Contracted Lorentz invariance for gravity with a preferred foliation

TL;DR

This paper develops a geometric framework for gravity theories with a preferred foliation, like Horava-Lifshitz gravity, by constructing nonrelativistic observer spaces that incorporate symmetry breaking from ISO(D) to SO(D).

Contribution

It introduces a nonrelativistic observer space geometry for preferred foliation gravity theories, extending the gauge theory approach to include symmetry breaking via the shift vector.

Findings

Constructed a nonrelativistic observer space with ISO(D) symmetry.

Formulated a torsion-free connection including derivatives of the shift vector.

Provided a geometric framework for analyzing preferred foliation gravity theories.

Abstract

In canonical gravity, the choice of a local time direction is not obviously compatible with local Lorentz invariance. One way to address this issue is to view gravity as a gauge theory on observer space, rather than spacetime. In a Lorentz covariant theory observer space is the space of unit future-directed timelike vectors; picking such a vector locally breaks the symmetry to a subgroup SO(D) of SO(D,1), so that on observer space the local symmetry group is SO(D). Observer space geometries naturally describe any gravitational theory that only assumes local invariance under SO(D). Here we construct nonrelativistic observer spaces for gravity with a fixed foliation, applicable to preferred foliation theories such as Horava-Lifshitz gravity. Different Horava-Lifshitz observers at a point are related by a change in the shift vector, leaving the preferred foliation invariant. Gravity is formulated on a non-relativistic frame bundle with structure group ISO(D); the shift vector acts as a symmetry breaking field breaking ISO(D) symmetry spontaneously to SO(D). A natural torsion-free connection is constructed, including the usual SO(D,1) connection of relativistic gravity plus terms depending on derivatives of the shift vector. This observer space geometry provides a novel geometric framework for the study of preferred foliation theories.