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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.

We study the thermodynamics of the near horizon of near extremal Kerr geometry (near-NHEK) within an AdS2/CFT1 correspondence. We do this by shifting the horizon by a general finite mass, which does not alter the geometry and the resulting solution is still diffeomorphic to NHEK, however it allows for a Robertson Wilczek two dimensional Kaluza-Klein reduction and the introduction of a finite regulator on the AdS2 boundary. The resulting asymptotic symmetry group of the two dimensional Kaluza-Klein reduction leads to a non-vanishing quantum conformal field theory on the respective AdS2 boundary. The s-wave contribution of the energy-momentum-tensor of the CFT, together with the asymptotic symmetries, generate a Virasoro algebra with calculable center and non-vanishing lowest Virasoro eigen-mode. The central charge and lowest eigen-mode reproduce the near-NHEK Bekenstein-Hawking entropy via the statistical Cardy Formula and our derived central charge agrees with the standard Kerr/CFT Correspondence. We also compute the Hawking temperature of the shifted near-NHEK by analyzing quantum holomorphic fluxes of the Robinson and Wilczek two dimensional analogue fields., Comment: 24 pages. References added and shortened introduction. Typos corrected, edits and revisions performed in accord with comments and suggestions received

An important operation in generalized complex geometry is the Courant bracket which extends the Lie bracket that acts only on vectors to a pair given by a vector and a p-form. We explore the possibility of promoting the elements of the Courant bracket to physical fields by constructing a geometric action based on the Kirillov-Kostant symplectic form. For the $p=0$ forms, the action generalizes Polyakov's two-dimensional quantum gravity when viewed as the geometric action for the Virasoro algebra. We show that the geometric action arising from the centrally extended Courant bracket for the vector and zero form pair is similar to the geometric action obtained from the semi-direct product of the Virasoro algebra with a U(1) affine Kac-Moody algebra. For arbitrary $p$ restricted to a Dirac structure, we derived the geometric action and exhibit generalizations for almost complex structures built on the Kirillov-Kostant symplectic form. In the case of p+1 dimensional submanifolds, we also discuss a generalization of a Kahler structure on the orbits., Comment: 31 pages, 2 figures Corrected various mistakes, added content on Kahler geometry and gneralizations of complex structures for orbits of the Courant bracket