  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.
