Post-Newtonian extension of the Newton-Cartan theory

TL;DR

This paper explores a post-Newtonian extension of the Newton-Cartan theory, analyzing its relation to the usual post-Minkowskian approximation and demonstrating their formal equivalence in certain contexts.

Contribution

It provides a consistent post-Newtonian extension of the Newton-Cartan theory and compares it with the post-Minkowskian approach, highlighting their formal equivalence.

Findings

Both frameworks are formally equivalent for field equations.

The scalar and Coriolis fields can be reduced to constants under global conditions.

The post-Newtonian extension aligns with Newton's original theory assumptions.

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.