Path Integral Formulation and Holonomy in Newton-Cartan Schwarzschild Geometry

Abstract

We use vielbein bundle's horizontal lift path integral formulation and gauge theory's holonomy map to compactly describe parallel transport and geodesic equations on a manifold. This is first applied to the geometry of general relativistic Schwarzschild as a review. The Newton-Cartan take of the Schwarzschild metric derived by previous literature is then adopted, and the analysis is repeated for both non-torsional and torsional geometry. Transport curves considered include a circular timeless loop, circular geodesic loop, radial geodesic loop, and a stationary loop. The findings on the three geometries are then contrasted, key differences summarized and the differing mechanisms discussed. In particular, we find a "performance tradeoff" between the torsional and non-torsional Newton-Cartan theory: the former yields more accurate equations of motion whereas the latter simulates relativistic dilation/contraction effects to an exact degree.