Homogeneous Nonrelativistic Geometries as Coset Spaces

TL;DR

This paper extends the coset construction method to non-Lorentzian geometries, especially nonrelativistic Newton-Cartan spacetimes, and explores their connections to relativistic geometries through contractions and reductions, aiding nonrelativistic holography.

Contribution

It introduces a generalized coset procedure for non-Lorentzian geometries and demonstrates how Newton-Cartan structures can be derived from relativistic algebras via contractions and reductions.

Findings

Constructed nonrelativistic coset spacetimes from symmetry algebras.

Connected nonrelativistic geometries to relativistic ones via Inönü-Wigner contraction.

Provided a framework for nonrelativistic limits in holographic dualities.

Abstract

We generalize the coset procedure of homogeneous spacetimes in (pseudo-)Riemannian geometry to non-Lorentzian geometries. These are manifolds endowed with nowhere vanishing invertible vielbeins that transform under local non-Lorentzian tangent space transformations. In particular, we focus on nonrelativistic symmetry algebras that give rise to (torsional) Newton-Cartan geometries, for which we demonstrate how the Newton-Cartan metric complex is determined by degenerate co- and contravariant symmetric bilinear forms on the coset. In specific cases, we also show the connection of the resulting nonrelativistic coset spacetimes to pseudo-Riemannian cosets via In\"on\"u-Wigner contraction of relativistic algebras as well as null reduction. Our construction is of use for example when considering limits of the AdS/CFT correspondence in which nonrelativistic spacetimes appear as gravitational backgrounds for nonrelativistic string or gravity theories.