Hořava–Lifshitz gravity

Definition

Let $M = \R \times \Sigma$ be a four-dimensional manifold with a fixed foliation by the slices $\set{t} \times \Sigma$, where $\Sigma$ is a three-manifold. The fields are, as in the ADM formalism, a lapse function $N$, a shift vector $N^i$ and a Riemannian metric $g_{ij}$ on each slice. The symmetry group is the group of foliation-preserving diffeomorphisms $t \mapsto t'(t)$, $x^i \mapsto x'^i(t, x)$. Hořava–Lifshitz gravity is the theory with the action $$S = \frac{2}{\kappa^2} \int dt \, d^3x \, \sqrt{g} \, N \left( K_{ij} K^{ij} - \lambda K^2 \right) - \int dt \, d^3x \, \sqrt{g} \, N \, V[g_{ij}],$$ where:

1. $K_{ij} = \frac{1}{2N} \left( \dot g_{ij} - \nabla_i N_j - \nabla_j N_i \right)$ is the extrinsic curvature of the slice and $K = g^{ij} K_{ij}$;
2. $\kappa$ and $\lambda$ are real coupling constants;
3. the potential $V[g_{ij}]$ is a scalar built from the spatial metric, its curvature and their spatial covariant derivatives, with up to six derivatives.

At short distances the theory is required to be invariant under the anisotropic scaling $t \mapsto b^3 t$, $x^i \mapsto b x^i$, that is, it has dynamical critical exponent $z = 3$.