Review on Non-Relativistic Gravity

Abstract

We review the history of Newton-Cartan gravity with an emphasis on recent developments, including the covariant, off-shell large speed of light expansion of general relativity. Depending on the matter content, this expansion either leads to Newton-Cartan geometry with absolute time or to Newton-Cartan geometry with non-relativistic gravitational time dilation effects. The latter shows that non-relativistic gravity includes a strong field regime and goes beyond Newtonian gravity. We start by reviewing early developments in Newton-Cartan geometry, including the covariant description of Newtonian gravity, mainly through the works of Trautman, Dautcourt, Künzle and Ehlers. We then turn to more modern developments, such as the gauging of the Bargmann algebra, and we describe why the latter cannot be used to find an off-shell covariant description of Newtonian gravity. We review recent work on the $1/c$ expansion of general relativity and show that this leads to an alternative `type II' notion of Newton-Cartan geometry. Finally, we discuss matter couplings, solutions and odd powers in $1/c$, and we conclude with a brief summary of related topics.

46+10 pages, review article; v2: minor changes, reference added; v3: reference added

References in corpus (22)

• Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time
• Newtonian Gravity and the Bargmann Algebra
• Spacetime Symmetries of the Quantum Hall Effect
• Non-relativistic conformal symmetries and Newton-Cartan structures
• Non-Relativistic Strings and Limits of the AdS/CFT Correspondence
• Lifshitz Gravity for Lifshitz Holography
• Expanding Lie (super)algebras through abelian semigroups
• Lifshitz Space-Times for Schroedinger Holography
• Warped Conformal Field Theory as Lower Spin Gravity
• On the intrinsic torsion of spacetime structures
• Torsional Newton-Cartan Geometry from the Noether Procedure
• The Eisenhart lift: a didactical introduction of modern geometrical concepts from Hamiltonian dynamics
• Lifshitz Hydrodynamics from Lifshitz Black Branes with Linear Momentum
• Eisenhart's theorem and the causal simplicity of Eisenhart's spacetime
• Teleparallel Newton--Cartan gravity
• Non-relativistic and Ultra-relativistic Expansions of Three-dimensional Spin-3 Gravity Theories
• Non-Relativistic and Ultra-Relativistic Scaling Limits of Multimetric Gravity
• A 3+1 formulation of the 1/c expansion of General Relativity
• Non-relativistic gravity theories in four spacetime dimensions
• Existence of families of spacetimes with a Newtonian limit
• Lagrangians for non-relativistic gravity
• The gauging procedure and carrollian gravity

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• Mixmasters in Wonderland: Chaotic dynamics from Carroll limits of gravity
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