Homogeneous Nonrelativistic Geometries as Coset Spaces

TL;DR

The paper develops a generalized coset construction for non-Lorentzian geometries, providing Newton–Cartan data from degenerate bilinear forms on cosets and linking these to Inönü–Wigner contractions and null reductions. It systematically builds NC spacetimes from Bargmann, Newton–Hooke, and Schrödinger algebras, and extends the framework to geometries with SO(3) isometries via an S^3-inspired construction. The results unify nonrelativistic holographic backgrounds with coset methods, including Lifshitz-Schrödinger-type NC spaces and TTNC variants, and suggest broad applicability toHořava–Lifshitz gravity and nonrelativistic string theory. The work also discusses how different coset choices and degenerate invariant forms yield physically meaningful NC geometries and potential holographic duals, while highlighting future directions (supersymmetry, higher-spin extensions, and entanglement entropy generalizations).

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 Inönü-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.