Connections and dynamical trajectories in generalised Newton-Cartan gravity II. An ambient perspective

Abstract

Connections compatible with degenerate metric structures are known to possess peculiar features: on the one hand, the compatibility conditions involve restrictions on the torsion; on the other hand, torsionfree compatible connections are not unique, the arbitrariness being encoded in a tensor field whose type depends on the metric structure. Nonrelativistic structures typically fall under this scheme, the paradigmatic example being a contravariant degenerate metric whose kernel is spanned by a one-form. Torsionfree compatible (i.e. Galilean) connections are characterised by the gift of a two-form (the force field). Whenever the two-form is closed, the connection is said Newtonian. Such a nonrelativistic spacetime is known to admit an ambient description as the orbit space of a gravitational wave with parallel rays. The leaves of the null foliation are endowed with a nonrelativistic structure dual to the Newtonian one, dubbed Carrollian spacetime. We propose a generalisation of this unifying framework by introducing a new non-Lorentzian ambient metric structure of which we study the geometry. We characterise the space of (torsional) connections preserving such a metric structure which is shown to project to (resp. embed) the most general class of (torsional) Galilean (resp. Carrollian) connections.

71 pages, 3 figures; v2: New material added, Sections 2.3 and 3.3 postponed to future work. Matches published version. 77 pages, 2 figures; v3: Added endnote [73] amending Proposition A.16

References in corpus (25)

• Conformal Carroll groups and BMS symmetry
• Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time
• Gauging the Carroll Algebra and Ultra-Relativistic Gravity
• Newtonian Gravity and the Bargmann Algebra
• Spacetime Symmetries of the Quantum Hall Effect
• Non-relativistic conformal symmetries and Newton-Cartan structures
• Conformal Carroll groups
• Horava-Lifshitz Gravity From Dynamical Newton-Cartan Geometry
• Dynamics of Carroll Particles
• The geometry of Schrödinger symmetry in non-relativistic CFT
• Thermal Hall Effect and Geometry with Torsion
• Lifshitz Space-Times for Schroedinger Holography
• Low-energy effective theory in the bulk for transport in a topological phase
• Curved non-relativistic spacetimes, Newtonian gravitation and massive matter
• Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view
• Field Theory on Newton-Cartan Backgrounds and Symmetries of the Lifshitz Vacuum
• General coordinate invariance in quantum many-body systems
• Effective theory of two-dimensional chiral superfluids: gauge duality and Newton-Cartan formulation
• The Symmetries of the Carroll Superparticle
• Kohn's Theorem, Larmor's Equivalence Principle and the Newton-Hooke Group
• Classical aspects of lightlike dimensional reduction
• Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux
• Conformal Killing Tensors and covariant Hamiltonian Dynamics
• Deformed Carroll particle from 2+1 gravity
• Dynamical construction of Horava-Lifshitz geometry

Cited by in corpus (51)

• Carrollian Physics at the Black Hole Horizon
• Carroll Structures, Null Geometry and Conformal Isometries
• Bridging Carrollian and Celestial Holography
• Carroll Expansion of General Relativity
• Conformal Carroll Scalars with Boosts
• Review on Non-Relativistic Gravity
• Brown-York charges at null boundaries
• Limits of JT gravity
• Spatially isotropic homogeneous spacetimes
• Embedding Galilean and Carrollian geometries I. Gravitational waves
• Carroll black holes
• Gauges in Three-Dimensional Gravity and Holographic Fluids
• Remarks on the non-Riemannian sector in Double Field Theory
• Flat from anti-de Sitter
• The Effective Action of Superrotation Modes
• Massless Scalars and Higher-Spin BMS in Any Dimension
• Carrollian fluids and spontaneous breaking of boost symmetry
• String Theory and non-Riemannian Geometry
• Asymptotic Shear and the Intrinsic Conformal Geometry of Null-Infinity
• Scaling and conformal symmetries for plane gravitational waves
• Non-Riemannian gravity actions from double field theory
• Hydro & Thermo Dynamics at Causal Boundaries, Examples in 3d Gravity
• Geometrizing non-relativistic bilinear deformations
• Homogeneous Nonrelativistic Geometries as Coset Spaces
• Three-dimensional exotic Newtonian gravity with cosmological constant
• Lie algebraic Carroll/Galilei duality
• Gravitation in flat spacetime from entanglement
• Generalized Maxwellian exotic Bargmann gravity theory in three spacetime dimensions
• p-brane Newton--Cartan Geometry
• Generalized Newton-Cartan Geometries for Particles and Strings
• Fractons, non-Riemannian Geometry, and Double Field Theory
• Ehlers, Carroll, Charges and Dual Charges
• Quantization of Carrollian conformal scalar theories
• Corner symmetry and quantum geometry
• Chern-Simons action and the Carrollian Cotton tensors
• Torsion and anomalies in the warped limit of Lifschitz theories
• Setting the connection free in the Galilei and Carroll expansions of gravity
• Carroll-Schrödinger Equation
• Asymptotic symmetries of projectively compact order one Einstein manifolds
• One-dimensional Carrollian fluids I: Carroll-Galilei duality
• Projective and Carrollian geometry at time/space-like infinity on projectively compact Ricci flat Einstein manifolds
• Possible ambient kinematics
• Relativistic Fluids, Hydrodynamic Frames and their Galilean versus Carrollian Avatars
• One-Dimensional Carrollian Fluids III: Global Existence and Weak Continuity in $L^{\infty }$
• Unique Carrollian manifolds emerging from Einstein spacetimes
• Non-relativistic quantum strings from gauged WZW models
• The gauging procedure and carrollian gravity
• Ti and Spi, Carrollian extended boundaries at timelike and spatial infinity
• Revisiting the Carrollian Scalar Field
• On Twisted Spacetimes: a new class of Galilean cosmological models
• One-dimensional Carrollian fluids II: $C^1$ blow-up criteria