds to the full non-linear massive gravity, and this determinant vanishes identically. In this case we find two first-class constraints, and present the corresponding gauge symmetry of the theory which eliminates both scalar degrees of freedom. In 3+1 dimensions the determinant of the kinetic matrix does not vanish identically and, for generic initial conditions, both scalar fields are propagating.