Curved non-relativistic spacetimes, Newtonian gravitation and massive matter

Abstract

There is significant recent work on coupling matter to Newton-Cartan spacetimes with the aim of investigating certain condensed matter phenomena. To this end, one needs to have a completely general spacetime consistent with local non-relativisitic symmetries which supports massive matter fields. In particular, one can not impose a priori restrictions on the geometric data if one wants to analyze matter response to a perturbed geometry. In this paper we construct such a Bargmann spacetime in complete generality without any prior restrictions on the fields specifying the geometry. The resulting spacetime structure includes the familiar Newton-Cartan structure with an additional gauge field which couples to mass. We illustrate the matter coupling with a few examples. The general spacetime we construct also includes as a special case the covariant description of Newtonian gravity, which has been thoroughly investigated in previous works. We also show how our Bargmann spacetimes arise from a suitable non-relativistic limit of Lorentzian spacetimes. In a companion paper [1] we use this Bargmann spacetime structure to investigate the details of matter couplings, including the Noether-Ward identities, and transport phenomena and thermodynamics of nonrelativistic fluids.Recently there has been a revival of interest in the Newton-Cartan description of nonrelativistic spacetimes in the condensed matter literature [2][3][4][5][6][7][8][9][10][11][12] where, it has been used with great effect to describe phenomena in the quantum Hall effect and various transport phenomena in condensed matter systems. Newton-Cartan spacetimes are used to describe matter fields and their interaction with general background geometries which are consistent with non-relativistic Galilean invariance. Newton-Cartan geometry also arises in the study of nonrelativistic holographic systems, where the boundary theory realizes a "twistless-torsionful" Newton-Cartan geometry [13][14][15][16][17][18][19]. On the other hand, in the gravitational physics literature (see [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] and also Ch.12 of [35] and Ch.4 of [36]) Newton-Cartan geometry has been well studied as a diffeomorphism-covariant, geometric way to describe Newtonian gravity. As such, the Newton-Cartan spacetimes considered there belong to a much more restricted class. This divergence of interests has lead to some conflicts in the construction (or at least in the interpretation) of Newton-Cartan spacetimes. One of the aims of this work is to alleviate these conflicts and set a clear stage for describing both Newtonian gravity and matter R I J := dω I J + ω I K ∧ ω K J (2.13b) 7Here we use the convention that the connection is a 1-form valued in the Lie algebra, instead of viewing it as 1-form components in a set of bases given by the generators of the Lie algebra.

Gravitational anomalies on the Newton-Cartan background

“…Non-relativistic spacetimes also do not have a preferred vector field τ µ and this leads to an additional invariance under Milne boosts [58,62] which are described by…”

Newton–Cartan, Galileo–Maxwell and Kaluza–Klein

“…Starting with a relativistic Kaluza-Klein ansatz and expanding it in powers of η leads to the non-relativistic ansatz (19)(20)(21), if one chooses the correct conformal frame. In the limiting procedure the conformal frame gets fixed by the demand of a non-singular limit through (43).…”

A Schrödinger approach to Newton-Cartan and Hořava-Lifshitz gravities

“…In fact, many condensed matter systems are described by non-relativistic field theories and coupling these systems to non-relativistic backgrounds provides appropriate external sources conjugated to their conserved currents. In the context of NC geometry, the appropriate way of performing this coupling has been studied from the theoretical point of view [29][30][31][32][33][34] and typical condensed matter applications include, e.g., the description of the quantum Hall effect [35][36][37][38] and even aging in materials [39][40][41][42][43][44][45][46].…”