Post-Newtonian extension of the Newton-Cartan theory

TL;DR

This paper investigates a post-Newtonian extension of Newton-Cartan theory as a singular GR limit for 1/c → 0, showing formal equivalence with the usual post-Minkowskian PN framework in terms of field and hydrodynamic equations. It derives explicit first post-Newtonian corrections within a covariant Newton-Cartan setting, including PN potentials and gauge-invariant relations, and explains how these coincide with Chandrasekhar’s PN results under appropriate gauges. The work highlights practical equivalence for PN predictions while noting that light-ray tests require Lorentz-covariant electrodynamics, implying no major practical advantage of the Newton-Cartan PN route over standard PN methods. Overall, the study clarifies the conceptual connection between Newtonian gravity, Newton-Cartan geometry, and GR in the PN regime, and delineates the role of harmonic time functions and gauge freedom in shaping the PN corrections.

Abstract

The theory obtained as a singular limit of General Relativity, if the reciprocal velocity of light is assumed to tend to zero, is known to be not exactly the Newton-Cartan theory, but a slight extension of this theory. It involves not only a Coriolis force field, which is natural in this theory (although not original Newtonian), but also a scalar field which governs the relation between Newtons time and relativistic proper time. Both fields are or can be reduced to harmonic functions, and must therefore be constants, if suitable global conditions are imposed. We assume this reduction of Newton-Cartan to Newton`s original theory as starting point and ask for a consistent post-Newtonian extension and for possible differences to usual post-Minkowskian approximation methods, as developed, for example, by Chandrasekhar. It is shown, that both post-Newtonian frameworks are formally equivalent, as far as the field equations and the equations of motion for a hydrodynamical fluid are concerned.