# Spatially covariant gravity: Perturbative analysis and field transformations

## TL;DR

This paper analyzes the degrees of freedom in spatially covariant gravity theories, showing that degenerate kinetic terms lead to a single scalar mode at linear order, but unwanted modes can reappear under certain conditions, requiring consistency constraints.

## Contribution

It provides a perturbative approach to understanding degrees of freedom in spatially covariant gravity, complementing Hamiltonian analysis and exploring field redefinitions.

## Key findings

- Degenerate kinetic terms yield one scalar mode at linear order.
- Unwanted modes reappear at nonlinear or inhomogeneous backgrounds.
- Consistency conditions are necessary to remove unwanted modes.

## Abstract

We make a perturbative analysis of the number of degrees of freedom in a large class of metric theories respecting spatial symmetries, of which the Lagrangian includes kinetic terms of both the spatial metric and the lapse function. We show that, as long as the kinetic terms are degenerate, the theory propagates a single scalar mode at the linear order in perturbations around a Friedmann-Robertson-Walker background. Nevertheless, an unwanted mode will reappear pathologically, either at nonlinear orders around the Friedmann-Robertson-Walker background, or at linear order around an inhomogeneous background. In both cases, it turns out that a consistency condition has to be imposed in order to remove the unwanted mode. This perturbative approach provides an alternative and also complementary point of view of the conditions derived in a Hamiltonian analysis. We also discuss the relation under field redefinitions, between theories with and without the time derivative of the lapse function.