Lower-dimensional Hořava–Lifshitz gravity

We consider Hořava–Lifshitz gravity in both $1+1$ and $2+1$ dimensions. These lower-dimensional versions of Hořava–Lifshitz gravity are simple enough to be explicitly tractable, but still complex enough to be interesting. We write the most general (nonprojectable) 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 nontrivial, 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 behavior as the scalar mode in (nonprojectable) Hořava–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 Hořava–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.