Newton-Cartan Gravity in Noninertial Reference Frames

TL;DR

This paper explores how Newton-Cartan gravity behaves under noninertial, nonrelativistic transformations, revealing the structure of fictitious forces, the behavior of the Ricci tensor, and analogies with Maxwell's equations in different reference frames.

Contribution

It demonstrates that noninertial effects are naturally incorporated into Newton-Cartan gravity via the Galilean line group and uncovers a Maxwell-like structure of the governing equations.

Findings

Fictitious forces are encoded in the Cartan connection and do not affect the Ricci tensor.

The Ricci tensor's 00-component relates to matter density, defining Newtonian ADM mass.

In rotating frames, an effective mass density differs from physical density due to simulated magnetic fields.

Abstract

We study properties of Newton-Cartan gravity under transformations into all noninertial, nonrelativistic reference frames. The set of these transformations has the structure of an infinite dimensional Lie group, called the Galilean line group, which contains as a subgroup the Galilei group. We show that the fictitious forces of noninertial reference frames are naturally encoded in the Cartan connection transformed under the Galilean line group. These noninertial forces, which are coordinate effects, do not contribute to the Ricci tensor which describes the curvature of Newtonian spacetime. We show that only the $00$-component of the Ricci tensor is non-zero and equal to ($4\pi$ times) the matter density in any inertial or noninetial reference frame and that it leads to what may be called Newtonian ADM mass. While the Ricci field equation and Gauss law are both fulfilled by the same physical matter density in inertial and linearly accelerating reference frames, there appears a discrepancy between the two in rotating reference frames in that Gauss law holds for an effective mass density that differs from the physical matter density. This effective density has its origin in the simulated magnetic field that appears in rotating frames, highlighting a rather striking difference between linearly and rotationally accelerating reference frames. We further show that the dynamical equations that govern the simulated gravitational and magnetic fields have the same form as Maxwell's equations, a surprising conclusion given that these equations are well-known to obey special relativity (and $U(1)$-gauge symmetry), rather than Galilean symmetry.