General theory of Galilean gravity

Abstract

We obtain the complete theory of Newton-Cartan gravity in a curved spacetime by considering the large $c$ limit of the vielbein formulation of General Relativity. Milne boosts originate from local Lorentzian transformations, and the special cases of torsionless and twistless torsional geometries are explained in the context of the larger locally Lorentzian theory. We write the action for Newton-Cartan fields in the first order Palatini formalism, and the large $c$ limit of the Einstein equations. Finally, we obtain the generalised Eisenhart-Duval lift of the metric that plays an important role in non-relativistic holography.

6 pages, no figures. V2: references added

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• Scaling and conformal symmetries for plane gravitational waves
• Galilean first-order formulation for the non-relativistic expansion of general relativity
• 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
• Setting the connection free in the Galilei and Carroll expansions of gravity
• A unified perspective on Poincaré and Galilei relativity: I. Special relativity
• Equivalence of a harmonic oscillator to a free particle and Eisenhart lift
• Extended Bargmann FDA and non-relativistic gravity