Non-Relativistic Gravity and its Coupling to Matter

Abstract

We study the non-relativistic expansion of general relativity coupled to matter. This is done by expanding the metric and matter fields analytically in powers of $1/c^2$ where $c$ is the speed of light. In order to perform this expansion it is shown to be very convenient to rewrite general relativity in terms of a timelike vielbein and a spatial metric. This expansion can be performed covariantly and off shell. We study the expansion of the Einstein-Hilbert action up to next-to-next-to-leading order. We couple this to different forms of matter: point particles, perfect fluids, scalar fields (including an off-shell derivation of the Schrödinger-Newton equation) and electrodynamics (both its electric and magnetic limits). We find that the role of matter is crucial in order to understand the properties of the Newton-Cartan geometry that emerges from the expansion of the metric. It turns out to be the matter that decides what type of clock form is allowed, i.e. whether we have absolute time or a global foliation of constant time hypersurfaces. We end by studying a variety of solutions of non-relativistic gravity coupled to perfect fluids. This includes the Schwarzschild geometry, the Tolman-Oppenheimer-Volkoff solution for a fluid star, the FLRW cosmological solutions and anti-de Sitter spacetimes.

arXiv comment101 pages (78 main text), Nordita preprint number 2020-008

References in corpus (79)

• On the Origin of Gravity and the Laws of Newton
• Emergent Gravity and the Dark Universe
• Nonlinear W(infinity) Algebra as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity
• Gravitation and quantummechanical localization of macroobjects
• Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time
• Non-Relativistic Closed String Theory
• Torsional Newton-Cartan Geometry and Lifshitz Holography
• Gauging the Carroll Algebra and Ultra-Relativistic Gravity
• The Schrödinger-Newton equation and its foundations
• Newtonian Gravity and the Bargmann Algebra
• Boundary Stress-Energy Tensor and Newton-Cartan Geometry in Lifshitz Holography
• Tabletop experiments for quantum gravity: a user's manual
• Spacetime Symmetries of the Quantum Hall Effect
• IIA/B, Wound and Wrapped
• Non-Relativistic Strings and Limits of the AdS/CFT Correspondence
• Action Principle for Newtonian Gravity
• Carroll versus Galilei Gravity
• Horava-Lifshitz Gravity From Dynamical Newton-Cartan Geometry
• `Stringy' Newton-Cartan Gravity
• Null Killing Vector Dimensional Reduction and Galilean Geometrodynamics
• Generating Lie and gauge free differential (super)algebras by expanding Maurer-Cartan forms and Chern-Simons supergravity
• Expanding Lie (super)algebras through abelian semigroups
• Lifshitz Space-Times for Schroedinger Holography
• Strings with Non-Relativistic Conformal Symmetry and Limits of the AdS/CFT Correspondence
• Schroedinger Invariance from Lifshitz Isometries in Holography and Field Theory
• On the coupling of Galilean-invariant field theories to curved spacetime
• Nonrelativistic String Theory and T-Duality
• Connection between Newtonian simulations and general relativity
• Perfect Fluids
• Gravitationally induced inhibitions of dispersion according to the Schrödinger-Newton Equation
• String Theory and String Newton-Cartan Geometry
• Field-dependent symmetries of a non-relativistic fluid model
• Torsional Newton-Cartan gravity from the large $c$ expansion of General Relativity
• Three-Dimensional Extended Bargmann Supergravity
• Problems with the Newton-Schrödinger Equations
• Relating non-relativistic string theories
• Newton-Cartan Supergravity
• Curved non-relativistic spacetimes, Newtonian gravitation and massive matter
• Non-Relativistic Chern-Simons Theories and Three-Dimensional Horava-Lifshitz Gravity
• Nonrelativistic String Theory in Background Fields
• Fields and fluids on curved non-relativistic spacetimes
• Newton-Cartan (super)gravity as a non-relativistic limit
• A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions
• Field Theory on Newton-Cartan Backgrounds and Symmetries of the Lifshitz Vacuum
• Aspects of hot Galilean field theory
• Gravity between Newton and Einstein
• Post-Newtonian extension of the Newton-Cartan theory
• Non-Relativistic Spacetimes with Cosmological Constant
• On the electrodynamics of moving bodies at low velocities
• Remark About Non-Relativistic String in Newton-Cartan Background and Null Reduction
• Non-relativistic conformal symmetries in fluid mechanics
• Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity
• Galilean Conformal Electrodynamics
• Lie Algebra Expansions and Actions for Non-Relativistic Gravity
• A worldsheet supersymmetric Newton-Cartan string
• Newton-Cartan Gravity and Torsion
• An Action for Extended String Newton-Cartan Gravity
• Torsional Newton-Cartan Geometry from the Noether Procedure
• Three-Dimensional Extended Newtonian (Super)Gravity
• Remarks on the non-Riemannian sector in Double Field Theory
• Non-relativistic symmetries in three space-time dimensions and the Nappi-Witten algebra
• Covariant formulation of the post-1-Newtonian approximation to General Relativity
• Non-Riemannian geometry of M-theory
• Hydrodynamic Modes of Homogeneous and Isotropic Fluids
• Nonrelativistic String Theory Sigma Model and Its Canonical Formulation
• Lifshitz Hydrodynamics from Lifshitz Black Branes with Linear Momentum
• Momentum and charge transport in non-relativistic holographic fluids from Hořava gravity
• Homogeneous Nonrelativistic Geometries as Coset Spaces
• Galilean free Lie algebras
• A Schrödinger approach to Newton-Cartan and Hořava-Lifshitz gravities
• (m,n)-String and D1-Brane in Stringy Newton-Cartan Background
• Three-dimensional exotic Newtonian gravity with cosmological constant
• (In)finite extensions of algebras from their Inonu-Wigner contractions
• Symmetries of post-Galilean expansions
• Zooming in on AdS${}_3$/CFT${}_2$ near a BPS Bound
• p-brane Newton--Cartan Geometry
• General theory of Galilean gravity
• Newton-Cartan, Galileo-Maxwell and Kaluza-Klein
• Stringy Newton Gravity with $H$-flux

Cited by in corpus (62)

• Carroll Expansion of General Relativity
• Aspects of Nonrelativistic Strings
• Conformal Carroll Scalars with Boosts
• Torsional string Newton-Cartan geometry for non-relativistic strings
• Non-relativistic and Carrollian limits of Jackiw-Teitelboim gravity
• Non-Relativistic Ten-Dimensional Minimal Supergravity
• Torsional Newton Cartan gravity from non-relativistic strings
• A non-relativistic limit of M-theory and 11-dimensional membrane Newton-Cartan geometry
• On the Non-Relativistic Expansion of Closed Bosonic Strings
• Nonrelativistic Approximations of Closed Bosonic String Theory
• Spin Matrix Theory String Backgrounds and Penrose Limits of AdS/CFT
• Carroll limit of four-dimensional gravity theories in the first order formalism
• Carrollian and Non-relativistic Jackiw-Teitelboim Supergravity
• Non-Riemannian gravity actions from double field theory
• Aspects of Non-Relativistic Quantum Field Theories
• Non-relativistic three-dimensional supergravity theories and semigroup expansion method
• Galilean first-order formulation for the non-relativistic expansion of general relativity
• Geometrizing non-relativistic bilinear deformations
• Matrix Theory Reloaded: A BPS Road to Holography
• The Supersymmetric Neveu-Schwarz Branes of Non-Relativistic String Theory
• The Non-Relativistic Geometric Trinity of Gravity
• Teleparallel Newton--Cartan gravity
• Three-dimensional Maxwellian Extended Newtonian gravity and flat limit
• Underdetermination in Classic and Modern Tests of General Relativity
• Lie Algebra Expansion and Integrability in Superstring Sigma-Models
• Classical string solutions in Non-Relativistic AdS${}_5\times$S${}^5$: Closed and Twisted sectors
• Generalized Newton-Cartan Geometries for Particles and Strings
• Non-relativistic and Ultra-relativistic Expansions of Three-dimensional Spin-3 Gravity Theories
• Non-Relativistic and Ultra-Relativistic Scaling Limits of Multimetric Gravity
• Carrollian Yang-Mills Theory
• Coset space actions for nonrelativistic strings
• Longitudinal Galilean and Carrollian limits of non-relativistic strings
• BMS Modular Diaries: Torus one-point function
• Anomalous chiral transport with vorticity and torsion: Cancellation of two mixed gravitational anomaly currents in rotating chiral $p+ip$ Weyl condensates
• A coupling prescription for post-Newtonian corrections in Quantum Mechanics
• Supersymmetric Galilean Electrodynamics
• Mixmasters in Wonderland: Chaotic dynamics from Carroll limits of gravity
• Null Dimensional Reduction of M2-Brane
• A 3+1 formulation of the 1/c expansion of General Relativity
• Uniqueness of Galilean and Carrollian limits of gravitational theories and application to higher derivative gravity
• Non-relativistic limit of gravity theories in the first order formalism
• Galilean Electrodynamics: Covariant formulation and Lagrangian
• Setting the connection free in the Galilei and Carroll expansions of gravity
• Lie Algebra Expansions, Non-Relativistic Matter Multiplets and Actions
• The classification of general affine connections in Newton--Cartan geometry: Towards metric-affine Newton--Cartan gravity
• Stable and Unstable Dp-branes in p-brane Newton-Cartan Background
• Geometric post-Newtonian description of massive spin-half particles in curved spacetime
• Lagrangians for non-relativistic gravity
• Phase space holography with no strings attached
• Hydrodynamics in the Carrollian regime
• Non-relativistic transport from frame-indifferent kinetic theory
• A unified perspective on Poincaré and Galilei relativity: I. Special relativity
• Dynamical de Sitter black holes in a quasi-stationary expansion
• Fate of $\kappa$-Minkowski space-time in non-relativistic (Galilean) and ultra-relativistic (Carrollian) regimes
• Noether Symmetry in Newtonian Dynamics and Cosmology
• On gauge transformations in twistless-torsional Newton--Cartan geometry
• The Newtonian limit of orthonormal frames in metric theories of gravity
• Conformal Mapping of Non-Lorentzian Geometries in SU(1,2) Conformal Field Theory
• Notes on su$(1,2)\oplus$u$(1)$ Chern-Simons theory and Torsional Newton-Cartan gravity
• On Carrollian and Galilean contractions of BMS algebra in 3 and 4 dimensions
• Remarks on Galilean electromagnetism
• Group Contractions via Infinite-Dimensional Lie Theory