Lower-dimensional Horava-Lifshitz gravity

TL;DR

This paper explores lower-dimensional versions of Horava-Lifshitz gravity in 1+1 and 2+1 dimensions, analyzing their dynamics and degrees of freedom to serve as simplified models for understanding the 3+1 theory.

Contribution

It provides explicit formulations and analysis of lower-dimensional Horava-Lifshitz gravity, including the equivalence with Einstein-aether theory and an algorithm for extending this equivalence.

Findings

1+1 dimensional case has no local degrees of freedom.

2+1 dimensional case has a single scalar degree of freedom.

The lower-dimensional models serve as useful simplified frameworks for the 3+1 theory.

Abstract

We consider Horava-Lifshitz gravity in both 1+1 and 2+1 dimensions. These lower-dimensional versions of Horava-Lifshitz gravity are simple enough to be explicitly tractable, but still complex enough to be interesting. We write the most general (non-projectable) action for each case and discuss the resulting dynamics. In the 1+1 case we utilize the equivalence with 2-dimensional Einstein-aether theory to argue that, even though non-trivial, the theory does not have any local degrees of freedom. In the 2+1 case we show that the only dynamical degree of freedom is a scalar, which qualitatively has the same dynamical behaviour as the scalar mode in (non-projectable) Horava-Lifshitz gravity in 3+1 dimensions. We discuss the suitability of these lower-dimensional theories as simpler playgrounds that could help us gain insight into the 3+1 theory. As special cases we also discuss the projectable limit of these theories. Finally, we present an algorithm that extends the equivalence with (higher order) Einstein-aether theory to full Horava-Lifshitz gravity (instead of just the low energy limit), and we use this extension to comment on the apparent naturalness of the covariant formulation of the latter.