Path Integral Formulation and Holonomy in Newton-Cartan Schwarzschild Geometry

TL;DR

This paper applies a path integral and holonomy approach to analyze parallel transport and geodesics in Newton-Cartan Schwarzschild geometries, comparing torsional and non-torsional cases and revealing a tradeoff in their accuracy and effects.

Contribution

It introduces a unified holonomy-based framework for Newton-Cartan Schwarzschild geometries, highlighting differences between torsional and non-torsional formulations.

Findings

Torsional Newton-Cartan yields more accurate equations of motion.

Non-torsional Newton-Cartan accurately models relativistic dilation.

Distinct transport curves reveal key geometric differences.

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.