Spatially covariant gravity theories with two tensorial degrees of freedom: the formalism

TL;DR

This paper develops a formalism within spatially covariant gravity theories to eliminate scalar degrees of freedom, ensuring only tensorial modes propagate, through Hamiltonian analysis and specific degeneracy conditions.

Contribution

It provides necessary and sufficient conditions to remove scalar degrees of freedom in spatially covariant gravity theories, advancing the understanding of their degrees of freedom structure.

Findings

Two conditions for eliminating scalar degrees of freedom identified

Degeneracy of lapse-extrinsic curvature sector is necessary

Phase space dimension must be even to exclude scalar mode

Abstract

Within the general framework of spatially covariant theories of gravity, we study the conditions for having only the two tensorial degrees of freedom. Generally, there are three degrees of freedom propagating in the theory, of which two are tensorial and one is of the scalar type. Through a detailed Hamiltonian analysis, we find two necessary and sufficient conditions to evade the scalar type degree of freedom. The first condition implies that the lapse-extrinsic curvature sector must be degenerate. The second condition ensures that the dimension of the phase space at each spacetime point is even, so that the scalar type degree of freedom is eliminated completely. We also compare our results with the previous studies, and apply our formalism to a simple example, in which the Lagrangian is quadratic in the extrinsic curvature.